The Connectivity of Boolean Satisfiability:Computational & Structural Dichotomies

Parikshit Gopalan Georgia Tech

Phokion Kolaitis IBM Almaden

Elitza Maneva UC Berkeley

Christos Papadimitriou UC Berkeley

Boolean Satisfiability Problems

• 3-SAT

(x1 Ç x2 Ç x3) Æ (:x2 Ç : x3 Ç x4)

• 2-SAT

(x1 Ç x2) Æ (x3 Ç x4)

• 3-LIN-SAT

(x1 + x2 + x3 = 1) Æ (x2 + x4 + x5 = 0)

• NotAllEqual-SAT

NAE(x1, x2, x3) Æ NAE(x2, x3, x4)

NP-complete

P

NP-complete

P

Boolean Satisfiability Problems

Q: What makes some SAT problems hard?

A: Classification of easy and hard SAT problems:

Schaefer’s Dichotomy Theorem [1978]

Easy Classes in P:1. 2-SAT

2. LIN-SAT

3. Horn-SAT

4. Dual-Horn-SAT

Everything else is NP-complete.

Boolean Satisfiability Problems

What is a Boolean SAT problem?

Boolean relation :

R1 R2

00

0110

11

Given a set of Boolean Relations S = {R1, …, Rk}.

CNF(S) consists of formulae:

x1,…,xn) = R1(x1, x16) Æ R2(x4, x5, x32) … Æ R1(xn-1, xn)

Boolean Satisfiability Problems

Example S = {R}.

R

000

111

Gives NAE-3SAT.

Schaefer’s Dichotomy Theorem

R is bijunctive if it is a 2-SAT formula.

R(x1,x2) :

: x1 Ç : x2

Schaefer’s Dichotomy Theorem

R is bijunctive if it is a 2-SAT formula.

R(x1, x2, x3) :

(x1 Ç x2) Æ (x2 Ç x3) Æ (x1 Ç x3)

S = {R1,…,Rk} is bijunctive if all Ris are bijunctive.

CNF(S) is easy, run any 2-SAT algorithm.

Schaefer’s Dichotomy Theorem

R is linear if it is a system of linear equations mod 2 .

S = {R1,…,Rk} is linear if all Ris are linear.

CNF(S) is easy, run Gaussian elimination.

R(x1,x2) :

x1 © x2 = 1 mod 2

Schaefer’s Dichotomy Theorem

The Schaefer sets of relations:

1. Bijunctive

2. Linear

3. Horn

4. Dual Horn

Satisfiability is in P.

For non-Schaefer sets , CNF(S) is NP-complete.

Dichotomy Theorems

Satisfiability Schaefer

Counting Creignou-Hermann

ApproximabilityKhanna-Sudan-Trevisan-Williamson, Creignou-Khanna-Sudan

3-valued SAT problems Bulatov

Minimal satisfiability Kirousis-Kolaitis

Inverse satisfiability Kavvadias-Sideri

Propositional abduction Creignou-Zanuttini

Structure of the Solution Space

Given (x1,…,xn) 2 CNF(S), its solutions induce a subgraph G() of the hypercube.

For which relations S does G() have nice structure ?

What is nice structure ?

Why is this interesting ?

Random formula with m clauses.• Phase transition in solution space structure as m increases.• Seems to control performance of heuristics.

Number of components for k-SAT :• 2-SAT: a single component • 3-SAT to 7-SAT: not known• 8-SAT above : exponential number of components

[Achlioptas, Ricci-Tersenghi `06]

[Mezard, Mora, Zecchina `05]

Solution Space Structure for Random Instances

Structure of the Solution Space

What is nice structure?

Structural Properties :

What kinds of graphs G() are possible?

Number of Components, Diameter, …

Computational Properties :

Is G() connected ?

Are two solutions in the same component ?

All give the same answer...

A New Dichotomy

Connectivity is PSPACE-complete.

Connectivty is in co-NP.

st-Connectivity is PSPACE-complete.

st-Connectivity is in P.

Diameter can be exponential in n.

Diameter is linear in number of variables n.

G() can be arbitrary.G() has nice structure.

Non-Tight SetsTight Sets

A New Dichotomy

Three kinds of Tight sets of relations:1. OR-free

2. NAND-free

3. Componentwise Bijunctive

Each has a nice structural property.

Inherited by all CNF formulas.

Results in small diameter, connectivity algorithms…

OR-free Relations

Cannot set variables in R(x1,…,xk) to constants and get x1 Ç x2.

G(R) does not contain :

OR-free Relations

Cannot set variables in R(x1,…,xk) to constants and get x1 Ç x2.

Example of an OR-free relation :

OR-free Relations

Cannot set variables in R(x1,…,xk) to constants and get x1 Ç x2.

Example of relation which contains OR:

NAE(x1, x2, 0) = x1 Ç x2

OR-free Relations

Cannot set variables in R(x1,…,xk) to constants and get x1 Ç x2.

G(R) does not contain :

Structural Property : Every component has a unique minimum.

If S is OR-free, every CNF(S) formula inherits this property.

OR-free Relations

Structural Property : Every component has a unique minimum.

Diameter of a Component, st-Connectivity:

NAND-free Relations

Cannot set variables in R(x1,…,xk) to constants and get : x1 Ç : x2.

G(R) does not contain :

Structural Property : Every component has a unique maximum.

Cup = xi Ç xj

Cdown = : xi Ç : xj

Each component is bijunctive. (2-SAT formula).

Examples : All bijunctive relations,

Componentwise Bijunctive Relations

C itself is not bijunctive.

C:

Each component is bijunctive. (2-SAT formula).

Componentwise Bijunctive Relations

Structural Property : Within each component, graph distance equals Hamming distance.

Each component is bijunctive. (2-SAT formula).

Componentwise Bijunctive Relations

Structural Property : Within each component, graph distance equals Hamming distance.

Not a characterization.

NAE has this property, it is not bijunctive.

Each component is bijunctive. (2-SAT formula).

Componentwise Bijunctive Relations

Structural Property : Within each component, graph distance equals Hamming distance.

Inherited by CNF formulas. Characterization

Results in small diameter, algorithms for st-Connectivity …

Relation to Schaefer Classes

Dual - HornHorn

OR-free NAND-freeC

om

p. B

ijun

ctiv

e

Linear Bijunctive

A New Dichotomy

Three kinds of Tight sets of relations:1. OR-free

2. NAND-free

3. Componentwise Bijunctive

Each has a nice structural property.

Inherited by all CNF formulas.

If S is not Tight, G() can be arbitrary.

Schaefer’s Expressibility Theorem

Theorem : If S is not Schaefer, every relation R is expressible as a CNF(S) formula.

Expressibility :

There is a formula 2 CNF(S) s.t.

R(x1,…, xk) = 9 y1, …, yt s.t. (x1,…,xk,y1,…, yt)

Taking R to be 3-SAT clauses; gives the hard part of Schaefer’s Dichotomy.

Need a notion of expressibility that is faithful to the solution space structure.

Faithful Expressibility

R(x1,…,xk)

y1,…,y3,x1,…,xk)

1. Projecting onto x gives R.

2. Witness space is connected.

3. Matching witness property.

Faithful Expressibility Theorem

Faithful Expressibility :

Preserves diameter up to polynomial factors.

Reduces Connectivity and st-Connectivity.

Theorem : If S is not tight, every relation R is faithfully expressible as a CNF(S) formula.

For 3-SAT formulas

• Diameter : exponential.

• Connectivty, st-Connectivity : PSPACE-complete.

A New Dichotomy

Connectivity is PSPACE-complete.

Connectivty is in co-NP.

st-Connectivity is PSPACE-complete.

st-Connectivity is in P.

Diameter can be exponential in n.

Diameter is linear in number of variables n.

G() can be arbitrary.G() has nice structure.

Non-Tight SetsTight Sets

Open Questions

2. Trichotomy for Connectivity ?

Known to be in coNP for Tight sets of relations.

Conjecture : In P for Schaefer sets; coNP-complete otherwise.

1. Are Tight but non-Schaefer sets easy on average?