Conditional Lower Bounds for Failed Literals and Related Techniques - - PowerPoint PPT Presentation

Conditional Lower Bounds for Failed Literals and Related Techniques

Matti Järvisalo Janne H. Korhonen*

• University of Helsinki, Department of Computer Science, and

Helsinki Institute for Information Technology HIIT

Background: Formula Simplification

1.

formula simplification CNF formula “simpler” formula

CNF formula SAT/ UNSAT SAT solver formula simplification

CNF formula SAT/ UNSAT SAT solver formula simplification

CNF formula SAT/ UNSAT SAT solver formula simplification simplification techniques faster stronger

Main Result: Lower Bound for Failed Literal Existence

2.

• Unit propagation
• apply until fixpoint
• we write F ⊢up (ℓ) if (ℓ) can be derived from F by

repeated application of unit resolution rule (¬ℓ1), (¬ℓ2), …, (¬ℓk) (ℓ)

}

(ℓ1∨…∨ℓk∨ℓ)

• Unit propagation
• apply until fixpoint
• we write F ⊢up (ℓ) if (ℓ) can be derived from F by

repeated application of unit resolution rule

• Failed literals
• a literal ℓ ∈ F is a failed literal if F∧(ℓ) ⊢up (ℓ’), (¬ℓ’)


for some ℓ’ ∈ F

• replace F with F∧(¬ℓ) if ℓ is a failed literal

(ℓ)

}

(¬ℓ1), (¬ℓ2), …, (¬ℓk) (ℓ1∨…∨ℓk∨ℓ)

Failed literal existence problem Input: CNF formula F Problem: Decide whether F has a failed literal

• Upper bounds (assuming bounded clause width)
• unit propagation O(n+m)
• failed literal existence O(n(n+m))
• failed literal elimination fixpoint O(n2(n+m))
• Can we do any better?

Failed literal existence problem Input: CNF formula F Problem: Decide whether F has a failed literal

• Theorem. If failed literal existence can solved in

O((n+m)2-ε) time on Horn-3-CNFs for some ε > 0, then CNF-SAT can be solved in time 2(1-ε/2)n poly(n,m)

• n formulas of unrestricted clause length.
• Recall that CNF formula is Horn if each clause has at

most one unnegated variable

• Horn-SAT is solvable in linear time

• Theorem. If failed literal existence can solved in

O((n+m)2-ε) time on Horn-3-CNFs for some ε > 0, then CNF-SAT can be solved in time 2(1-ε/2)n poly(n,m)

• n formulas of unrestricted clause length.
• We do not know how to solve CNF-SAT in time


2(1-ε)n poly(n,m) for any ε > 0

• This would give exponential speed-up for CNF-SAT!

lim inf {δ : k-SAT can be solved in time O(2δn)} = 1

n→∞

[Calabro, Impagliazzo, and Paturi 2009]

CNF-SAT with unrestricted clause length cannot be solved in time 2(1-ε)n poly(n,m) for any ε > 0

The Strong Exponential Time Hypothesis

• Corollary. Failed literal existence restricted to Horn-3-

CNFs cannot be solved in time O((n+m)2-ε) for any ε > 0 unless SETH fails.

• Corollary. Failed literal existence restricted to Horn-3-

CNFs cannot be solved in time O((n+m)2-ε) for any ε > 0 unless SETH fails.

• Compare with other similar results: for any ε > 0 we

cannot solve

• k-dominating set for k ≥ 3 in time O((n+m)k-ε)
• 2-SAT with O(n) clauses and two unrestricted

length clauses in time O(n2-ε)

• Local alignment of two binary strings in time O(n2-ε)

[Pătraşcu and Williams 2010] [Abboud, Vassilevska Williams, and Weimann 2014]

Proof of the Failed Literal Existence Lower Bound

3.

Proof Overview

3a.

• Theorem. If failed literal existence can solved in

O((n+m)2-ε) time on Horn-3-CNFs for some ε > 0, then CNF-SAT can be solved in time 2(1-ε/2)n poly(n,m)

• n formulas of unrestricted clause length.

Horn-3-CNF O((n+m)2-ε) failed literal / no failed literals n variables m clauses F

CNF formula 2(1-ε/2)n poly(n,m) F n variables m clauses SAT / UNSAT

CNF formula n variables m clauses F

CNF formula Reduction 2n/2 poly(n,m) N = 2n/2 poly(n,m) M = 2n/2 poly(n,m) n variables m clauses Horn-3-CNF F F’

CNF formula Reduction 2n/2 poly(n,m) O((N+M)2-ε) = 2(1-ε/2)n poly(n,m) N = 2n/2 poly(n,m) M = 2n/2 poly(n,m) n variables m clauses Horn-3-CNF failed literal / no failed literals F F’

CNF formula Reduction 2n/2 poly(n,m) O((N+M)2-ε) = 2(1-ε/2)n poly(n,m) SAT / UNSAT F N = 2n/2 poly(n,m) M = 2n/2 poly(n,m) n variables m clauses F’ Horn-3-CNF failed literal / no failed literals

Reduction from CNF-SAT to Failed Literal Elimination

3b.

CNF formula Reduction 2n/2 poly(n,m) O((N+M)2-ε) = 2(1-ε/2)n poly(n,m) SAT / UNSAT F N = 2n/2 poly(n,m) M = 2n/2 poly(n,m) n variables m clauses F’ Horn-3-CNF failed literal / no failed literals

n variables input formula F

n variables n/2 “upper” variables n/2 “lower” variables input formula F

n variables n/2 “upper” variables n/2 “lower” variables 2n/2 partial truth assignments P 2n/2 partial truth assignments Q input formula F

constructing the output formula F’ P Q ~ m clauses in F c1 c2 c3 … cm ~ 2n/2 partial truth assignments in Q ~ 2n/2 partial truth assignments in P yp yq

constructing the output formula F’ P Q c1 c2 c3 … cm yp yq add clause (yp → ci) if p(Ci) = 1

constructing the output formula F’ P Q yp yq add clause ((⋀i∈S ci)→ yq) where S = { i : q(Ci) ≠ 1 }

⋀

add clause (yp → ci) if p(Ci) = 1

constructing the output formula F’ P Q yp yq add clause ((⋀i∈S ci)→ yq) where S = { i : q(Ci) ≠ 1 }

⋀

add clause (yp → ci) if p(Ci) = 1 F∧(yp) ⊢up (yq) pq(F) = 1 if and only if

Extensions and Open Questions

4.

• Assuming SETH, for any ε > 0 we cannot solve
• asymmetric tautology existence on Horn-3-CNFs in

time O((n+m)2-ε)

• asymmetric literal existence on Horn-3-CNFs in time

O((n+m)2-ε)

• singleton arc consistency on (3,2)-CSPs in time


O((n+m)2-ε)

• k-step lookahead lower bound?
• Fix values for k variables, do unit propagation
• We can probably show lower bound vs. O((n+m)k+1-ε) time

Extensions of the Main Result

• Failed literal existence on 2-CNFs?
• CNF version requires clause length 3
• CSP version requires domain size 3
• Maybe one can do better on 2-CNFs?
• Failed literal elimination fixpoint?
• Lower bound O((n+m)3-ε)?

Open Questions

Questions, comments?

Thank you!

• clause C = (ℓ1∨…∨ℓk) ∈ F is an asymmetric tautology

if (F \ C)∧(¬ℓ1)∧…∧(¬ℓk) ⊢up (ℓ’), (¬ℓ’) for some ℓ’ ∈ F

• replace F with F \ C
• literal ℓ in a clause C ∈ F is an asymmetric literal if

F∧(ℓ) ⊢up (ℓ’) for some ℓ’ ∈ C \ {ℓ}

• replace F with (F \ C) ∧ (C \ ℓ)

Definitions: Asymmetric Tautologies and Literals