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!

BIRS 2014 Stefan Szeider

The Gap Theory n<250 vs Practice n>1.000.000

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BIRS 2014 Stefan Szeider

Structure Matters!

real-world instance (SW verification)

random instance

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BIRS 2014 Stefan Szeider

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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BIRS 2014 Stefan Szeider

• Introduction

• Parameterized Complexity

• How to Parameterize?

• Parameterized Proof Complexity

• Complexity Barrier Breaking Red’s

• Summary

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BIRS 2014 Stefan Szeider

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

BIRS 2014 Stefan Szeider

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

BIRS 2014 Stefan Szeider

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

BIRS 2014 Stefan Szeider

W-Hierarchy

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W[1]

FPT

PTIME

W[2]

W[3]

... XP

BIRS 2014 Stefan Szeider

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

BIRS 2014 Stefan Szeider

• Introduction

• Parameterized Complexity

• How to Parameterize?

• Parameterized Proof Complexity

• Complexity Barrier Breaking Red’s

• Summary

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BIRS 2014 Stefan Szeider

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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BIRS 2014 Stefan Szeider

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]

BIRS 2014 Stefan Szeider

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

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(#)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

BIRS 2014 Stefan Szeider

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]

BIRS 2014 Stefan Szeider

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

BIRS 2014 Stefan Szeider

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).

BIRS 2014 Stefan Szeider

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]

BIRS 2014 Stefan Szeider

• Introduction

• Parameterized Complexity

• How to Parameterize?

• Parameterized Proof Complexity

• Complexity Barrier Breaking Red’s

• Summary

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BIRS 2014 Stefan Szeider

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

• Parameterized Complexity

• How to Parameterize?

• Parameterized Proof Complexity

• Complexity Barrier Breaking Red’s

• Summary

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BIRS 2014 Stefan Szeider

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

BIRS 2014 Stefan Szeider

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

BIRS 2014 Stefan Szeider

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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BIRS 2014 Stefan Szeider

• 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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BIRS 2014 Stefan Szeider

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

BIRS 2014 Stefan Szeider

• Introduction

• Parameterized Complexity

• How to Parameterize?

• Parameterized Proof Complexity

• Problems Beyond NP

• Summary

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BIRS 2014 Stefan Szeider

Summary• Theory and Practice Gap

• One-dimensional framework not adequate

• use parameters to capture structure

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

• Parameterized Proof Complexity

• fpt-reductions to SAT

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