Fixed-Parameter Algorithms forArtificial Intelligence, Constraint

Satisfaction, and Database Problems

Georg Gottlob1, Stefan Szeider2

1Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford, OX13QD, England, UK. 2Durham University, Department of Computer Science, Science Labs,

South Road, DH1 3LE Durham, England, UK.

Email: georg.gottlob@comlab.ox.ac.uk, stefan.szeider@durham.ac.uk

We survey the parameterized complexity of problems that arise in artificialintelligence, database theory, and automated reasoning. In particular, weconsider various parameterizations of the constraint satisfaction problem, theevaluation problem of Boolean conjunctive database queries, and the propositionalsatisfiability problem. Furthermore, we survey parameterized algorithms forproblems arising in the context of the stable model semantics of logic programs, fora number of other problems of non-monotonic reasoning, and for the computation

of cores in data exchange.

1. INTRODUCTION

Many problems that arise in artificial intelligence,database theory, and automated reasoning are at leastNP-hard. In such cases, one often applies heuristics orapproximation algorithms in order to obtain suboptimalsolutions in a reasonable amount of time. Theframework of parameterized complexity, initiated byRod Downey and Mike Fellows, provides a meansfor developing optimal algorithms which are feasiblein practice. The main idea is to associate with aproblem instance a parameter, usually a non-negativeinteger k, which is small for instances of relevance,and to develop algorithms with running times that arepossibly exponential in the parameter but uniformlypolynomial in the instance size. In particular, analgorithm is a fixed-parameter algorithm if it solvesinstances of size n and parameter k in time O(f(k)nc);here f denotes a computable function and c denotesa constant which is independent of the parameter k.Although the function f can be exponential (and isexponential for non-trivial cases), a fixed-parameteralgorithm remains feasible for large instances providedthat the parameter k is small. This feature of fixed-parameter algorithms makes them often preferable toalgorithms with a running time of the form O(nf(k)),since the latter become impractical for large instanceseven when f(k) = k and k is small.

Example 1.1. A vertex cover of a graph G is aset S of vertices of G such that every edge of Ghas at least one of its ends in S. Given a graph Gand a non-negative integer k, the problem of deciding

whether G admits a vertex cover of size at most k, is aclassical NP-complete problem [49]. However, severalfixed-parameter algorithms are known that solve thefollowing parameterized version of the problem:

VC

Instance: A graphG and a non-negative integer k.

Parameter: k.

Question: Does G admit a vertex cover of size atmost k?

Indeed, VC is probably the best studied problemin parameterized complexity with a long history ofimprovements [18]. The currently fastest fixed-parameter algorithm for VC is due to Chen, Kanj,and Xia [19] and requires time O(1.273k + nk) andpolynomial space for graphs of order n.

In this paper we focus on positive results, i.e.,the existence of fixed-parameter algorithms for theproblems under consideration. The presented negativeresults (i.e., hardness results) have merely the purposeof carving out territories that are very likely to beinaccessible to fixed-parameter algorithms.

The majority of combinatorial problems studied inthe framework of parameterized complexity offer a“natural parameter,” e.g., it is natural to parameterizethe vertex cover problem by the size of the vertex coversas in VC above. However, the problems consideredin this survey lack of a natural parameter; there arenumerous possibilities for parameters. An important

task is the identification of parameters that are asgeneral as possible (i.e., for many instances of practicalimportance one can expect that the parameter issmall), and which are still accessible to fixed-parameteralgorithms. Thus, problems of that kind open up a racebetween more and more general parameters.

We have tried to cover many relevant results under aunified framework. However, there are certainly severalinteresting fixed-parameter algorithms known which arenot covered in this survey. We apologize to all theauthors whose results are not included.

The remainder of the paper is organized as follows.We close this section with a brief review of thebasics of parameterized complexity. In Section 2 weconsider various parameterizations of the constraintsatisfaction problem and the evaluation problem forBoolean conjunctive database queries. In Section 3 weconsider several parameterizations of the propositionalsatisfiability problem. Finally, in Section 4, we considerproblems arising in the context of the stable modelsemantics of logic programs, problems related to otherforms of non-monotonic reasoning, and we discusscomplexity problems related to the computation ofcores in data exchange.

1.1. Parameterized complexity in a nutshell

Next we give a brief and rather informal review of themost important concepts of parameterized complexity.For an in-depth treatment of the subject we refer thereader to other sources [29, 30, 35, 41, 45, 81].

An instance of a parameterized decision problemis a pair (I, k) where I is the main part and k isthe parameter ; the latter consists usually of one orseveral non-negative integers. A parameterized decisionproblem is fixed-parameter tractable if it can be solvedby a fixed-parameter algorithm, i.e., if instances (I, k)can be decided in time O(f(k)‖I‖c) where f is acomputable function, c is a constant, and ‖I‖ representsthe size of I with respect to a reasonable encoding.FPT denotes the class of all fixed-parameter tractabledecision problems.

The framework of parameterized complexity offersa completeness theory, similar to the theory ofNP-completeness, that allows the accumulation ofstrong theoretical evidence that a parameterizedproblem is not fixed-parameter tractable. Thiscompleteness theory is based on the weft hierarchyof complexity classes W[1],W[2], . . . ,W[P]. Eachclass is the equivalence class of certain parameterizedproblems under fpt-reductions. Such a reduction is astraightforward extension of a polynomial-time many-one reduction that ensures a parameter for one problemmaps into the parameter for another (see [30] fordetails). All reductions considered in this survey areof this type. The class XP consists of parameterizedproblems that can be solved in polynomial time if theparameter is considered as a constant. The above

classes form the chain

FPT ⊆W[1] ⊆W[2] ⊆ · · · ⊆W[P] ⊆ XP

where all inclusions are assumed to be proper; it isknown that FPT 6= XP [30, 45]. The followingparameterized clique-problem is W[1]-complete; thisproblem is the basis for several hardness resultsconsidered below.

CLIQUE

Instance: A graphG and a non-negative integer k.

Parameter: k.

Question: Does G contain a clique on k vertices?

A good reason for assuming that FPT and W[1]are distinct is provided by an analog of Cook’sTheorem: FPT = W[1] implies the existence of afixed-parameter algorithm for deciding whether a non-deterministic Turing machine accepts an input withink non-deterministic computation steps [30]. AlthoughXP contains problems which are very unlikely tobe fixed-parameter tractable, it is often a significantimprovement to show that a problem belongs to thisclass, in contrast to, e.g., k-SAT which is NP-hard forevery constant k ≥ 3.

2. CONSTRAINT SATISFACTION ANDCONJUNCTIVE DATABASE QUERIES

2.1. Preliminaries

Constraint Satisfaction is a central problem of artificialintelligence and provides a powerful and generalformalism for the encoding of various special-purposeproblems. An instance of the constraint satisfactionproblem (CSP) consist of a set of variables that rangeover a finite domain of possible values, together witha collection of constraints. Each constraint imposes arestriction on the combination of values for specifiedvariables. The question is whether one can assign valuesto variables complying with all the restrictions imposedby the constraints.

Constraint satisfaction can be rephrased as theevaluation problem for Boolean conjunctive queriesover relational databases. Actually, constraintsatisfaction and Boolean conjunctive query evaluationare essentially the same problem [74]. Below wewill state concepts and results in terms of constraintsatisfaction; we will occasionally point out aspectsthat arise from the database theoretic point of view.Furthermore, the evaluation problem for Booleanconjunctive queries can be considered as the modelchecking problem for first-order formulas in prenex formwhose only connectives are ∃ and ∧. We note in passingthat first-order model checking problems provide analternative way of defining the complexity classes of theweft hierarchy; this approach was initiated by Downey,

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Fellows, and Regan [31] and refined and extended byFlum and Grohe [44].

Formally, a constraint satisfaction instance is a triple(V,D, C) where V is a set of variables, D is a set ofvalues, the domain, and C is a set of constraints. Eachconstraint in C is a pair (S,R) where S, the constraintscope, is a sequence of distinct variables of V , andR, the constraint relation, is a relation over D whosearity matches the length of S (that is, R ⊆ Dr). Aconstraint satisfaction instance with domain {0, 1} iscalled Boolean. We denote by var(C) the set of variablesthat occur in the scope of a constraint C. We definethe size of an instance I = (V,D, C) as the length ofa string that encodes the instance; more specifically,the size ‖C‖ of a constraint C = (S,R) is given by|var(C)| · (|R|+1); the size of the instance I is given by|V |+ |D|+

∑C∈C ‖C‖.

Consider a constraint satisfaction instance I =(V,D, C). An assignment α : V → D satisfies aconstraint ((x1, . . . , xr), R) ∈ C if (α(x1), . . . , α(xr)) ∈R. An assignment satisfies I if it satisfies all theconstraints of I simultaneously; in that case we alsosay that α is a solution of I. A constraint satisfactioninstance is satisfiable or consistent if it has at least onesolution. CSP is the problem of deciding whether agiven instance is consistent.

Example 2.1. Graph 3-colorability can be expressedas a constraint satisfaction problem: for a given graphG = (V,E) we form the constraint satisfaction instanceI = (V, {red, green, blue}, C) where for every edgeuv ∈ E the set C contains the constraint ((u, v), R) withR = {(red,green), (red,blue), (green,red), (green,blue),(blue,red), (blue,green)}.

This example shows that the constraint satisfactionproblem is NP-hard.

Example 2.2. Similarly to Example 2.1, the popularSudoku puzzle can be described as a constraintsatisfaction problem [98]. As variables we take the 81squares of the game, the domain is the set of integersfrom 1 to 9. Each clue number gives raise to a unaryconstraint that fixes the value for a variable. For eachset of nine variables x1, . . . , x9 that form a row, column,or a 3× 3 subgrid we take a constraint ((x1, . . . , x9), R)where R contains all tuples that correspond to apermutation of the integers 1, . . . , 9.

The next two examples show how the clique problemcan be encoded as as a constraint satisfaction problem.These two examples are the base for several hardnessresults considered below.

Example 2.3 ([87]). Let G = (V,E) be a graphand let k be a positive integer. We consider theconstraint satisfaction instance I = ({x1, . . . , xk}, V ,{ ((xi, xj), R) : 1 ≤ i < j ≤ k }) where R ={ (u, v) : uv ∈ E }. It is easy to verify that G containsa clique on k vertices if and only if I is consistent.

Example 2.4 ([93]). Let G = ({v1, . . . , vn}, E) bea graph and let k be a positive integer. Each edgee = vivi′ of G can be encoded by a tuple te =(d1, . . . , dn, d

′1, . . . , d

′n) where di = d′i′ = 1 and all

other values are 0. We consider the Boolean constraintsatisfaction instance I with kn variables xj

i (1 ≤i ≤ n, 1 ≤ j ≤ k) and

(k2

)many constraints Cj,j′

(1 ≤ j < j′ ≤ k). The constraints are defined byCj,j′

= ((xj1, . . . , x

jn, x

j′

1 , . . . , xj′

n ), R), where R containsall tuples tvivi′ for vivi′ ∈ E and i < i′. Again, it iseasy to verify that G contains a clique on k vertices ifand only if I is consistent.

Example 2.5. Consider a relational database withthe following relational schemes.

works(Emp#,Proj#)

manages(Emp#,Proj#)

relative(Emp1,Emp2)

The following Boolean conjunctive query checkswhether an employee works in a project managed byhis or her relative.

ans ←− works(E,P) ∧ manages(M,P)∧ relative(E,M).

Formulating this query as a constraint satisfactioninstance we get I = ({xE , xM , xP }, D, {((xE , xP ), Rw),((xM , xP ), Rm), ((xE , xM ), Rr)}), where D is the set ofall persons contained in the database, and Rw, Rm, Rr

are the binary relations that correspond to the databaserecords.

2.2. Parameterized constraint satisfaction

A constraint satisfaction parameter is a computablefunction p that assigns to every constraint satisfactioninstance I a non-negative integer p(I). For constraintsatisfaction parameters p1, . . . , pr we consider thefollowing generic parameterized problem:

CSP(p1, . . . , pr)

Instance: A constraint satisfaction instance Iand non-negative integers k1, . . . , kr with p1(I) ≤k1, . . . , pr(I) ≤ kr.

Parameters: k1, . . . , kr.

Question: Is I consistent?

Note that we formulate this problem as a “promiseproblem” in the sense that for solving the problemwe do not need to verify the assumption p1(I) ≤k1, . . . , pr(I) ≤ kr. However, for most of the casesconsidered below the verification of the assumptionp1(I) ≤ k1, . . . , pr(I) ≤ kr is fixed-parameter tractable.For a constraint satisfaction instance I = (V,D, C) wehave the basic parameters vars(I) = |V |, the numberof variables of I, length(I) =

∑C∈C |var(C)|, the sum

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of arities of constraints, and dom(I) = |D|, the domainsize.

In the context of Boolean conjunctive queries, theparameter length is known as the query size. Forqueries it is natural to assume a small query size incontrast to a large size of the database.

If we parameterize by the domain size, then theproblem remains NP-hard, since, e.g., 3-colorabilitycan be expressed as a constraint satisfaction problemwith constant domain (see Example 2.1 above). ThusCSP(dom) is not fixed-parameter tractable unlessP = NP. On the other hand, if we parameterizeby the number of variables and the domain size,i.e., CSP(vars,dom), then we have a triviallyfixed-parameter tractable problem: we can decidethe consistency of an instance I by checking alldom(I)vars(I) possible assignments.

Somewhat surprisingly, if we do not bound thedomain size, we get a W[1]-complete problem.

Theorem 2.1 ([87]). The problems CSP(vars) andCSP(length) are W[1]-complete.

The hardness part of this theorem follows fromExample 2.3 which gives a reduction from CLIQUE.

Let A be a deterministic polynomial-time algorithmthat applies simplification and propagation rules toa constraint satisfaction instance I without changingconsistency. For example, arc consistency is animportant simplification and propagation rule usedin constraint solvers. Let A(I) denote the instanceobtained form I by applying algorithm A, and letkernA(I) denote the size of A(I) (if the size of A(I)depends on a particular ordering of variables andconstraints, let kernA(I) denote the largest size of A(I)over all such orderings).

It follows now that the problem CSP(kernA) isfixed-parameter tractable, since after the polynomial-time preprocessing we are left with an instance A(I)whose size is bounded in terms of the parameter, andtherefore any brute-force algorithm applied to A(I)yields fixed-parameter tractability. In other words,CSP(kernA) is fixed-parameter tractable by definition.

Next we will consider nontrivial constraint satis-faction parameters that give rise to tractable classesof constraint satisfaction instances. There are twomain approaches for obtaining tractable classes. Thefirst possibility for achieving tractability is to restrictthe constraint relations to a certain class of rela-tions, a constraint language. For example, Schaefer’sDichotomy Theorem [96] is a classical result which clas-sifies Boolean constraint languages. For further resultson tractable constraint languages, see, e.g., Cohen andJeavon’s survey [21].

The second possibility is to impose structuralrestrictions on the way how variables and constraintsinteract. To this end one associates with a constraintsatisfaction instance certain (hyper)graphs that modelthe overall structure of the instance.

As observed by Feder and Vardi [39], the constraintsatisfaction problem can be rephrased as the question ofwhether there exists a homomorphism from a relationalstructure A into a relational structure B. This point ofview reveals an interesting symmetry between the abovetwo possibilities for achieving tractability: structuralrestrictions correspond to restrictions on the structureA, whereas restrictions on the constraint relationscorrespond to restrictions on the structure B.

In this survey we will focus mainly on parametersthat impose structural restrictions. To this end weassociate the following (hyper)graphs with a constraintsatisfaction instance I.

The primal graph GI is the graph whose vertices arethe variables of I; two variables are joined by an edgeif and only if the variables occur together in the scopeof a constraint of I. The incidence graph G∗I is thebipartite graph whose vertices are the variables and theconstraints of I; a variable x and a constraint C arejoined by an edge if and only if x ∈ var(C). Finally,the constraint hypergraph HI is the hypergraph whosevertices are, as for primal graphs, the variables of I; thehyperedges of HI are the sets var(C) for constraints Cof I.

2.3. Instances of bounded treewidth

The graph parameter treewidth measures in a certainsense the “tree-likeness” of a graph. Treewidthis famous for its central role in Robertson andSeymour’s Graph Minor Project; however, the conceptwas independently discovered by other authors.Many otherwise NP-hard graph problems such asHamiltonicity and 3-colorability are fixed-parametertractable if parameterized by the treewidth of the inputgraph. It is generally believed that many practicallyrelevant problems actually do have low treewidth [11].Treewidth has also been fruitfully applied to severalareas of AI and automated reasoning [59]. Beforewe discuss applications of treewidth to constraintsatisfaction we briefly recall its definition.

Let G be a graph and let T be a tree. To improvereadability we will refer to the vertices of T as nodes.Let χ be a labeling of the nodes of T by sets vertices ofG; the sets χ(t) are called bags. The pair (T, χ) is a treedecomposition of G if the following conditions hold:

(i) every vertex of G belongs to χ(t) for some node tof T ;

(ii) for every edge vw of G there is some node t of Tsuch that v, w ∈ χ(t);

(iii) for any three nodes t1, t2, t3 of T , if t2 lies on thepath from t1 to t3, then χ(t1) ∩ χ(t3) ⊆ χ(t2).

The width of a tree decomposition (T, χ) is themaximum |χ(t)| − 1 over all nodes t of T ; thetreewidth tw(G) of G is the minimum width over allits tree decompositions. Determining the treewidthof a graph is an NP-hard problem, as shown by

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Arnborg, Corneil, and Proskurowski [5]. However,as shown by Bodlaender [15], if k is fixed, then fora given graph G one can find in linear time a treedecomposition of width at most k, or decide that thetreewidth of G exceeds k. Bodlaender’s algorithm is notfeasible in practice as its linear running time involves ahuge constant factor. For practical applications otheralgorithms are preferred [14, 16].

The importance of treewidth lies in the fact thatmany hard graph problems can be solved in polynomialtime for graphs of bounded treewidth [12, 23]. So itis obvious to consider constraint satisfaction instanceswith primal or incidence graphs of bounded treewidth.Indeed, it was observed by Freuder [48] that instanceswhose primal graphs have bounded treewidth can besolved in (non-uniformal) polynomial time.

We define the constraint satisfaction parameterstw(I) = tw(GI) and tw∗(I) = tw(G∗I) where GI

and G∗I are the primal and incidence graphs of I,respectively. Since tw(I) ≤ vars(I) holds by trivialreasons for every constraint satisfaction instance I, theW[1]-hardness of CSP(tw) is a direct consequence ofTheorem 2.1.

Corollary 2.1. The problem CSP(tw) is W[1]-hard.

However, as observed by Gottlob, Scarcello, andSideri [60], additionally bounding the domain sizerenders the problem fixed-parameter tractable:

Theorem 2.2 ([60]). The problem CSP(tw,dom)is fixed-parameter tractable.

Fixed-parameter tractability of CSP(tw,dom) canbe established by means of dynamic programming alongtree decompositions. Given an instance I = (V,D, C)we first obtain a tree decomposition ((VT , ET ), χ) ofwidth k of the primal graph GI . Next we producean instance I ′ = (V,D, C′) that is solution-equivalentto I, taking for every node t of T a new constraintCt = (χ(t), Rt) whose relation Rt contains those tuplesover D that are consistent with all constraints C of Iwith var(C) ⊆ χ(t). Since the domain size is bounded,the size of Ct is bounded as well. Now I ′ is an “acyclic”instance that can be solved by a bottom-up traversalof T . This procedure for acyclic instances was firstdescribed by Yannakakis [109].

Bounding the treewidth of incidence graphs insteadof the treewidth of primal graphs yields a more generalconstraint satisfaction parameter tw∗. Namely, thereare classes of constraint satisfaction instances withbounded tw∗ and arbitrarily large tw. However,the treewidth of the incidence graph G∗I exceeds thetreewidth of the primal graph GI at most by one [74].Hence we have the following corollary to Theorem 2.1.

Corollary 2.2. The problem CSP(tw∗) is W[1]-hard.

However, unlike for CSP(tw), bounding the domainsize by an additional parameter does not yield fixed-parameter tractability. This follows from the reductionfrom CLIQUE as described in Example 2.4. In fact,for Boolean instances I that encode the existence of aclique of size k in a graph, as described in Example 2.4,we have tw∗(I) ≤

(k2

). This can be seen by taking a

tree decomposition (T, χ) of the incidence graph whereT is a path (i.e., (T, χ) is a path decomposition) wherebags contain all the

(k2

)constraints plus one of the

variables. Hence we have a reduction from CLIQUEto CSP(tw∗,dom).

Theorem 2.3 ([93]). The problem CSP(tw∗,dom)is W[1]-hard.

This result implies W[1]-hardness of CSP(p,dom)for every constraint satisfaction parameter p that ismore general than tw∗, i.e., if p(I) ≤ f(tw∗(I)) forsome computable function f .

Next we present a decomposition method which isbased on feedback vertex sets and was proposed byDechter [27]. This method is less general than themethod of bounded treewidth. However, it is oftenpreferred since it is significantly simpler to implementand requires less space. This method provides anexample for the “backdoor set approach” to constraintsolving.

A set S of vertices of a graph G is a cycle cut setor feedback vertex set of G if G − S is acyclic (hereG− S denotes the graph obtained from G by removingall vertices in S and all the edges that are incident withvertices in S). Although it is NP-hard to determine,given G and k, whether G has a feedback vertex set ofsize at most k [49], the following parameterized problemis fixed-parameter tractable.

FVS

Instance: A graph G, a non-negative integer k.

Parameter: k.

Question: Does G have a feedback vertex set ofsize at most k?

Downey and Fellows [30] have proposed two fixed-parameter algorithms for FVS, one is based on boundedsearch trees, one is based on a structure theorem due toBodlaender [13]. An efficient fixed-parameter algorithmfor FVS based on the iterative compression techniqueis due to Guo et al. [64].

Assume that S is a feedback vertex set of the primalgraph of a constraint satisfaction instance I = (V,D, C).For each partial assignment α : S → D we obtain fromI the instance I[α] = (V \ S,D, C′), the restriction ofI to α, as follows: First we remove all constraints fromI that are satisfied by α. Next we remove from theremaining constraints all tuples that disagree in at leastone variable with α, and we eliminate the variables of Sfrom the scopes and update the relations accordingly.

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It is not difficult to see that I is consistent if and onlyif I[α] is consistent for at least one partial assignmentα : S → D. However, since S is a feedback vertex setof the primal graph of I, the primal graphs of all theinstances I[α] are acyclic. Whence each of the |D||S|instances I[α] can be solved in polynomial time. Thisrenders the problem fixed-parameter tractable if we take|D| and |S| as parameters.

Therefore, denoting by fvs(I) the size of a smallestfeedback vertex set of the primal graph of I, we havethe following result.

Theorem 2.4. The problem CSP(fvs,dom) isfixed-parameter tractable.

As noted above, this result is less general thanTheorem 2.2 since graphs with small feedback vertexsets have small treewidth, but the converse is not thecase.

Partial assignments and restrictions as consideredabove provide a general framework for parameterizingthe constraint satisfaction problem. Let Γ be a class ofconstraint satisfaction instances for which consistencycan be decided in polynomial time. Consider anarbitrary constraint satisfaction instance I = (V,D, C)and a subset B ⊆ V . The set B is called a strong Γ-backdoor set of I if for all partial assignments α : B →D, the restriction I[α] belongs to the base class Γ; thisnotion of backdoor sets was introduced by Williams,Gomes, and Selman [106]. Let bΓ(I) denote the sizeof a smallest strong Γ-backdoor set of I. If we know astrong Γ-backdoor set B of an instance I = (V,D, C),then we can solve I by considering all |D||B| restrictionsI[α] for α : B → D. Hence, if the detection ofa strong Γ-backdoor set of size at most k is fixedparameter tractable with respect to the parameter k,then CSP(bΓ,dom) is fixed-parameter tractable. Notethat, if Γ0 denotes the class of constraint satisfactioninstances with acyclic primal graphs, then feedbackvertex sets of primal graphs, as considered above, arestrong Γ0-backdoor sets.

The identification of base classes Γ with fixed-parameter tractable backdoor set detection is aninteresting research objective. In the next section wewill consider the concept of backdoor sets with respectto propositional satisfiability.

Grohe [61] has established an interesting result underthe complexity theoretic assumption FPT 6= W[1]; seealso the conference paper of Grohe, Schwentick, andSegoufin [63]: if a class of polynomial time solvableconstraint satisfaction instances is solely characterizedin terms of primal graphs, then the primal graphsof the instances in the class have bounded treewidth.This result rests on Robertson and Seymour’s deepExcluded Grid Theorem [90]. For stating Grohe’stheorem more specifically, we introduce the followingnotation. For a class G of graphs let CSP[G] denote theconstraint satisfaction problem restricted to instanceswhose primal graphs belongs to G. We say that G has

bounded treewidth if there exists a constant k such thatall graphs in G have treewidth at most k.

Theorem 2.5 ([61]). Unless FPT = W[1], thefollowing statements are equivalent for every recursivelyenumerable class G of graphs:

(i) The problem CSP(length)[G] is fixed-parametertractable.

(ii) The problem CSP[G] can be solved in polynomialtime.

(iii) G has bounded treewidth.

2.4. Beyond treewidth

Constraints of large arity can be helpful, as illustratedby the following example.

Example 2.6. Consider constraint satisfactioninstance I = (V,D, C) that contains a constraint Cbig

with var(Cbig) = V . We can solve the instance in poly-nomial time, since we only need to check whether oneof the tuples of the constraint relation of Cbig defines asolution.

A constraint whose scope contains n variablesproduces a clique on n vertices in the primal graph.The primal graph, however, does not provide enoughinformation to conclude from the presence of a cliqueon n vertices that the corresponding n variables occurtogether in the scope of a constraint (the clique could bethe result of, say, several binary constraints). Thereforeit is beneficial to consider the constraint hypergraph, asdefined above, which contains this additional structuralinformation.

Let us fix some hypergraph terminology beforewe discuss properties of constraint hypergraphs. Ahypergraph H is a pair (V,E) where V is a set of verticesand E is a set of hyperedges; a hyperedge is a subsetof V . The primal graph or 2-section of a hypergraphH = (V,E) is the graph G = (V,E2) where E2 containsall pairs of distinct vertices that occur together insome hyperedge of H. Thus the primal graph of aconstraint satisfaction instance I is the primal graphof the constraint hypergraph of I.

A constraint satisfaction instance is acyclic if itsconstraint hypergraph is acyclic, where a hypergraphis called acyclic if it can be reduced to the emptyhypergraph (∅, ∅) by multiple applications of thefollowing rules (we refer here to the most generalconcept of hypergraph acyclicity, also known as α-acyclicity [36]).

(i) Remove a vertex that belongs to at most onehyperedge.

(ii) Remove a hyperedge that is empty or is a subsetof another hyperedge.

Several equivalent definitions of hypergraph acyclicityare known [8]. For example, a hypergraph is acyclicif and only if there exists a tree decomposition of

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its primal graph such that the bags of the treedecomposition are hyperedges of the hypergraph. Sucha tree decomposition, rooted at one of its nodes, is alsoknown as a join tree of the acyclic hypergraph. Thefollowing is a well-known result of Yannakakis [109].

Theorem 2.6 ([109]). Acyclic constraint satisfac-tion instances can be solved in polynomial time.

Actually, in our above sketch of the algorithm thatestablishes Theorem 2.2 we have already encountered ajoin tree: the tree decomposition of the given instance Iis nothing but a join tree of the constraint hypergraphof the equivalent instance I ′.

Next we state a related fixed-parameter tractabilityresult that is based on a mixture of restrictions onboth the structure and the constraint relations. Oftenconstraint satisfaction instances involve several non-equality constraints, i.e., constraints that impose therestriction that two variables cannot receive the samevalue. For a constraint satisfaction instance I let I6=denote the instance obtained from I by removing allnon-equality constraints. Furthermore, let CSPacyc

6=denote the constraint satisfaction problem restricted toinstances I for which I6= is acyclic. The question arisesof whether CSPacyc

6= is easier than CSP. Interestingly,as shown by Papadimitriou and Yannakakis [87], theanswer is no for the non-parameterized setting, but yesfor the parameterized setting:

Theorem 2.7 ([87]). The problem CSPacyc6= is NP-

complete.

Theorem 2.8 ([87]). The problem CSPacyc6= (vars)

is fixed-parameter tractable.

In a series of papers, Gottlob, Leone, and Scarcellohave developed a concept of hypergraph decompositionswhich allows a slice-wise generalization of acyclicconstraint satisfaction instances [56]. Let H = (V,E)be a hypergraph. A generalized hypertree decompositionof H is a triple (T, χ, λ) where (T, χ) is a treedecomposition of the primal graph of H and λ is amapping that assigns to every node t of T a set λ(t)of hyperedges of H such that each vertex x ∈ χ(t)is contained in some hyperedge e ∈ λ(t). Thus λ(t)is a set cover of χ(t). The width of a generalizedhypertree decomposition (T, χ, λ) is the size of a largestset λ(t) over all nodes t of T . The generalizedhypertree width ghw(H) of H is the minimum widthover all generalized hypertree decompositions ofH. Thehypertree width hw(H) is defined similarly, consideringhypertree decompositions that satisfy an additionaltechnical condition. Recently Adler, Gottlob, andGrohe [1] have shown that hypertree width andgeneralized hypertree width are related by a smallmultiplicative constant, namely ghw(H) ≤ hw(H) ≤3 ·ghw(H)+1. For fixed k one can decide in polynomialtime whether hw(G) ≤ k (see below); for generalized

hypertree width, however, deciding ghw(H) ≤ 3 is NP-hard [57].

We define the constraint satisfaction parametersghw and hw via the (generalized) hypertree width ofconstraint hypergraphs. Gottlob, Leone, and Scarcello[55] have shown that for any constant k one can decideconstraint satisfaction instances I with hw(I) ≤ k inpolynomial time; the problem is even in the complexityclass LOGCFL and so highly parallelizable.

Theorem 2.9 ([55]). The problems CSP(hw) andCSP(ghw) are in XP.

The polynomial time algorithm of Gottlob et al. [55]can be outlined as follows. Let I be an instance ofsize s with ghw(I) ≤ k and let HI = (V,E) be itsconstraint hypergraph; by the above result of Adleret al., hw(I) ≤ 3k + 1 =: k′. First we compute ahypertree decomposition of H of width at most k′. Thiscan be accomplished in time O(|E|2k′ |V |2). Second,using the decomposition, we transform I into a solution-equivalent acyclic instance I ′. This can be accomplishedin time O(sk′

). Third, we solve the acyclic instance I ′

by Yannakakis’ polynomial-time algorithm as outlinedabove. In total we have the overall time complexityO(s2(k

′+1) log s).This algorithm does not render the problem

CSP(ghw) fixed-parameter tractable. A recent resultof Gottlob, Grohe, Musliu, Samer, and Scarcello [54]shows that the first step of the above procedure involvesa W[2]-hard problem:

Theorem 2.10 ([54]). The problem of decidingwhether ghw(H) ≤ k (or, hw(H) ≤ k) holds for agiven hypergraph H is W[2]-hard with respect to theparameter k.

Chen and Dalmau [17] have shown that it is possibleto avoid the decomposition step. They presenta game-theoretic algorithm that decides consistencyin polynomial time for instances whose generalizedhypertree width is bounded by a constant. Theiralgorithm, however, is not a fixed-parameter algorithmeither. Furthermore, a similar argument as used forthe proof of Theorem 2.3 shows that the problemsCSP(hw,dom) and CSP(ghw,dom) are W[1]-hard.

Grohe and Marx [62] have proposed two newhypergraph parameters that give raise to new classesof polynomial-time solvable constraint satisfactioninstances. A fractional edge cover of a hypergraphH = (V,E) is a mapping ψ : E → [0,∞] such that∑

e∈E,v∈e ψ(e) ≥ 1 holds for all vertices v ∈ V . Theweight of ψ is given by

∑e∈E ψ(e). The fractional edge

cover number ρ∗(H) is the minimum of the weights of alledge covers of H. For a constraint satisfaction instanceI let ρ∗(I) = ρ(HI). Using Shearer’s Lemma [20],Grohe and Marx have shown that, given a constraintsatisfaction instance I of size s and k = ρ∗(I), allsolutions of I can be listed in time sk+O(1). This yieldsthe following result.

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Theorem 2.11 ([62]). The problem CSP(ρ∗) is inXP.

This result is interesting since there are classes ofconstraint satisfaction instances of constant ρ∗ witharbitrary high ghw, and classes of instances witharbitrarily high ρ∗ and constant ghw [62]. Furthermore,Grohe and Marx have defined fractional hypertreedecompositions of hypergraphs, combining the conceptsof generalized hypertree decompositions and fractionaledge covers. This new decomposition concept givesrise to the hypergraph parameter fractional hypertreewidth fhw(H). For every hypergraph H, fhw(H) isbounded by both ρ∗(H) and ghw(H); hence fractionalhypertree width generalizes both other parameters. Itis not known whether for a given hypergraph H withfhw(H) ≤ k for a fixed constant k, one can finda fractional hypertree decomposition of H of widthat most k in polynomial time. However, if such adecomposition is provided, then the instance can besolved in polynomial time. For a classH of hypergraphslet CSP[H] denote the constraint satisfaction problemrestricted to instances whose constraint hypergraphsbelong to H.

Theorem 2.12 ([62]). The problem CSP(vars)[H]is fixed-parameter tractable for every class H ofhypergraphs with bounded fractional hypertree width.

Since fhw is more general than tw∗, Theorem 2.3implies that the problem CSP(fhw,dom) is W[1]-hard.

3. PROPOSITIONAL SATISFIABILITY

3.1. Preliminaries

Propositional satisfiability is the classical problemof determining whether a propositional formula inconjunctive normal form has a satisfying truthassignment. Cook’s famous proof that this problemis NP-complete placed satisfiability as the cornerstoneof complexity theory. Despite its seemingly specializednature, satisfiability has proved to be extremely usefulin a wide range of different disciplines such as hardwareand software verification [10, 76, 105] and planning [71].

We consider propositional formulas in conjunctivenormal form (CNF) presented as a finite sets of clauseswhere a clause is a finite set of literals, and a literalis a negated or unnegated propositional variable. Forexample,

F = {{¬x, y, z}, {¬y,¬z}, {x,¬y}}

represents the propositional formula (¬x∨y∨z)∧(¬y∨¬z)∧ (x∨¬y). For a literal ` we denote by ` the literalof opposite polarity, i.e., x = ¬x and ¬x = x. Similarly,for a set L of literals, we put L = { ` : ` ∈ L }. We saythat two clauses C,D overlap if C ∩D 6= ∅, and we saythat C and D clash if C and D overlap.

For a clause C we denote by var(C) the set ofvariables that occur (negated or unnegated) in C; for a

CNF formula F we put var(F ) =⋃

C∈F var(C). Wemeasure the size of a CNF formula F by its length∑

C∈F |C|.A truth assignment is a mapping τ : X → {0, 1}

defined on some set X of variables. We extend τ toliterals by setting τ(¬x) = 1 − τ(x) for x ∈ X. F [τ ]denotes the formula obtained from F by removing allclauses that contain a literal ` with τ(`) = 1 and byremoving from the remaining clauses all literals `′ withτ(`′) = 0; F [τ ] is the restriction of F to τ . Notethat var(F [τ ]) ∩ X = ∅ holds for every assignmentτ : X → {0, 1} and every CNF formula F . A truthassignment τ : X → {0, 1} satisfies a CNF formulaF if F [τ ] = ∅. A formula F is satisfiable if thereexists a truth assignment that satisfies F ; otherwise F isunsatisfiable. SAT is the problem of deciding whethera given CNF formula is satisfiable.

Example 3.1. A classical application of SAT is thetautology testing problem. Consider the propositionalformula Ψ = (x → (y → x)). We transform Ψinto a CNF formula using a technique which is due toTseitin [104]. First we replace the subformula (y → x)by a new “extension variable” u and produce the setof clauses F1 = {{y, u}, {¬x, u}, {¬y, x,¬u}} whichasserts the equivalence u ≡ (y → x). Next, we replace(x→ u) by a new extension variable v and produce theset of clauses F2 = {{x, v}, {¬u, v}, {¬x, u,¬v}} whichasserts the equivalence v ≡ (x → u). It is easy to seethat Ψ is a tautology if and only if the CNF formulaF = F1 ∪ F2 ∪ {{¬v}} is unsatisfiable.

In general, for every propositional formula Ψ we canobtain in linear time a CNF formula F such that Ψ isa tautology if and only if F is unsatisfiable.

Obviously, we can decide the satisfiability ofa CNF formula F with n variables by checkingall 2n truth assignments τ : var(F ) → {0, 1};surprisingly, no algorithm is known that performs, inthe worst case, significantly better than the bruteforce search. More precisely, no algorithm is knownthat decides satisfiability of CNF formulas with nvariables in time 2o(n), and it is believed that nosuch algorithm exists [69]. State-of-the-art satisfiabilitysolvers, however, are capable of solving large CNFformulas originating from real-world applications withthousands of variables. This successful performanceof solvers is usually explained by the presence of a“hidden structure” in problem instances that arise frompractical applications. The parameterized approachto satisfiability entails the identification of parametersthat allow fixed-parameter algorithms and exploit thehidden structure.

Similarly as for constraint satisfaction above, weconsider as a satisfiability parameter any computablefunction p that assigns to a CNF formula Fa non-negative integer p(F ). For satisfiabilityparameters p1, . . . , pr we consider the following genericparameterized decision problem:

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SAT(p1, . . . , pr)

Instance: A CNF formula F , non-negativeintegers k1, . . . , kr with p1(F ) ≤ k1, . . . , pr(F ) ≤kr.

Parameters: k1, . . . , kr.

Question: Is F satisfiable?

As for constraint satisfaction, we can consider thesatisfiability parameter kernA(F ), the size of the CNFformula F ′ obtained from F by a polynomial-timealgorithm A. The algorithm simplifies a given instanceby applying simplification and data-reduction rulesand outputs a satisfiability-equivalent CNF formula F ′.Clearly SAT(kernA) is fixed-parameter tractable bydefinition.

Unit propagation and pure literal elimination are themain simplification and data reduction methods usedin SAT solvers. If a CNF formula F contains a unitclause {`}, we can simplify it by unit propagation to thesatisfiability-equivalent CNF formula F [` = 1]. Notethat F [` = 1] may contain new unit clauses that wherenot present in F . A literal ` is a pure literal of aCNF formula F if some clauses of F contain ` but noclause contains `. If ` is a pure literal of F , then wecan simplify F by obtaining the satisfiability-equivalentCNF formula F [` = 1]. Also pure literal eliminationpropagates.

3.2. Treewidth and related parameters

We associate certain graphs and hypergraphs with CNFformulas similarly as for constraint satisfaction. Theprimal graph GF of a CNF formula is the graph withvertex set var(F ) where two variables x, y are joinedby an edge if and only if F contains a clause C withx, y ∈ var(C). The incidence graph G∗F is the bipartitegraph whose vertices are the variables and the clausesof F ; a variable x and a clause C are joined by an edgeif and only if v ∈ var(C). The directed incidence graphD∗

F is obtained from G∗F by orienting an edge xC fromx to C if ¬x ∈ C and from C to x if x ∈ C. Finally, theformula hypergraph HF is the hypergraph with vertexset var(F ) and hyperedges var(C) for all C ∈ F . Wedefine the satisfiability parameters tw(F ) = tw(GF )and tw∗(F ) = tw(G∗F ).

If a CNF formula F has clauses of bounded size,then we can transform F into a constraint satisfactioninstance I such that F is satisfiable if and only if I isconsistent. Namely, we replace each clause C with aconstraint whose constraint relation contains 2|C| − 1tuples in the obvious way. However, if the size ofclauses is unbounded, then this transformation yieldsan exponential blow-up of the instance size. It is knownthat the treewidth of a graph G plus 1 is an upperbound for the size of a clique in G. Note also that aclause of size s yields a clique of size s in the primalgraph. Hence, if the treewidth of the primal graph of

a CNF formula F is at most k, then the size of clausesof F is bounded by k + 1. In other words, boundingthe treewidth of primal formula graphs automaticallybounds the size of clauses. Consequently, the abovetransformation does not cause an exponential blow-upfor CNF formulas whose primal graphs have boundedtreewidth. Thus Theorem 2.2 immediately gives us thefollowing result.

Theorem 3.1. The problem SAT(tw) is fixed-parameter tractable.

Alekhnovich and Razborov [4] have pointed out thesignificance of the hypergraph parameter branchwidthfor satisfiability. We briefly recall the definition ofbranchwidth. Let H be a hypergraph, T = (V,E) aternary tree (i.e., all vertices of T have either degree1 or 3), and τ a one-to-one mapping from the set ofleaves of T to the set of hyperedges of H; (T, τ) iscalled a branch decomposition of H. The order ofan edge e of T is the number of vertices of H whichbelong to hyperedges τ(t1), τ(t2) for nodes t1 and t2that belong to different components of T −e. The widthof a branch decomposition (T, τ) is the maximum orderof all edges of T ; the branchwidth bw(H) of H is thesmallest width over all its branch decompositions. Thebranchwidth of a CNF formula is the branchwidth of itsformula hypergraph, that is, bw(F ) = bw(HF ). Thefollowing is a consequence of the results of Alekhnovichand Razborov [4].

Theorem 3.2 ([4]). The problem SAT(bw) is fixed-parameter tractable.

This result has been obtained via a modificationof Robertson and Seymour’s algorithm for computingbranch-decompositions [92]. Bacchus, Dalmao, andPitassi [7] have proposed an efficient fixed-parameteralgorithm for CNF formulas of bounded branchwidth.This algorithm, which actually computes the number ofsatisfying assignments, is based on the DPLL procedureand uses caching techniques for an efficient reuse ofsolutions for subproblems.

Using a result of Robertson and Seymour [91], onecan show the following lemma [100].

Lemma 3.1. For every CNF formula F without pureliterals we have

bw(F ) ≤ tw(F ) + 1 ≤ 2 · bw(F ).

Thus Theorems 3.1 and 3.2 are essentially equivalentsince pure literals can be eliminated in polynomial timewithout increasing the treewidth.

3.3. Treewidth and clique-width

Similarly as for constraint satisfaction (see the previoussection) bounding the treewidth of incidence graphsgives larger classes of CNF formulas than boundingthe treewidth of primal graphs. Indeed satisfiability

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is fixed-parameter tractable for the more generalparameter.

Theorem 3.3. The problem SAT(tw∗) is fixed-parameter tractable.

This theorem can be shown by means of linearprogramming on tree decompositions of incidencegraphs [42, 94]. However, using general results ofdescriptive complexity theory, one can easily derivefixed-parameter tractability of SAT(tw∗). Next wewill outline this approach (a variant was proposed bySzeider [100]).

Courcelle [23] has shown that if a graph property canbe expressed in a certain fragment of monadic secondorder logic, MSO2, then this property can be checkedin linear time for graphs of bounded treewidth; see alsoDowney and Fellows’ book [30] for an elegant proofof this result. MSO2 admits quantification over setsof vertices and over sets of edges. MSO1 denotes theweaker fragment of monadic second order logic whereonly quantification over sets of vertices is allowed; forour purposes it suffices to consider MSO1.

Example 3.2. Whether a graph G = (V,E) is 3-colorable can be expressed by the MSO1 formula

(∃R ⊆ V )(∃G ⊆ V )(∃B ⊆ V )[partition(R,G,B)∧proper(R) ∧ proper(G) ∧ proper(B)]

using the abbreviations

partition(R,G,B) ≡R ∩G = ∅ ∧R ∩B = ∅∧B ∩G = ∅ ∧R ∪G ∪B = V,

proper(X) ≡∀u, v ∈ V (uv ∈ E → ¬(u ∈ X ∧ v ∈ X)).

Hence, it follows from Courcelle’s Theorem that theproblem of deciding whether a graph of treewidth k is3-colorable is fixed-parameter tractable with respect tothe parameter k.

We establish Theorem 3.3 using the following graphtheoretic formulation of satisfiability. For a CNFformula F we obtain the graph G4(F ) with vertexset {x,¬x : x ∈ var(F ) } ∪ F ∪ { aC , bC : C ∈ F };edges are defined such that (i) each literal is adjacentwith its complement and with all clauses that containthe literal, and (ii) each clause C forms a trianglewith the vertices aC and bC . The purpose of the“auxiliary” vertices aC , bC is to distinguish betweenvertices representing clauses and vertices representingliterals. It is straightforward to state an MSO1 formulaΨ that expresses the following property of G4(F ):

“There exists a set V1 of literals such that for everyliteral ` ∈ V1 its complement ` is not in V1, and everyclause is adjacent with at least one literal in V1.”

It is not difficult to see that F is satisfiable if and onlyif Ψ is true for G4(F ), and that tw(G4(F )) is bounded

in terms of tw(G∗(F )). Hence, Theorem 3.3 follows byCourcelle’s Theorem.

Courcelle’s Theorem provides a very general and pow-erful means for showing fixed-parameter tractability.However it is important to note that the running timesof algorithms obtained by this result involve huge hid-den constants which make an implementation difficultand limits the practical applicability. However, once thefixed-parameter tractability of a problem is established,one can try to develop made-to-measure combinatorialalgorithms for the problem under consideration.

Clique-width is a graph parameter that is moregeneral than treewidth in the sense that every graphclass of bounded clique-width has also boundedtreewidth, but there are graph classes of boundedclique-width and unbounded treewidth. Since aresult similar to Courcelle’s Theorem holds for clique-width, clique-width gives raise to a more generalparameterization of the satisfiability problem. Beforewe state the relevant results in detail we give thedefinition of clique-width.

Let k be a positive integer. A k-graph is a graphwhose vertices are labeled by integers from {1, . . . , k}.We consider an arbitrary graph as a k-graph with allvertices labeled by 1. We call the k-graph consisting ofexactly one vertex v, labeled by i ∈ {1, . . . , k}, an initialk-graph and denote it by i(v). The clique-width cwd(G)of a graph G is the smallest integer k such that G can beconstructed from initial k-graphs by means of repeatedapplication of the following three operations.

(i) Disjoint union (denoted by ⊕);(ii) Relabeling : changing all labels i to j (denoted by

ρi→j);(iii) Edge insertion: connecting all vertices labeled by

i with all vertices labeled by j, i 6= j (denoted byηi,j).

A construction of a k-graph using the above operationscan be represented by an algebraic term composed ofi(v), ⊕, ρi→j , and ηi,j , (i, j ∈ {1, . . . , k}, and i 6= j).Such a term is called a k-expression defining G. Thus,the clique-width of a graph G is the smallest integer ksuch that G can be defined by a k-expression.

Example 3.3. The cliqueK4 on the vertices u, v, w, xis defined by the 2-expression

ρ2→1(η1,2(ρ2→1(η1,2(ρ2→1(η1,2(

2(u)⊕ 1(v)))⊕ 2(w)))⊕ 2(x))).

Hence cwd(K4) ≤ 2. In general, every clique on two ormore vertices has clique-width 2.

Similarly one can define the clique-width of a directedgraph D as the smallest integer k such that D canbe defined by a k-expression where the operation ηi,j

connects all vertices labeled by i with all vertices labeledby j by edges directed from i-labeled vertices to j-labeled vertices.

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Courcelle, Makowsky, and Rotics [24] have shownthat any property for graphs (or directed graphs)definable in MSO1 can be checked in linear time ifa k-expression of the given graph is provided as aninput; here k is considered as a constant. Thisresult immediately applies to satisfiability via directedincidence graphs. Namely, a CNF formula F issatisfiable if and only if the following property holdsfor the directed incidence graph D∗(F ):

“The set of variables can be partitioned into two setsV0, V1 such that for every clause C there exists either(i) a variable x ∈ V0 and a directed edge (x,C) or(ii) a variable x ∈ V1 and a directed edge (C, x).”

Note that the partition V0, V1 corresponds to asatisfying assignment τ with τ(x) = i for x ∈ Vi,i = 0, 1. Considering the satisfiability parameterdcwd∗(F ) = cwd(D∗(F )) we do not have yet fixed-parameter tractability of SAT(dcwd∗) since the resultof Courcelle et al. [24] requires a k-expression as aninput. Determining the clique-width of a given graphis NP-hard, as shown by Fellows, Rosamond, Rotics,and Szeider [40]. However, Oum and Seymour providea fixed-parameter algorithm for approximating clique-width.

Theorem 3.4 ([84]). Given a graph G of order nand clique-width k one can find a (23k+2−1)-expressionfor G in time O(n9 log n).

As observed by Fischer, Makowsky, and Ravve [42],the proof of Theorem 3.4 carries over to directedgraphs. Hence, combining the MSO1 result of [24] withTheorem 3.4, one obtains fixed-parameter tractabilityof SAT(dcwd∗).

Theorem 3.5 ([42]). The problem SAT(dcwd∗) isfixed-parameter tractable.

From the theoretical point of view, Theorem 3.5subsumes Theorems 3.1 and 3.3 since there exists acomputable function f such that tw∗(F ) ≤ k impliesdcwd∗(F ) ≤ f(k). The latter follows from a result ofCourcelle and Olariu [25].

We note that Theorem 3.5 can also be obtainedusing Theorem 3.4 for undirected graphs since it iseasy to show that cwd(G4(F )) is bounded in termsof cwd(D∗(F )).

3.4. Maximum deficiency and falsum number

Matchings in incidence graphs give raise to thesatisfiability parameter maximum deficiency, firstconsidered by Franco and Van Gelder [47]. A set Mof edges of a graph G is a matching if each vertex ofG is incident with at most one edge in M . A matchingof largest size of a graph G is a maximum matching ofG. It is well known that a maximum matching can befound in polynomial time. The following observation isdue to Aharoni and Linial [2].

Lemma 3.2. If the incidence graph of a CNF formulaF has a matching M such that every clause of F isincident with an edge of M , then F is satisfiable.

Namely, given such a matching, we can satisfy eachclause independently by choosing the right truth valuefor the variable to which the clause is matched. Themaximum deficiency md(F ) of a CNF formula F isthe number of clauses of F that are not incidentwith an edge of an arbitrary but fixed maximummatching of the incidence graph of F . Lemma 3.2covers the case md(F ) = 0. Defining the deficiencyof a CNF formula F as the difference d(F ) =|F | − |var(F )|, the following equation, which can beestablished using Hall’s Theorem, explains the originof the term “maximum deficiency”

md(F ) = maxF ′⊆F

d(F ′).

Fleischner, Kullmann, and Szeider [43] have shownthat SAT(md) is in XP. Szeider [99] has improvedthis result to fixed-parameter tractability.

Theorem 3.6 ([99]). The problem SAT(md) isfixed-parameter tractable.

This result can be applied to the recognition of min-imal unsatisfiable CNF formulas (i.e., CNF formulasthat are unsatisfiable but become satisfiable by remov-ing any clause). In general, recognition of minimalunsatisfiable CNF formulas is DP-complete [86]. Aproblem is in DP if and only if it can be formulatedas the conjunction of an NP-problem and a co-NP-problem [85]. A canonical DP-problem is the following:given two CNF formulas F1, F2, determine whether F1 issatisfiable and F2 is unsatisfiable. Deficiency and max-imum deficiency agree for minimal unsatisfiable CNFformulas [2, 43, 99]. Hence, using the algorithm of The-orem 3.6 as a subroutine, one can obtain the followingresult.

Theorem 3.7 ([99]). The recognition of minimalunsatisfiable CNF formulas is fixed-parameter tractablewith respect to the parameter deficiency.

Note that the satisfiability parameters md anddcwd∗ are incomparable in the following sense. Thereare CNF formulas with constant md and arbitrarilylarge dcwd∗, and conversely, there are CNF formulaswith arbitrarily large md and constant dcwd∗ [99]. Ofrelated interest is a recent paper of Szeider [102] wherethe relationship between maximum deficiency and thesize of backdoor sets is considered.

Next we consider propositional formulas that are notin conjunctive normal form. A propositional formula ψis called f -implicational if → (implication) is the onlyconnective of ψ; however, ψ may contain the constantf (falsum) that is always mapped to 0 by any truthassignment. Franco, Goldsmith, Schlipf, Speckenmeyer,and Swaminathan [46] have shown the following.

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Theorem 3.8 ([46]). Satisfiability of f -implicationalformulas with at most two occurrences of each variableand k occurrences of f is fixed-parameter tractable withrespect to the parameter k.

Heusch, Porschen, and Speckenmeyer [68] haveimproved the algorithm of Franco et al. using dynamicprogramming techniques.

Szeider [100] has considered a canonical translation ofCNF formulas F into f -implicational formulas ψF withat most two occurrences of each variable. The falsumnumber fn(F ) of a CNF formula F is the number ofoccurrences of f in ψF . Theorem 3.8 yields directly thefixed-parameter tractability of SAT(fn).

Theorem 3.9 ([100]). The problem SAT(fn) isfixed-parameter tractable.

Since md(F ) ≤ fn(F ) holds for CNF formulas Fwithout pure literals [100], Theorem 3.6 subsumesTheorem 3.9.

3.5. Backdoor sets

We have already considered the concept of backdoorsets in the section on constraint satisfaction. Thisconcept also applies to satisfiability [107]. Consider aclass Γ of CNF formulas. A set B of variables of a CNFformula F is a strong Γ-backdoor set of F if for all truthassignments τ : B → {0, 1}, the restriction F [τ ] belongsto Γ. Similarly as above, let bΓ(F ) denote the size of asmallest strong Γ-backdoor set of F . For a base class Γwe consider the following parameterized problem.

SBDS(Γ)

Instance: A CNF formula F , a non-negativeinteger k.

Parameter: k.

Question: Does F have a strong Γ-backdoor setof size at most k? I.e., is bΓ(F ) ≤ k?

Let Γ be a class of CNF formulas for whichsatisfiability can be decided in polynomial time.Assume that SBDS(Γ) is fixed-parameter tractablein the sense that a fixed-parameter algorithm finds astrong Γ-backdoor set of size at most k if such setexists. Then, clearly, SAT(bΓ(F )) is fixed-parametertractable.

For the base classes HORN (CNF formulas whereeach clause contains at most one positive literal) and2CNF (CNF formulas with clauses of size at most 2),Nishimura, Ragde, and Szeider [82] have shown thefollowing.

Theorem 3.10 ([82]). The problems SBDS(Γ) andSAT(bΓ(F )) are fixed-parameter tractable for Γ ∈{HORN, 2CNF}.

The algorithms of Nishimura et al. make use of theconcept of deletion backdoor sets. For a CNF formula

F and a set of variables B ⊆ var(F ), let F −B denotethe CNF formula {C \ (B ∪B) : C ∈ F }. We say thatB is a deletion Γ-backdoor set of F if F − B ∈ Γ. Wedenote the size of a smallest deletion Γ-backdoor set ofF by dbΓ(F ).

Consider a base class Γ with the property that allsubsets of a CNF formula of Γ also belong to Γ; inthat case we say that Γ is clause-induced. Since F [τ ] ⊆F − B holds for every truth assignment B → {0, 1},it follows that for a clause-induced base class Γ, eachdeletion Γ-backdoor set is also a strong Γ-backdoorset. Often it is easier to search for deletion backdoorsets than for strong backdoor sets; however, smallestdeletion backdoor sets can be larger than smalleststrong backdoor sets.

For the base classes HORN and 2CNF we are lucky:it can be shown that strong and deletion backdoor setsfor these two classes coincide [82]. Consequently, thesearch for strong HORN or 2CNF-backdoor sets can beaccomplished by a simple search-tree algorithm. Let Fbe a CNF formula that contains a non-Horn clause C;i.e., C contains at least two positive literals x, y. Everydeletion HORN-backdoor set of F must contain eitherx or y. Thus we can branch into cases F − {x} andF−{y} and repeat recursively with the parameter k−1.The branching step for the detection of deletion 2CNF-backdoor sets is similar: we take a clause C of size atleast 3 and pick three distinct variables x, y, z ∈ var(C).We branch into cases F − {x}, F − {y}, and F − {z},and repeat recursively with the parameter k − 1.

In a recent paper, Samer and Szeider [95] have gener-alized the notion of backdoor sets to Quantified BooleanFormulas (QBF) and have show that Theorem 3.10 alsoholds in this more general setting.

Consider the class UP of CNF formulas that canbe decided by unit propagation; that is, by repeatedapplication of unit propagation to F ∈ UP we caneither reduce F to the empty CNF formula (F issatisfiable) or we can reduce F to a CNF formulathat contains the empty clause (F is unsatisfiable). Itis well known that every unsatisfiable Horn formulabelongs to UP [72]. Thus, a possible generalization ofTheorem 3.10 is to consider strong UP-backdoor sets.However, Szeider [101] has shown that the detection ofstrong UP-backdoor sets is a W[P]-complete problem.The same holds true for the class PL of CNF formulasdecidable by pure literal elimination, and the classUP + PL of CNF formulas decidable by a combinationof unit propagation and pure literal elimination.

Theorem 3.11 ([101]). The problems SBDS(UP),SBDS(PL), and SBDS(UP + PL) are W[P]-complete.

Note that for the classes UP, PL, and UP + PL,deletion backdoor sets are not always strong backdoorsets (the classes are not clause-induced). Hence deletionbackdoor sets do not provide a way for overcoming thelimits of Theorem 3.11.

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Let RHORN denote the class of renamable HornCNF formulas, i.e., of CNF formulas F for which thereexists a set X ⊆ var(F ) such that, replacing in theclauses of F the literal x by ¬x and the literal ¬x byx whenever x ∈ X, yields a Horn formula. Obviously,RHORN properly contains HORN. In turn, from anunsatisfiable formula F ∈ RHORN we can derive theempty clause by unit propagation, i.e, F ∈ UP [72].The parameterized complexity of SBDS(RHORN) isan interesting open question, as it lies in a certain sensebetween the fixed-parameter tractable SBDS(HORN)and the W[P]-complete SBDS(UP). Since RHORN isclause-induced, one can also study the relaxed problemof deciding whether a given CNF formula has a deletionRHORN-backdoor set of size at most k. The latterproblem is equivalent (under parameterized reductions)to the problem of deciding whether a graph G witha perfect matching M has a vertex cover of size atmost |M |/2 + k. Note that for such a graph G, |M |/2is a guaranteed value for the size of a vertex cover;parameterizations above guaranteed value have recentlyreceived a lot of attention [77, 66, 65].

A further interesting base class is the class CLU ofcluster formulas. A CNF formula is a cluster formula ifit is the variable-disjoint union of hitting formulas whichare CNF formulas where any two clauses of clash [73].

The next lemma is due to an observation ofIwama [70].

Lemma 3.3. A hitting formula F with n variableshas exactly 2n −

∑C∈F 2n−|C| satisfying assignments

τ : var(F )→ {0, 1}.

Consequently, we can decide the satisfiability ofhitting formulas, and in turn, the satisfiability of clusterformulas, in polynomial time.

Nishimura, Ragde, and Szeider [83] have consideredthe parameterized complexity of backdoor set detectionfor the base class CLU. Note that CLU is clause-induced, hence both strong and deletion backdoor setsare relevant.

Theorem 3.12 ([83]). The problem SBDS(CLU) isW[2]-hard.

This negative result follows by a reduction fromthe parameterized hitting set problem. The relaxedparameter dbCLU admits a fixed-parameter algorithm.

Theorem 3.13 ([83]). The problem SAT(dbCLU) isfixed-parameter tractable.

This result is obtained by means of an algorithm thatsystematically destroys certain obstructions that consistof pairs or triples of clauses. To this end, the obstructiongraph of a CNF formula F is considered. The vertex setof this graph is the set of variables of F ; two variablesx, y are joined by an edge if and only if at least one ofthe following conditions hold.

(i) F contains two clauses C1, C2 that do not clash,x ∈ var(C1 ∩ C2), and y ∈ var(C1 \ C2);

(ii) F contains three clauses C1, C2, C3 such that C1

and C3 do not clash, x ∈ var((C1 \ C3) ∩ C2), andy ∈ var((C3 \ C1) ∩ C2).

Vertex covers of obstruction graphs are closely relatedto backdoor sets: Every deletion CLU-backdoor set ofa CNF formula F is a vertex cover of the obstructiongraph of F . Conversely, every vertex cover of theobstruction graph of a CNF formula F is a strong CLU-backdoor set of F .

The satisfiability parameter clu(F ), the clustering-width, is defined as the size of a smallest vertex coverof the obstruction graph of F . Hence the followinginequalities hold for every CNF formula F .

bCLU(F ) ≤ clu(F ) ≤ dbCLU(F ).

Finding a size k vertex cover of a given graph (ordeciding that such a vertex cover does not exist) isfixed-parameter tractable, see Example 1.1. Since everyvertex cover of the obstruction graph of a CNF formulaF is a strong CLU-backdoor set of F , we have thefollowing.

Theorem 3.14 ([83]). The problem SAT(clu) isfixed-parameter tractable.

Moreover, Theorem 3.13 follows, since we alwayshave clu(F ) ≤ dbCLU(F ). There are classes of CNFformulas with constant clu and unbounded dcwd∗,and classes of CNF formulas with unbounded clu andconstant dcwd∗ [83]. Hence the clustering number isincomparable with any of the satisfiability parameterstw, tw∗, bw, and dcwd∗.

4. LOGIC PROGRAMMING ANDNON-MONOTONIC REASONING

4.1. Computing stable models

A well-studied form of non-monotonic reasoning islogic programming under the stable model semantics.Before we survey parameterized problems that arisein this context we briefly review some concepts anddefinitions. Here we restrict our scope to propositionallogic programming.

A literal is an atom A or a negated atom ¬A. Adisjunctive logic program (DLP) P is a finite set of rulesof the form

A′1 ∨ · · · ∨A′r ←− A1 ∧ · · · ∧Am ∧ ¬Am+1 ∧ · · · ∧ ¬An

where A,A1, . . . , An, A′1, . . . , A

′r are atoms [88]. Here

the head H(r) of a rule r of P is the set of atomsappearing at the left-hand-side of r, and the body B(r)is the set of atoms at the right-hand-side of r. We alsowrite H(P ) (respectively B−(P )) for the union of H(r)(respectively B−(r)) for all rules r of P . If |H(r)| = 1for all rules r of a disjunctive logic program P , we call P

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a normal logic program. A DLP is positive if B−(r) = ∅holds for all rules r. A positive normal logic program isa Horn program.

The body B(r) of a rule is partitioned into the set ofunnegated atoms B+(r) and the set of negated atomsB−(r). The universe U(P ) of such a program P is theset of all atoms occurring in P , and an interpretation Iof P is a subset of U(P ) which fixes a set of atoms thatare true according to I, while the atoms in U(P )\ I arefalse according to I. An interpretation I of a disjunctivelogic program P satisfies a rule r if H(r) ∩ I 6= ∅ orB+(r) \ I 6= ∅ or B−(r) ∩ I 6= ∅. If I satisfies all therules of P we call I a model of P .

The reduct P I of a disjunctive logic program Punder an interpretation I is obtained by performingthe following two steps: first we remove every rule rsuch that B−(r)∩ I 6= ∅; second we remove all negatedatoms from the bodies of the remaining rules. Thereduct of a disjunctive logic program is a positive logicprogram, but not necessarily a Horn program, given thepossible disjunctions in the rule heads. Thus the reductP I may have several minimal models with respect toset-inclusion. However, the reduct of a normal logicprogram is a Horn program. Since the intersection oftwo models of a Horn program P is again a model ofP , it follows that a satisfiable Horn program P has aunique smallest model with respect to set-inclusion, theleast model of P .

Now, an interpretation I is called a stable model ofthe disjunctive logic program P if I happens to coincidewith a minimal model of P I (or the minimal model ofP I , if P is normal). We say that a formula ϕ is a braveconsequence of a disjunctive logic program P if thereis at least one stable model of P such that M satisfiesϕ, in symbols, M |= ϕ. We say that a formula ϕ is acautious consequence of P if M |= ϕ for every stablemodel M of P .

Example 4.1 ([60]). Consider the normal logicprogram P consisting of the rules

p ←− q ∧ ¬rr ←− q ∧ ss ←− ¬q ∧ rq ←− ¬uu ←− ¬q.

The interpretation I = {u} is a stable model for P ,since the reduct P I consisting of the rules

p ←− qr ←− q ∧ ss ←− ru ←−

is the least model of P I . The only further stable modelof P is the interpretation {p, q}.

4.2. Small and large stable models

The following parameterized problems were consideredby Truszczynski [103].

SmallStableModel

Instance: A normal logic program P and a non-negative integer k.

Parameter: k.

Question: Does P admit a stable model of size atmost k?

LargeStableModel

Instance: A normal logic program P and a non-negative integer k.

Parameter: k.

Question: Does P admit a stable model of size atleast |P | − k?

For solving SmallStableModel it suffices to checkall sets of atoms of P of size at most k. The numberof such subsets is bounded by O(nk) for n = |U(P )|,thus the problem is in XP. A more refined analysis[75] puts it into W[2]; a reduction from weighted CNFsatisfiability determines the problem as W[2]-complete.

Theorem 4.1 ([75, 103]). The problem Small-StableModel is W[2]-complete.

Next we consider the problem LargeStableModel.Consider a normal logic program P with m = |P |rules. Let k be a non-negative integer. We apply twopreprocessing steps: We obtain a logic program P1 fromP by removing from the bodies of rules of P all literals¬A for which A /∈ H(P ) holds. This removal does noteffect the set of stable models. Furthermore, we obtaina logic program P2 from P1 by removing all rules r fromP1 for which |B−(r)| > k holds. The following can beshown:

(i) An interpretation I containing at leastm−k atomsis a stable model of P1 if and only if it is a stablemodel of P2.

(ii) If a stable model I of P2 contains at least m − katoms, then |B−(P2)| ≤ k + k2.

Thus, if |B−(P2)| > k + k2 we can reject the instance.However, each stable model of P2 is determined bya subset of B−(P2). Consequently, if |B−(P2)| ≤k + k2, then the number of such subsets is boundedby O(2k+k2

). Hence we have the following result.

Theorem 4.2 ([103]). The problem LargeStable-Model is fixed-parameter tractable.

Lonc and Truszczynski [75] have determined theparameterized complexity of several model searchproblems for general logic programs, definite Hornprograms, and purely negative programs, distinguishing

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the search for models, supported models, and stablemodels (see Lonc and Truszczynski’s paper for therespective definitions).

4.3. Structural parameters for logic programs

Similarly as for satisfiability and constraint satisfactionabove, we define a generic parameterized problemfor stable models; here p1, . . . , pr denote computablemappings that assign to a logic program P a non-negative integer pi(P ).

SM(p1, . . . , pr)

Instance: A normal logic program P andnon-negative integers k1, . . . , kr with p1(P ) ≤k1, . . . , pr(P ) ≤ kr.

Parameters: k1, . . . , kr.

Question: Does P admit a stable model?

Gottlob, Scarcello, and Sideri [60] have considered(directed) graphs to model the structure of logicprograms. The dependency graph D(P ) is the directedgraph with vertex set U(P ) and two types of arcsdefined as follows. There is a positive arc from Ato B if there exists a rule r ∈ P with {A} = H(r)and B ∈ B+(r); there is a negative arc from A to Bif there exists a rule r ∈ P with {A} = H(r) andB ∈ B−(r). From D(P ) one obtains the undirecteddependency graph G(P ) by replacing each positive arcrunning from A to B by an edge between A and B,and by replacing each negative arc from A to B by aninduced path of length two from A to B, introducinga new vertex vAB . The parameter fvs(P ) is defined asthe size of a smallest feedback vertex set of G(P ).

Theorem 4.3 ([60]). The problem SM(fvs) is fixed-parameter tractable.

The proof of this result follows from the followingconsiderations. First, it is easy to see that if fvs(P ) ≤ k,then the undirected dependency graph G(P ) has afeedback vertex set S of size at most k containingonly atoms, since each vertex of the form vAB canbe replaced by one of its two neighbors. Given sucha feedback vertex set S ⊆ U(P ) one considers, oneafter the other, all assignments τ : S → {0, 1} and thecorresponding restriction P [τ ] of P to τ . The restrictionP [τ ] is defined similarly as for CNF formulas, seeSection 3. It can be shown that if I is a stable model ofP , then it is also a stable model of P [τ ]. Moreover, sincethe undirected dependency graph of P [τ ] is acyclic, P [τ ]is a “stratified” program and has a unique stable modelMτ which can be computed in linear time. We checkwhether Mτ is a stable model of P , and, if so, we addit to a cumulating set Σ. When we have consideredall assignments τ : S → {0, 1} we are left with the setΣ of all stable models of P . Hence SM(fvs) is fixed-parameter tractable. As we have actually computed all

stable models, we can now determine whether a givenatom belongs to some stable model (brave reasoning)and whether a given atom belongs to all stable models(cautious reasoning).

The fixed-parameter tractability of the SM problemhas also been explored under the treewidth parameter.This was actually done in the more general context ofdisjunctive logic programs.

Example 4.2 (adapted from [26]). Consider thefollowing disjunctive logic program P , describing thebehavior of a reviewer while reviewing a paper:

good ∨ bad ← papertired ← ¬angry ∧ ¬happy

happy ← goodangry ← badsmoke ← happysmoke ← angrypaper ←

Intuitively, here paper means that the reviewer currentlyreviews a paper, good means that the paper is good, bad,that it is bad, and so on. As one can easily verify, thereare only two stable models of P , which are:

M1 = {paper, good, happy, smoke} andM2 = {paper, bad, angry, smoke}.

Thus, for example, happy and angry are both braveconsequences of P , but none of the two is a cautiousconsequence of P . However happy∨ angry is a cautiousconsequence of P , and so is smoke.

Note that each propositional logic program is alsoa disjunctive logic program, and thus disjunctive logicprogramming is a true generalization of propositionallogic programming as introduced before. It immediatelyfollows that all fixed-parameter tractability results fordisjunctive logic programs also apply to normal logicprograms.

Let us denote the problem of checking whether adisjunctive logic program has a stable model by DSM,and the problem of checking whether for a givendisjunctive logic program P and propositional formulaϕ in CNF, whether ϕ is a brave (cautious) consequenceof P by BRAV (CAUT). Eiter and Gottlob [34]have determined the complexity of DSM, BRAV, andCAUT: Both DSM and BRAV are Σp

2-complete, andCAUT is Πp

2-complete. Moreover, BRAV remainsΣp

2-complete even if ϕ consists of a single atom andCAUT remains Πp

2-complete even if ϕ consists of asingle negated atom.

For a computable mapping p that assigns a non-negative integer to a disjunctive logic program, or, inthe case of BRAV and CAUT, to a pair consistingof a disjunctive logic program and a formula ϕ, we candefine in the obvious way the parameterized versionsDSM(p), BRAV(p), and CAUT(p) of DSM, BRAV,and CAUT, respectively. For example, BRAV(p) has

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as instance a disjunctive logic program P , a formulaϕ in CNF, and a non-negative integer k, such thatp(P,ϕ) ≤ k. The question is whether P admits a stablemodel satisfying ϕ.

Since each disjunctive logic program is syntacticallyin CNF, the parameters tw and tw∗ are well definedfor such programs, and thus for the problem DSM.For each instance (P,ϕ) of the problems BRAV andCAUT, we define tw∗(I) as tw∗(P ∧ϕ), and tw(I) astw(P ∧ ϕ), where P ∧ ϕ is simply the CNF formularesulting from conjoining the two CNF formulas Pand ϕ. The following positive result was recently shown:

Theorem 4.4 ([59]). The problems DSM(tw∗),BRAV(tw∗), and CAUT(tw∗) are all fixed-parametertractable and actually solvable in linear time for fixedparameter tw∗.

The proof uses Courcelle’s Theorem by formulatingthe problems DSM, BRAV, and CAUT as MSO1

sentences which are evaluated over structures ofbounded incidence treewidth encoding the instancesP , and (P,ϕ), respectively. The interesting aspect ofthese results (and proofs) is that the non-parameterizedversions of these problems are complete for classes at thesecond level of the Polynomial Hierarchy. Thus, whenkeeping the tw∗ parameter fixed, we jump down twolevels the Polynomial Hierarchy. In this sense, the FPTresults are somewhat more drastic than FPT resultsfor NP-complete problems. In fact, we also have aslightly more complex MSO1 formula which containsan alternation of second-order quantifiers. More FPTresults of this sort will be discussed in the next section.

Since, as already noted, bounded tw implies boundedtw∗, the fixed-parameter tractability of DSM(tw),BRAV(tw), and CAUT(tw) trivially follows fromTheorem 4.4.

It would be interesting to explore the problems DSM,SM, BRAV, and CAUT for yet other structuralparameters and to develop a backdoor set approach asdescribed in Sections 2 and 3 above for various baseclasses.

4.4. Closed World Reasoning

Gottlob, Pichler, and Wei [59] also study parameterizedversions of a number of other forms of non-monotonicreasoning problems, and show that they are fixed-parameter tractable w.r.t. the parameter treewidth. Inthis section we give a short overview of these results.

Closed world reasoning is an important techniqueused in AI and database theory. Its simplest form, theClosed World Assumption (CWA) [89] is based on theobservation that in many situations it is common useto express a data or knowledge base T just by listing“positive knowledge”, and assuming that what does notlogically follow from T is false.

Example 4.3. Consider the data relation STU-DENT that stores the students enrolled in a paint-ing class, could be STUDENT= {emma, john,leila, mary, zoe}. This relation could be logi-cally formalized as a conjunction T of logical atoms:T ≡ student(emma) ∧ student(john) ∧ student(leila) ∧student(mary)∧ student(zoe). In this example, we haveone atom for each data tuple. This formalization is,however, somewhat problematic given that we usuallyassume that all students enrolled in the course arelisted in the STUDENT relation. Thus, in the rela-tional database view, STUDENT is a model whose trueatoms are precisely those listed. For example, we wouldnormally infer from the knowledge present in the STU-DENT relation that Irene and Tom are not students ofthe painting class. However, first-order logic does notallow us to do so. In fact, neither ¬student(irene) nor¬student(tom) is a logical consequence of T .

The idea that led to the CWA was thus toadd the negation of all facts (atoms) that are notlogical consequences of T to T , getting a new theoryCWA(T ), and then take CWA(T ) as the intendedlogical semantics of the given data relations. Formallywe put

CWA(T ) = T ∪ {¬K : K is a positive literal suchthat T 6|= K }.

For Example 4.3 it now clearly holds that CWA(T ) |=¬student(irene) and CWA(T ) |= ¬student(tom)(assuming that irene and tom belong to some suitablydefined universe, e.g., of all students of the university).Obviously, reasoning under the CWA is non-monotonic.For example if we add the atom student(irene) to T ,then we can no longer infer ¬student(irene).

The CWA interpretation CWA(T ) of a logical theoryT works well as long as T does not contain disjunctiveinformation. It fails dramatically for theories as simpleas D ≡ a ∨ b, where a and b are propositional atoms.In fact, in this case, we have CWA(D) |= ¬a andCWA(D) |= ¬b and thus CWA(D) is inconsistent. Ingeneral, CWA(T ) is consistent if and only if T has aunique minimal model M , and, in this case, CWA(T )has precisely M as its unique model.

On input T , the complexity of reasoning under theCWA, i.e., of checking whether CWA(T ) is consistent,is currently an open problem. It is known thatthis problem is co-NP-hard and can be solved inpolynomial time with a logarithmic number of callsto an oracle in NP [32]. Several more sophisticatedclosed world reasoning mechanisms that redress theconsistency problem of CWA at least partially havebeen designed. Examples for such mechanisms areGeneralized CWA (GCWA) [80], Extended GCWA(EGCWA) [108], Careful CWA (CCWA) [50]. andExtended CWA (ECWA), also known as propositionalcircumscription [51, 79]. For definitions of thesemechanisms we refer to [32]. Note that, just like the

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CWA, all these reasoning methods are non-monotonic.For C ∈ {CWA, GCWA, EGCWA, CCWA, ECWA },deciding whether C(T ) |= ϕ holds is Πp

2-complete orharder [32].

For C ∈ {CWA, GCWA, EGCWA, CCWA, ECWA }we consider the following parameterized problems.

C-Deduction(tw∗)

Instance: Propositional CNF formulas T and ϕ,and a non-negative integer k with tw∗(T ∧ϕ) ≤ k.Parameter: k.

Question: Does C(T ) |= ϕ hold?

By means of Tseitin’s transformation (see Example 3.1)this problem generalizes in a natural way to instanceswhere T and ϕ are not in CNF.

Gottlob et al. [59] note that the deduction problemfor all these forms of non-monotonic reasoning canbe encoded in terms of MSO sentences over acombined structure encoding both the theory T andthe formula ϕ. We thus have the following result.

Theorem 4.5. For each of the CWA reasoningmechanisms C ∈ {CWA, GCWA, EGCWA, CCWA,ECWA }, the problem C-Deduction(tw∗) is fixed-parameter tractable and actually linear time solvable forconstant tw∗.

4.5. Propositional Abduction

Another form of non-monotonic reasoning is abduction.Abduction is a kind of “reverse reasoning” where oneseeks possible implicants of a manifestation or, in otherterms, for causes of a symptom.

In the following we will consider a simple form of plainpropositional abduction. For the abduction problemsbelow, an instance is given by a tuple P = (V,H,M, T )where V is a set of propositional variables, H is asubset of V (the “hypotheses”), M is a subset of V (the“manifestations”), and T is a consistent propositionalformula (the “theory”). A solution of P is a subset S ⊆H, such that T ∪S is consistent and T ∪S |= M . Givenan instance P, the basic problems of propositionalabduction are the following:• Solvability: Does there exist a solution of P?• Relevance: Given h ∈ H, is h contained in at

least one solution of P?• Necessity: Given h ∈ H, is h contained in every

solution of P?Often, a refined version of abduction is used, where

one is not interested in all solutions, but only insolutions that are minimal w.r.t. to some preorder � onthe powerset 2H . In this case, the problems Relevanceand Necessity are adapted accordingly, and we speakof �-Relevance (respectively, �-Necessity). Plainabduction as above corresponds to the special casewhere � is equality. The following preorders have alsobeen considered.

• Subset-minimality “⊆.”• Prioritization “⊆P ” for a fixed number p of

priorities: H is partitioned into “priorities”H1, . . . , Hp. Then A ⊆P B, if and only if A = Bor there exists a k s.t. A∩Hi = B∩Hi for all i < kand A ∩Hk ⊂ B ∩Hk.

• Minimum cardinality “≤”: A ≤ B if and only if|A| ≤ |B|.

• Penalization “vp” (also referred to as “weightedabduction”): To each element h ∈ H, a weightw(h) is attached. Then A vp B if and only if∑

h∈A w(h) ≤∑

h∈B w(h).

Example 4.4 ([33], inspired by [22]). Consider thefollowing theory T , set of hypotheses H, and set ofmanifestations M .

T = {¬(rich mixture ∧ lean mixture),rich mixture→ high fuel consumption,

lean mixture→ overheating,

low oil→ overheating,

low water→ overheating},H = { rich mixture, lean mixture,

low oil, low water },M = {high fuel consumption, overheating }.

Then, {rich mixture, low oil} and {rich mixture,low water} are solutions w.r.t. the preorders =, ⊆, and≤, but the set of hypotheses {rich mixture, low oil,low water} is a solution only for the preorder =. Notethat {rich mixture, lean mixture} is ruled out as asolution, since it is inconsistent with T .

It is easy to see that logic-based abduction is aform of non-monotonic reasoning. For instance, addingthe formula ¬low oil to the theory T of the aboveexample has the effect that {rich mixture, low oil} is nolonger a solution. Thus, from stronger theories we maysometimes abduce less.

In different contexts, different preorders may bethe most appropriate. A detailed discussion can befound in the paper of Eiter and Gottlob [33]. Thecomplexity of �-Relevance and �-Necessity for allthese relations (including =) was studied by Eiterand Gottlob [33] who show that these problems alllie on the second and third level of the PolynomialHierarchy. Gottlob et al. [59] consider parameterizedversions of the above propositional abduction problems.For instances (V,H,M, T ) where T is a propositionalformula in CNF, the parameter is the bound on thetreewidth of the incidence graph of T .

Theorem 4.6 ([59]). Each of the following prob-lems is fixed-parameter tractable: Solvability(tw∗),Relevance(tw∗), and Necessity(tw∗), as well as �-Relevance(tw∗) and �-Necessity(tw∗) with � ∈ {=,⊆,⊆P ,≤,vp}. Moreover, if � ∈ {=,⊆,⊆P }, then theproblems can be solved in linear time for constant tw∗.

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The proofs for preorders � ∈ {=,⊆,⊆P } useCourcelle’s Theorem. The theory T is encoded in afinite structure of the same incidence treewidth as T ,and the �-Relevance and �-Necessity problems areformulated as an MSO sentence with second-orderquantifier alternations. However, for the preorders ≤and vp, such a MSO-formalization is not possible. Infact, set cardinality comparisons are known not to beexpressible in MSO. Therefore the proof of Theorem 4.6for the preorders ≤ and vp used a result by Arnborg,Lagergreen, and Seese [6] stating that the evaluation offormulas that extend MSO with weight functions, sumsand minimum/maximum-functions is fixed-parametertractable (but not necessarily linear) with respect tothe parameter treewidth.

5. CORES AND DATA EXCHANGE

In this section we deal with the problem of computingcores of graphs, and, more generally, of relationalstructures. Roughly, a core of a relational structure A isthe smallest relational structure contained in A that ishomomorphically equivalent to A. This core is uniqueup to isomorphism. Cores were considered in graphtheory [67], and more recently in the context of dataexchange [37, 38, 52, 58], where methods are studiedfor transferring data from one database to anotherdatabase having a different structure.

If G = (V,E) is a graph, then an endomorphism ofG is a mapping h : V → V such that for each edgexy ∈ E, also h(x)h(y) ∈ E. A subgraph G′ of a graphG is called a core of G if G′ is a minimal endomorphicimage of G (see [67]) . For example, if a graph G is3-colorable and, in addition, contains a triangle, thenthis triangle is a core of G. Since all cores of a graphare isomorphic, we usually speak about the core of agraph. It is well known that computing cores of graphsis NP-hard [67].

The concept of core can be extended as follows tofinite relational instances with constants and variables,i.e., finite structures whose universe elements are of twosorts: Constants and variables. Let I be an instancesover the universe U = C∪V , where C are the constantsand V are the variables. Then an endomorphism is amapping h : U → U such that the following hold.

(i) h preserves constants, i.e., h(c) = c holds for allc ∈ C, and

(ii) for each relation R of the structure, andfor each tuple (x1, . . . , xk) ∈ R, we have(h(x1), . . . , h(xk)) ∈ R.

Relational instances with variables arise when dealingwith unknown data in form of marked null values. Suchmarked null values are to be considered existentiallyquantified. If we identify the tuples of a relationaldatabase with logical atoms, then the logical semanticsof a database D with variables is given by aprenex formula ϕD whose quantifier prefix existentially

quantifies over all variables of D and whose matrixconsists of the conjunction of all atoms of D.

Example 5.1. Suppose we know that the employeesJulia and Sue have the same phone number, but we donot know which one. We can use a specific variable (ormarked null), say x1, to represent this phone number ina relational instance, and write x1 both in the field ofSue’s phone and in the field of Julia’s phone. Considerthe database D = {phone(sue,x1), phone(bill,x2),phone(julia,x3), phone(bill,73859)} and the formulaϕD = (∃x1 ∃x2)(phone(sue, x1) ∧ phone(bill, x2) ∧phone(julia, x1) ∧ phone(bill, 73859)). The database Dis obviously redundant, as the tuple phone(bill,x2) canbe eliminated from D without changing the semantics.By eliminating this tuple we actually obtain the coreof D.

More generally, it is very easy to see and well known(cf. [38]) that the core of a finite relational instanceI is (up to isomorphism) the smallest sub-instance Jof I which is logically equivalent to I, i.e., such thatϕJ ≡ ϕI .

Let us consider the following two problems:

(i) GraphCoreIdentification (GCI): Given agraph G = (V,E), and a subgraph G′ of G, decidewhether G′ is the core of G.

(ii) InstanceCoreIdentification (ICI): Given aninstance I and a sub-instance J of I (i.e. aninstance whose relations consist of subsets of thoseof I), decide whether J is the core of I.

Note that ICI is at least as hard as GCI, since forevery instance G of GCI we can trivially obtain anequivalent instance of ICI by simply identifying the setV of vertices of G with a set of variables; we then getan instance of ICI whose set of constraints is empty.

Gottlob and Fermuller [53] have shown that theClause Condensation Problem, which is immediatelyseen to be equivalent to ICI, is DP-complete (seeSection 3.4 for more information on DP). More recently,Fagin, Kolaitis, and Popa [38] showed by a somewhatmore involved proof, that also GCI is DP-complete.

The Gaifman graph of an instance I of ICI has asvertices the variables of I and an edge between twovariables x and y if x and y appear in a same tupleof I. The blockwidth blw(I) of I is the size of thelargest connected component of the Gaifman graph of I.The parameterized problem ICI(blw) is defined in theobvious way.

Theorem 5.1 ([52]). The problem ICI(blw) isW[1]-hard.

This problem is most likely not in any class W[t] ofthe W-hierarchy because ICI is DP-hard and thus alsoco-NP-hard.

Data exchange aims at materializing in a targetdatabase data stemming from some source database.While data exchange has been recognized as an

18

important problem for several decades, systematicresearch on foundational and algorithmic issues of thisproblem has started only a few years ago with thefundamental work of Fagin, Kolaitis, Miller, and Popa[37]. The basic and most fundamental data exchangeproblem for relational databases, as defined by Fagin etal. [37], is as follows. Given a source database schema σ,a target database schema τ , a source database instanceS, and a set of constraints Σ, find a target databaseinstance T such that S and T satisfy all constraints inΣ, denoted by (S, T ) |= Σ. Fagin et al. [37, 38] restricttheir attention to the following types of constraints

(i) Tuple generating dependencies (TGDs) which arefirst-order implications of the form∀u (ϕ(u) =⇒ ∃v ψ(u, v)), where ϕ and ψ areconjunctions of atoms and u and v are lists ofvariables. If v is empty, then we speak about afull TGD.

(ii) Equation generating dependencies (EGDs)which are first-order implications of the form∀u (ϕ(u) =⇒ v = w), where ϕ is a conjunctionof atoms, u is a list of variables, and v and w aresingle variables from the list u.

Source-to-target constraints are those where (i) thepremise ϕ(u) contains atoms whose predicate symbolsare relation names of the source signature σ only, andwhere (ii) the conclusion ψ(u, v) is made of atomswhose predicate symbols are relation names of thetarget signature τ only. All atoms occurring in targetconstraints refer to the target signature only.

In particular, Fagin et al. [37, 38] consider the casewhere the set Σ = Σst ∪ Σt of constraints consists ofsource-to-target constraints Σst encoding conditions onthe mapping between source and target data, and thetarget constraints Σt which express data dependencieson the target database.

This setting allows us to formulate a large class ofconstraints.

Example 5.2. Let a source database contain arelation

student(STUDNAME,BIRTHDATE,SSN,ZIPCODE)

and let the target database contain two relations

person(NAME,BORN,SSN,ZIPCODE,PHONE),zc(ZIPCODE,STATE),

then a typical source-to-target constraint would be

st1 : (∀u1, u2, u3, u4) (student(u1, u2, u3, u4)=⇒ (∃v) person(u1, u2, u3, u4, v)),

while the target constraints could be

t1 : (∀u1, u2, u3, u4, u5) (person(u1, u2, u3, u4, u5)=⇒ (∃v) zc(u5, v))

t2 : (∀u, v) (zc(u, v) ∧ zc(u,w) =⇒ v = w).

Here st1 and t1 are inclusion dependencies, and t2 isa functional dependency. Note that t1 and t2 togetherexpress a “foreign key” constraint.

It is easy to see that, in addition to functional,inclusion dependencies, and foreign key constraints, alsomultivalued dependencies and even join dependenciescan be expressed by formulas in this setting. Thus,this setting is very general and encompasses all majordependencies used in database design and for databasemaintenance.

The problem is to check whether a target instance Texists, and if so, to compute a good one. Vari-ables (or, equivalently, labeled null values) are allowedto appear in target instances. In Example 5.2,a tuple (doe, 19880203, 1234567, 94305) of the stu-dent source relation could be translated into a tuple(doe, 19880203, 1234567, 94305, x1) of the person rela-tion, where x1 is a variable representing a null value.For further, more detailed examples, we refer to thepaper of Fagin et al. [37].

A universal solution of a data exchange problem is atarget instance T which is more general than all othersolutions, i.e., such that for each other solution T ′ thereexists a homomorphism T → T ′. Fagin, Kolaitis, Miller,and Popa [37] have shown that universal solutions ofdata exchange problems can be obtained via the well-known chase procedure [9, 3, 78]. One first chasesthe set Σst of source-to-target TGDs over the sourceinstance S and obtains in polynomial time an initialtarget instance T = SΣst . Then one chases the targetconstraints Σt over T . If this chase terminates, oneobtains a finite universal solution. In order to guaranteetermination, Fagin et al. [37, 38] restrict themselves totarget TGDs which are weakly acyclic. Weak acyclicityis a syntactic condition on TGDs ensuring that thereare no cyclic dependencies among argument positionsinvolving existential constraints (see [38] for a precisedefinition). Weak acyclicity has been so far the mostgeneral known sufficient condition for termination of thechase. This concept was developed by Deutsch [28] andPopa [37]. Fagin and his colleagues [37] have shown thatuniversal solutions are very useful for query answering.In particular, any universal solution can be used toobtain the certain answers tuples to a conjunctive queryover the target schema, i.e., those answer tuples that arecontained in all solutions of the data exchange problem.

If a data exchange problem P is solvable, then it canhave several universal solutions, and these solutions cannoticeably differ in size. However, as observed in [38],the cores of all universal solutions to a data exchangeproblem are all mutually isomorphic, and thus there is,up to isomorphism, one single smallest solution, whichis, up to isomorphism, equal to the core of each universalsolution. To compute this smallest solution, one canfirst compute an arbitrary universal solution J and thencompute the core of J . Fagin, Kolaitis, and Popa [38]strongly argue that this should be the solution of choice.

Fagin et al. [38] formulated the following problem:Given a (solvable) data exchange problem whosesource-to-target constraints are TGDs and whosetarget constraints consist of weakly-acyclic TGDs and

19

arbitrary EGDs, can the core of a universal solution becomputed in polynomial time? Here the source schemaσ, the target schema τ and the set of constraints Σ areall considered to be fixed, and the problem instance thusconsists of a source instance S.

Unfortunately, even though (in case of solvabledata exchange problems) a universal solution canbe computed in polynomial time via the chaseprocedure, the solutions can be of unbounded blockwidth. In fact, even very simple TGDs such as(∀x, y) ((P (x) ∧ P (y))→ P (x, y)) can “lump” variablesfrom different components (blocks) together and thusmerge several small components into very largecomponents of connected variables.

Thus, at a first glance, Theorem 5.1 may suggest anegative answer to the above problem raised by Fagin,Kolaitis, and Popa. However, it turned out that it hasa positive answer.

Theorem 5.2 ([58]). The core of a universalsolution of a solvable data exchange problem whosesource-to-target constraints are TGDs and whosetarget constraints consist of weakly-acyclic TGDs andarbitrary EGDs can be computed in polynomial time.

The proposed solution to the problem is technicallyrather involved. The reader is referred to the conferencepaper [58] for further details.

Acknowledgment

We thank Mike Fellows for valuable comments on apreliminary version of the paper and Marko Samerfor careful proof reading. Georg Gottlob’s work wassupported by a Royal Society Wolfson Research MeritAward. Stefan Szeider’s work was supported bythe EPSRC, Project EP/E001394/1 Fixed-parameteralgorithms and satisfiability.

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