parameterized complexity sat · the gentle revolution of parameterized complexity •...

Parameterized Complexity &

SAT

Stefan Szeider Vienna University of Technology, Austria

Theoretical Foundations of Applied SAT Solving BIRS, Banff, 2014

BIRS 2014 Stefan Szeider

3SAT time bounds

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2n trivial

1.333n 1999 (Schöning)

1.3302n 2002

1.3290n 2003

1.3280n 2003

1.324n 2010 (Hertli)

For n=250 that exceeds the number of nano seconds that passed since the big bang!


The Gap Theory n<250 vs Practice n>1.000.000

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Structure Matters!

real-world instance (SW verification)

random instance

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Aims of Talk

• New theoretical framework: Parameterized Complexity

• Gentle (and subjective) introduction

• What questions can be asked

• Survey of some recent results

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• Introduction

• Parameterized Complexity

• How to Parameterize?

• Parameterized Proof Complexity

• Complexity Barrier Breaking Red’s

• Summary

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Two-Dimensional Theoretical Model?

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• Parameter can represent various structural aspects of input

Q: Suitable mathematical correlation between n and k?

size n

parameter k

Inputs

“Str

uctu

redn

ess”

Input Size


Fixed-Parameter Tractability [Downey, Fellows 1999]

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• running time: f(k)·nc

• “Battle of function classes”

vs f(k)·nc nf(k)

FPT XP

• hardness theoryFPT ⊆W[1] ⊆W[2] ⊆ … XP


Famous Examples from Combinatorial Optimization

Given a graph G,

1. find a vertex cover of size k

2. find an independent set of size k

3. find a dominating set of size k

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• All three problems are NP-complete

• All three problems can be solved in XP-time (nk)

• The problems are of different practical hardness

• The problems are of different parameterized complexity

vertex cover

independent set

dominating set

FPT

W[1]-c

W[2]-c


W-Hierarchy

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W[1]

FPT

PTIME

W[2]

W[3]

... XP


Canonical Problems for W-Hierarchy

• Instance: Boolean circuit C, integer k

• Parameter: k

• Question: Is C satisfied by an assignment of weight k?

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W[1] C is 3CNF

W[2] C is CNF

W[3] C is of weft 2

W[4] C is of weft 3

fpt-transformations: (x,k) → (x’,k’) with k’≤ f(k)

Methods and Tools

W[i]-hardnessHardness T

ools

kernel lower bounds

Exponential Time Hypothesis

Algori

thmic T

ools

kernelization

bounded search trees

logic meta-theorems

color coding

iterative compression

graph minors

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The Gentle Revolution of Parameterized Complexity

• Bioinformatics, Operations Research, Optimization, Automated Reasoning, etc.

• 4 Monographs

-[Downey&Fellows 1999, 2014]

-[Flum&Grohe 2006]

-[Niedermeier 2006]

• ERC grants (Sz., Fomin, Marx, Saurabh)

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Papers containing “parameterized complexity” or “fixed-parameter tractable” published per year. Source: Google Scholar

!!!Papers containing “parameterized complexity” or “fixed-parameter tractable” published per year. Source: Google Scholar

0

225

450

675

900

2000 2012


• Introduction





• Summary

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ParametersA. Solution size of an

optimization problem

B. Measure hidden structure(e.g., tree likeness, density, etc)

C.Explicit parameters(e.g., clause size, variable degree, domain size)

D. Combination of parameters (e.g., clause size+tree likeness)

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How to parameterize (|) Decompositions

• Idea: decompose the problem into subproblems, and combine solutions to subproblems to a global solution

• Parameter: overlap between subproblems

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Width Measures

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treewidth and branchwidth [Robertson&Seymour 1986]

clique-width [Courcelle, Engelfriet, Rozenberg 1990], can also be defined for directed graphs

rank-width [Oum & Seymour 2006]

hypertree width [Gottlob, Leone, Scarcello 2002]


Graphical Models

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F=(x ⋁¬y)⋀(y ⋁¬z⋁¬w)⋀(¬x⋁w)

wyx z

(x ⋁¬y) (y ⋁¬z⋁¬w) (¬x⋁w)

(directed) incidence graph

w

yx

z

primal graphw

yx

zhypergraph

BIRS 2014 Stefan Szeider��19

(#)SAT

incidence tw

dir incidence cw

primal tw

Courcelle, Makowsky, Rotics 2001

undir incidence cw

hypertree-widthOrdyniak, Paulusma, Sz. 2010, Slivovsky, Sz. 2013

Freuder 1985 Alekhnovich&Razborov 2002 Bacchus, Dalamao, Pitassi 2003

Fischer, Makowsky, Ravve 2008 Samer, Szeider 2010

NP-h

XP

FPT

d

Horn

2CNF

disguised Horn

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How to parameterize (II) Distance to an Island of Tractability

Tree-like

• What, if instance is outside an island?

• Parameter: Distance to island

• Backdoor: small set of variables such that instantiating the variables makes the instance tractable


Backdoors into a tractable class C

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Given a CNF formula F and a set X of k variables. Let t1, ...t2^k be the partial truth assignments on X.

F

...F[t1] F[t2^k]F[t2]

X is a weak backdoor if some F[ti] belongs to C and is satisfiable

F

...F[t1] F[t2^k]F[t2]X is a strong backdoor if all the F[ti] belong to C

[Williams, Gomes, Selman 2003]

[variant: deletion bd]


BD Detection

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Survey Article, [Gaspers, Sz ’12]

base class weak bd strong bd del bd

Schaefer W[2]-h (FPT) FPT FPT

subsolver W[P]-c W[P]-c n/a

Acycl W[2]-h (FPT) open (open) FPT

ren Horn W[2]-h W[2]-h (open) FPT

Clustering W[2]-h (FPT) W[2]-h (FPT) FPT

Green: if complexity is different for 3CNF inputs


Decomps vs Backdoorsin general incomparable parameters:

• ∃ Horn formulas with arbitrary high treewidth

• take many disjoint copies of formula

Idea: combine by taking TW[t] (class of formulas with incidence treewidth ≤ t) as base class, t is constant.

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Strong backdoors into TW[t] can be found in FPT time. [Gaspers, Sz. FOCS’13]

size of returned backdoor is ≤2k where k=size of smallest backdoor

BIRS 2014 Stefan SzeiderSAT Interactions 2012 Stefan Szeider��24

assignment: 010100110101101 (k=3 variables flipped)

k-Opt Max SAT: flip k vars to increase # of sat clauses

assignment: 010110100111101

How to parameterize (IV) Parameterized Local Search

Theorem: [Sz. SAT’09] k-OPT MAX-SAT is

• W[1]-hard in general

• fixed-parameter tractable for 3-CNFs with bounded number of occurrences of variables (both restrictions are necessary).


Further results on parm’d heuristics

• Several papers on k-step local search (TSP, Stable Matchings, Min-Ones-CSP, vertex cover…)

• List Coloring parameterised by “conservation” [Hartung&Niedermeier 2013]

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BIRS 2014 Stefan SzeiderSAT Interactions 2012 Stefan Szeider

how to parameterize (4)

Max-SAT above a Guaranteed Value

• MAX-SAT: One can always satisfy half of the clauses

• Hence m/2 is a guaranteed value

• Can we satisfy m/2+k clauses?

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} k

m/2

BIRS 2014 Stefan SzeiderSAT Interactions 2012 Stefan Szeider

Results

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MAX-SAT parameterized above m/2 is fixed-

parameter tractable. [Mahajan, Raman 1999]

For MAX-r-SAT there is a better guaranteed value: (1-2-r)m [by Johnson’s Algorithm 1973]

MAX-r-SAT parameterized above (1-2-r)m is fixed-parameter tractable. [Alon,Gutin,Kim,Sz.,Yeo SODA’10]


• Introduction





• Summary

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Param’d Proof Complexity

• refutations of FPT-size?

• parameterized contradictions “F has no satisfying assignment of weight ≤k’’)

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vs f(k)·nc nf(k)

FPT XP

[Dantchev, Martin, Sz.; FOCS’07] gap-theorem for tree-like Res [Beyersdorff, Galesi, Lauria; SAT’11] tree-like Res. of k-Clique [Beyersdorff, Galesi, Lauria, Razborov; ICALP’11] bounded depth Frege of PH

[Dantchev, Martin; CSR ’13] Res(1) vs Res(2) separation

• Introduction





• Summary

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Above NP

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DNF-Minimization

Shortest Implicant-Core

Schaefer-Umans 2002

Skeptical ASP-reasoning

… etc …

Abductive Reasoning

Eiter-Gottlob 199x

… etc …

Idea: Use Fixed-Parameter Tractability not to solve

the problem, but to reduce it to SAT!

SAT


Answer Set ProgrammingInstance: set P of rules of the form

!

Answer set: set of atoms that is a minimal model of the “GL-reduct” of P

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Notions

Answer Set Programs

• (Disjunctive) program P set of rules of the formr : x1 _ . . . _ xl| {z }

H(r)

y1, . . . , yn| {z }B+(r)

,¬ z1, . . . ,¬zm| {z }B�(r)

• Normal: class of all programs where H(r) 1• Tight: class of all programs where a certain graph

representation of the program is acyclic

ASP Problems

• Consistency: Does P have an answer set?• Brave (Cautious) Reasoning : Given P and an atom a⇤ of P.

Does a⇤ belong to some (all) answer set of P?

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• checking whether x is in all answer sets is Πp2-complete (Skeptical ASP-reasoning)

• If P is normal, then check drops to co-NP-c


Backdoors for ASP [Fichte, Sz ’11,’12]

• take Normal (class of normal programs) as base class

• ASP-reasoning parameterised by size of backdoor to Normal can be fpt-reduced to SAT

• hence we have an fpt-reduction that breaks the complexity barrier between 1st and 2nd levels of the Polynomial Hierarchy!

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• Similar approach works for Abductive reasoning [Pfandler, Rümmele, Sz. IJCAI’13]

• Some problems appear to disallow an fpt-reduction to SAT…

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Hardness Theory [De Haan, Sz ’14]

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∀k∃*

• Instance: Quantified Boolean circuit ∀X∃Y C(X,Y), integer k

• Parameter: k

• Question: Is C true where assignment to X has weight k?

∀*∃k

• Instance: Quantified Boolean circuit ∀X∃Y C(X,Y), integer k

• Parameter: k

• Question: Is C true where assignment to Y has weight k?

TM-characterizations, FO-MC characterization Classified several problems (Schaefer-Umans 2002, Eiter-Gottlob 199x) for natural parameters Compendium: http://arxiv.org/abs/1312.1672

http://arxiv.org/abs/1312.1672


• Introduction




• Problems Beyond NP

• Summary

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Summary• Theory and Practice Gap

• One-dimensional framework not adequate

• use parameters to capture structure

• Different parameters: backdoors, decompositions, locality, above guarantee


• fpt-reductions to SAT

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parameterized complexity sat · the gentle revolution of parameterized complexity •...

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