a rendezvous of complexity, logic, algebra, and ......graph-theoretic problems spatial and temporal...

A Rendezvous of Complexity, Logic, Algebra, andCombinatorics

Hubie Chen

Universitat Pompeu FabraBarcelona, Spain

BCAM - September 2011

Hubie Chen A Rendezvous

3-Colorability

Given an undirected graph...

...is it 3-colorable?

Idea: divide a set of objects into three groups so that no twoincompatible objects are placed into same group


Boolean satisfiability

Given a conjunction of propositional clauses...

(¬a ∨ ¬b ∨ c) ∧ (¬c ∨ d) ∧ (a ∨ ¬d)

...is there a satisfying assignment to the variables?

a → 1b → 0c → 0d → 1


Digraph acyclicity

Given a directed graph, does it contain a cycle?


Efficient algorithms

A broad goal of computer science:

Develop fast, usable algorithms for computational problems

A broad goal of theoretical computer science:

Classify problems according to whether or notthey have “efficient” algorithms

That is, classify problems according to their inherentcomplexity...


Classifying problems

Theoretical computer science: place problems into complexityclasses...

P: polynomial time

Theoretical notion of efficiency/tractabilityProblems for which there’s an algorithm running in time O(nk)Ex: digraph acyclicity

NP: non-deterministic polynomial time

P ⊆ NP, believed that NP is “much bigger”Idea: problems for which a solution can be efficiently verified

Exs: all three problems seen

NP-hard: problems “as hard as”/that “can simulate”all problems in NP

Exs: 3-colorability, boolean sat.

Generally believed that

P = NP ⇔ no NP-hard problem is in P


A dream

(Garey & Johnson ’79): Compendium of 100s of NP-hardproblems

Dream: to have an encyclopedia – a general classificationtheorem that, given a problem, would tell us if tractable ornot (if in P or if NP-hard)

This talk: discuss (my) research towards this dream...


Classification

Classification projects

Identify a broad family of problemsTry to classify each problem as tractable/intractableOften, obtain generic conditions for tractability/intractability


Exhibit #1: Constraint Satisfaction


Examples

Given an undirected graph, is it 3-colorable?

Given a conjunction of propositional clauses, is there asatisfying assignment?

(¬a ∨ ¬b ∨ c) ∧ (¬c) ∧ (a)

Given a directed graph, does it contain a cycle?


Animals

Are these problems inherently different animals?

...or can we place them into an organizational scheme?


Constraint Satisfaction

All of these problems are constraint satisfaction problems!

The constraint satisfaction problem (CSP)

Given a primitive positive (pp) sentence

φ = ∃v1 . . . ∃vn (R(vi1 , . . . , vik ) ∧ . . .)

and a relational structure B, decide if B |= φ.

Idea: decide if there is an assignment to variables satisfying acollection of constraints

Examples:

graph coloring (vi = vertices, B = colors,one constraint E (vi , vj) for each edge)boolean satisfiability (vi = variables, B = 0, 1,one constraint for each clause)



For each structure B, we can define:

CSP(B)

Given a primitive positive (pp) sentence φ, decide if B |= φ.

Can capture many studied problems:

Graph 3-Coloring (B = K3), generalizationsBoolean sat. problems: 3-SAT, 2-SAT, Horn-SATDigraph acyclicity (B = (Q;<))Temporal Reasoning (e.g. B = (Q;≤, <, =))Systems of equations over various algebraic structures



Unified framework, captures many problems

Numerous application areas:

Assignment/scheduling problemsBoolean sat. problems & generalizationsGraph-theoretic problemsSpatial and temporal reasoningInterval reasoningAlgebraic equations problems

Research question:

Which problems CSP(B) are tractable (in poly-time), and why?

Known:

(Schaefer ’78): Classification for 2-element structures(Bulatov ’02): Classification for 3-element structures


Constraint Satisfaction and Consistency

Consistency methods: simple, efficient, well-studied,used-in-practice inference methods that (in general) candetect unsatisfiability

What are the scope of various consistency methods?Which problems CSP(B) can they solve?

My work: study ofarc consistency and extensions(Bodirsky & Chen ’10), (Chen, Dalmau & Grussien ’11)local-to-global consistency(Bodirsky & Chen ’09), (Bodirsky, Chen & Wrona, ongoing)


An Algebraic Approach

Algebraic approach: associate to each struct. B an algebra AB

An operation f : Bm → B is a polymorphism of a relationU ⊆ Bk if for any choice of m tuples from U, applying fcoordinatewise yields a tuple (s1, . . . , sk) also in U.

(t11, t12, . . . t1k) ∈ U...

.... . .

...(tm1, tm2, . . . tmk) ∈ U

f ↓ f ↓ . . . f ↓

(s1, s2, . . . sk) ∈ U

An operation f is a polymorphism of a structure if it is apolymorphism of all relations (of the structure)


An Algebraic Approach

For a struct. B, define the algebra of polymorphisms AB to bethe algebra

with universe B , andwhose operations are all polymorphisms of B

Key Theorem (Geiger/Bodcharnuk et al. ’60s, Jeavons et al. ’90s):

If finite structs. B,B have AB = AB , thenB,B can pp-define (∃,∧) the same relations, andCSP(B), CSP(B) have same complexity(up to polytime reduction)

Use properties of algebra AB to understand struct. B


Exhibit #2: Quantified Constraint Satisfaction


Quantified Constraint Satisfaction

The quantified constraint satisfaction problem (QCSP)

Given a conjunctive positive (cp) sentence

φ = Q1v1 . . .Qnvn (R(vi1 , . . . , vik ) ∧ . . .)

(with Qi ∈ ∀, ∃), and a relational structure B, decide if B |= φ.

Can model situations with uncertainty, two-party interaction,... that cannot be modelled with CSP(complexity is higher: generally PSPACE-complete)

Wide range of applications: verification, game playing,adversarial reasoning, planning, ...


Quantified Constraint Satisfaction

QCSP(B)

Given a conjunctive positive sentence φ, decide if B |= φ.

Framework goes back to:

(Aspvall, Plass, Tarjan ’79) - Quantified 2-SAT

(Karpinski, Kleine Buning, Schmitt ’87) - Quantified HornSAT

Research issues:

Search for generic sources of tractability / hardness

Develop algebraic approach

Attempt to establish systematic classifications


Polynomially Generated Powers (PGP)

(Chen ’08, ’11): study of polynomially generated powers(PGP) property – algebraic property

A combinatorial property of algebras concerning the size ofgenerating sets for An

(Wiegold ’70s-’80s), (Quick & Ruskuc ’10), (Awang, Garrido,McLeman, Quick & Ruskuc), ...



Def: An algebra A has polynomially generated powers if thereis a polynomial p ∈ N[X ] such that, for each n ≥ 1,the algebra An has a generating set of size ≤ p(n)

Example: let G be a finite group with identity elt. eConsider tuples where all entries [with one possible exception]are = e. In G 4:

(g1, e, e, e)(e, g2, e, e)(e, e, g3, e)(e, e, e, g4)↓ ↓ ↓ ↓

(g1, g2, g3, g4)

Have generating set for Gn of size ≤ |G |n: linear growth



(Chen ’11): Gave broad conditions for reducing QCSP to CSPbased on PGP

Observation: Suppose that algebra AB has the PGP, and agenerating set for An

B can be computed in poly time (in n)

Then, Π2-QCSPc(B) reduces to CSPc(B):

Consider an instance Φ = ∀y1 . . . ∀yn∃x1 . . . ∃xmψCompute a generating set t1, . . . , tk for An

BLet Φ[ti ] be the CSP instance where the vars. yj arereplaced with the values in tiΦ is equivalent to the CSP instance Φ[t1] ∧ · · · ∧ Φ[tk ]



Thm (Chen ’11): Let AB be a 3-elt algebra[that’s idempotent and where CSP(B) tractable].

Either:

AB is switchable, has the PGP, and QCSP(B) tractable; orAB has exponentially generated powers (EGP):generating sets require size Ω(cn), with c > 1

Dichotomy theorem: all studied algebras have PGP or EGP

Reduces the classification of tractable 3-elt QCSP(B) to aparticular, robust class of algebras

Open questions:

Full QCSP classification for 3-elt structuresClassification of PGP/EGP on all finite algebras


Exhibit #3: Constraint Satisfaction,from the other side


Constraint Satisfaction, the other side

Recall...

The Constraint Satisfaction Problem (CSP)

Given a primitive positive sentence φ = ∃v1 . . . ∃vn ∧ αj anda structure B, decide if B |= φ.

Before, we restricted the structure. Now we restrict the formulas.

Remark: for a fixed sentence φ, the CSP on φ is in poly time

Question: on which sets of formulas is CSP tractable?What structural restrictions can we place on formulas, toensure tractability?


Graph-based approach

Graph-based approach: given a CSP instance, look at graph wherevertices are variables, two variables linked if in a commonconstraint

Example:∃a∃b∃c∃d∃e∃f ∃g (R(a, b, c) ∧ S(b, c , d) ∧ T (d , e, f )

∧U(a, g) ∧W (f , g))

↓

a

b

c

d

e

f

g


Treewidth

Thm (Freuder ’90):When the graphs have bounded treewidth,the CSP is tractable.

Tree, tw=1 Cycle, tw=2 k-Clique,tw=(k-1)

Thm (Grohe ’07): This theorem is optimal–unbounded treewidth implies CSP intractability.

In fact, bounded treewidth essentially determines tractabilityon formulas of bounded arity.


Hypergraph-based approach

Hypergraph-based approach: given a CSP instance, look athypergraph where vertices are variables, have a hyperedge for eachconstraint

Example:

∃a∃b∃c∃d∃e∃f ∃g (R(a, b, c) ∧ S(b, c , d) ∧ T (d , e, f )∧U(a, g) ∧W (f , g))

↓

a

b

c

d

e

f

g


Hypergraph-based approach

Basic question: classify sets of hypergraphs as tractable /intractable

(Gottlob et al., ’90s): Introduction of hypergraph complexitymeasures – hypertree width, generalized hypertree width

Showed that CSP is tractable when hypertree width is bounded

Thm (Chen & Dalmau ’05): CSPs are tractable when thegeneralized hypertree width is bounded.(Resolved primary open question of Gottlob et al. work)

Still open: which hypergraphs can be solvable in poly time?


Exhibit #4: Quantified Constraint Satisfaction,from the other side


Quantified Constraint Satisfaction, the other side

Recall...

The Quantified Constraint Satisfaction Problem (QCSP)

Given a conjunctive positive sentence φ = Q1v1 . . .Qnvn ∧ αj anda structure B, decide if B |= φ.

Before, we restricted the structure. Now we restrict the formulas.

Question: on which sets of formulas is QCSP tractable?What structural restrictions can we place on formulas, toensure tractability?


Graph-based approach

Graph-based approach: given a QCSP instance, look atprefixed graph containing quantifier prefix & graph

Example:∀a∃b∃c∀d∃e∀f ∃g (R(a, b, c) ∧ S(b, c , d) ∧ T (d , e, f )

∧U(a, g) ∧W (f , g))

↓

(∀a∃b∃c∀d∃e∀f ∃g ,

a

b

c

d

e

f

g)


Prefixed Graphs

Prefixed graph perspective:now have graph PLUS quantifier prefix

Examples:

X1

X2

X3

Y

∃X1∃X2∃X3 . . . ∃Xn∀YTractable!

Y1

Y2

Y3

X

∀Y 1∀Y 2∀Y 3 . . . ∃Yn∃XIntractable!


Prefixed Graphs

(Chen & Dalmau ’05): Notion of treewidth for prefixedgraphs, proof that bounded treewidth ⇒ tractability

Pulina and Tacchella (’10) study this treewidth measure,showing that an approximation

“is a marker of empirical hardness”, and “it is theonly parameter that succeeds consistently in beingso among several other syntactic parameters whichare plausible candidates”

Close connection between

treewidth notions ↔ variable elimination algorithms(used in practice)


Prefixed Graphs

(Chen & Dalmau, recent):

New & refined notion of width for prefixed graphs(assigns a natural number w ∈ N to each prefixed graph)

Generalization of treewidth

Complete classification of all prefixed graphs:A set of prefixed graphs is tractable iff it has bounded width

Open: Classification for formulas of bounded arity.

Perhaps the following will help:

Theorem (Chen, Madelaine & Martin ’08):Entailment and equivalence are decidable (computable) on QCSP(∀, ∃,∧) sentences.

(New decidability result for entailment!Optimal in that entailment undecidable in positive (∀, ∃,∧,∨).)


Closing


Themes

Study the computational complexity of fundamentalcomputational problems

Dream: to have an encyclopedic “master” classificationtheoremLook at broad families of problems,attempt to obtain comprehensive classification theoremsOften obtain general conditions for tractability/intractability

Use ideas from / interface with many areas: algorithms,constraint satisfaction, universal algebra, graph theory,combinatorics, logic ...


a rendezvous of complexity, logic, algebra, and ......graph-theoretic problems spatial and temporal...

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