The Connectivity of Boolean Satisfiability: The Connectivity of Boolean Satisfiability: Structural and Computational DichotomiesStructural and Computational Dichotomies

Elitza Maneva (UC Berkeley)Elitza Maneva (UC Berkeley)

Joint work with Joint work with Parikshit Gopalan, Phokion Kolaitis and Christos PapadimitriouParikshit Gopalan, Phokion Kolaitis and Christos Papadimitriou

Features of our dichotomyFeatures of our dichotomy

• Refers to the structure of the entire space of solutions

• The dichotomy cuts across Boolean clones

• Motivated by recent heuristics for random input CSP.

Space of solutionsSpace of solutions1111111111

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n-dimensional hypercube

Space of solutionsSpace of solutions1111111111

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Space of solutionsSpace of solutions1111111111

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Space of solutionsSpace of solutions

Connectivity of graph of solutions?Connectivity of graph of solutions?

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Our dichotomyOur dichotomy • Computational problems

– CONN: Is the solution graph connected?– st-CONN: Are two solutions connected?

• Structural property– Possible diameter of components

PSPACE-complete

PSPACE-complete

exponential

NP-complete

CONN

st-CONN

diameter

SAT

Tight Tight CSPCSP Non-tight Non-tight CSP CSP

in co-NP

in P

linear

P and NP-complete

Motivation for our studyMotivation for our studyHeuristics for random CSP are influenced by the structure

of the solution space

Random 3-SAT with parameter :n variables, n clauses are chosen at random

4.154.15 4.274.2700

Easy Hard Unsat

Motivation for our studyMotivation for our studyHeuristics for random CSP are influenced by the structure of

the solution space

Survey propagation algorithm [Mezard, Parisi, Zecchina ‘02]• designed to work for clustered random problems• very successful for such random instances• based on statistical physics analysis

Clustering in random CSPClustering in random CSPWhat is known?What is known?

2-SAT: a single cluster up to the satisfiability threshold 2-SAT: a single cluster up to the satisfiability threshold

3-SAT to 7-SAT: not known, but conjectured to have 3-SAT to 7-SAT: not known, but conjectured to have clusters before the satisfiability thresholdclusters before the satisfiability threshold

8-SAT and above: exponential number of clusters8-SAT and above: exponential number of clusters[Achlioptas, Ricci-Tersenghi `06][Achlioptas, Ricci-Tersenghi `06][Mezard, Mora, Zecchina `05] [Mezard, Mora, Zecchina `05]

Our dichotomyOur dichotomy

PSPACE-complete

PSPACE-complete

exponential

CONN

st-CONN

diameter

Tight Tight CSPCSP Non-tight Non-tight CSP CSP

in coNP

in P

linear

OR-free CSPsOR-free CSPs

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Distance preserving CSPsDistance preserving CSPs

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Graph distance = Hamming distanceGraph distance = Hamming distance

TightTight

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OR / NAND-free CSPOR / NAND-free CSP

• Set of relations neither of which can express OR by substituting constants• Includes Horn• Includes some NP-complete CSP, e.g. POS-1-in-k SAT

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Graph distance = Hamming distanceGraph distance = Hamming distance

Distance preserving CSPDistance preserving CSP

• Set of relations, for which every component is a 2-SAT formula (component-wise bijunctive)• Includes bijunctive • Includes some NP-complete CSP

Proof for the hard side of the dichotomyProof for the hard side of the dichotomy

• Proof for 3-SAT• Expressibility theorem like Schaefer’s

Schaefer expressibilitySchaefer expressibilityA relation is expressible from set of relations S if there is a CNF(S) formula , s.t. : (x1, …, xn) = w1, … ,wt (x1, …, xn, w1, …, wt)

FaithfulFaithful expressibility expressibilityA relation is faithfully expressible from set of relations S if there is a CNF(S) formula , s.t. : (x1, …, xn) = w1, … ,wt (x1, …, xn, w1, …, wt)

FaithfulFaithful expressibility expressibilityA relation is faithfully expressible from set of relations S if there is a CNF(S) formula , s.t. : (x1, …, xn) = w1, … ,wt (x1, …, xn, w1, …, wt)

FaithfulFaithful expressibility expressibilityA relation is faithfully expressible from set of

relations S if there is a CNF(S) formula , s.t.: (x1, …, xn) = w1, … ,wt (x1, …, xn, w1, …, wt)and

(1) For every a {0,1}n with (a)=1, the graph of solutions of (a, w) is connected.

(2) For every a, b {0,1}n with (a)= (b)=1, |a-b|=1, there exists w s.t. (a, w)=(b, w)=1

Lemma: For 3-SAT (a) Exist formulas with exponential diameter(b) CONN and st-CONN are PSPACE-complete

Lemma: Faithful expressibility:(a) preserves diameter up to a polynomial factor(b) Is a poly time reduction for CONN and st-CONN

Faithful Expressibility Theorem: If S is not tight, every relation is faithfully expressible from S.

Proof for the hard side of the dichotomyProof for the hard side of the dichotomy

Theorem : If S is not tight, every relation is faithfully expressible from S.

Proof in 4 steps.

Step 0: Express 2-SAT clauses.

• Some relation can express OR (NAND).

• Other 2-SAT clauses by resolution:

(x1 x2) = w (x1 w) (w x2)

Faithful Expressibility TheoremFaithful Expressibility Theorem

_ _ _

(x1 x3) =

Faithful Expressibility TheoremFaithful Expressibility Theorem

Theorem : If S is not tight, every relation is faithfully expressible from S.

Proof in 4 steps.

Step 1 : Express a relation where some distance expands. Use R which is not component-wise bijunctive.

Faithful Expressibility TheoremFaithful Expressibility Theorem

Theorem : If S is not tight, every relation is faithfully expressible from S.

Proof in 4 steps.

Step 2 : Express a path of length 4 between vertices at distance 2.

Theorem : If S is not tight, every relation is faithfully expressible from S.

Proof in 4 steps.

Step 3: Express all 3-SAT clauses from such paths. [Demaine-Hearne ‘02]

Faithful Expressibility TheoremFaithful Expressibility Theorem

Theorem : If S is not tight, every relation is faithfully expressible from S.

Proof in 4 steps.

Step 4: Express all relations from 3-SAT clauses.

Faithful Expressibility TheoremFaithful Expressibility Theorem

Open questionsOpen questions

• Trichotomy for CONN? Trichotomy for CONN? – P for component-wise bijunctiveP for component-wise bijunctive– coNP-complete for non-Schaefer tight relationscoNP-complete for non-Schaefer tight relations– open for Horn/dual-Hornopen for Horn/dual-Horn

• Which Boolean CSPs have a clustered phase?Which Boolean CSPs have a clustered phase?

Thank youThank you