Faster Exact Solving of SAT Formulae with a Low Number of Occurrences per Variable

Wahlström, Magnus

doi:10.1007/11499107_23

Cited by 11 publications

(7 citation statements)

References 14 publications

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“…the assumption that w 4 = 2w 3 (for Step 11, the worst branching vector in [18] is [4,8]). The branching factor for these three steps will decrease if the value of w 3 increases.…”

Section: Discussionmentioning

confidence: 99%

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A Fast Algorithm for SAT in Terms of Formula Length

Peng¹,

Xiao²

2021

Preprint

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In this paper, we prove that the general CNF satisfiability problem can be solved in O * (1.0646 L ) time, where L is the length of the input CNF-formula (i.e., the total number of literals in the formula), which improves the current bound O * (1.0652 L ) given by Chen and Liu 12 years ago. Our algorithm is a standard branch-and-search algorithm analyzed by using the measure-and-conquer method. We avoid the bottleneck in Chen and Liu's algorithm by simplifying the branching operation for 4-degree variables and carefully analyzing the branching operation for 5-degree variables. To simplify case-analyses, we also introduce a general framework for analysis, which may be able to be used in other problems.

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“…the assumption that w 4 = 2w 3 (for Step 11, the worst branching vector in [18] is [4,8]). The branching factor for these three steps will decrease if the value of w 3 increases.…”

Section: Discussionmentioning

confidence: 99%

“…All variables are 3-variables now. We apply the O * (1.1279 n )-time algorithm by Wahlström [18] to solve this special case, where n is the number of variables. For this case, we have that n = µ(F )/w 3 .…”

Section: Step 11mentioning

confidence: 99%

A Fast Algorithm for SAT in Terms of Formula Length

Peng¹,

Xiao²

2021

Preprint

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“…In [ 13 , 14 ], M. Wahlström presented a definition of (

)-variable to classify all variables in a CNF formula, and designed two algorithms for solving a CNF formula with at most d occurrences per variable. Here, an (

)-variable is a variable which occurs positively in a clauses and negatively in b clauses.…”

Section: Introductionmentioning

confidence: 99%

Uniquely Satisfiable d-Regular (k,s)-SAT Instances

2020

Entropy

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Unique k-SAT is the promised version of k-SAT where the given formula has 0 or 1 solution and is proved to be as difficult as the general k-SAT. For any k ≥ 3 , s ≥ f ( k , d ) and ( s + d ) / 2 > k − 1 , a parsimonious reduction from k-CNF to d-regular (k,s)-CNF is given. Here regular (k,s)-CNF is a subclass of CNF, where each clause of the formula has exactly k distinct variables, and each variable occurs in exactly s clauses. A d-regular (k,s)-CNF formula is a regular (k,s)-CNF formula, in which the absolute value of the difference between positive and negative occurrences of every variable is at most a nonnegative integer d. We prove that for all k ≥ 3 , f ( k , d ) ≤ u ( k , d ) + 1 and f ( k , d + 1 ) ≤ u ( k , d ) . The critical function f ( k , d ) is the maximal value of s, such that every d-regular (k,s)-CNF formula is satisfiable. In this study, u ( k , d ) denotes the minimal value of s such that there exists a uniquely satisfiable d-regular (k,s)-CNF formula. We further show that for s ≥ f ( k , d ) + 1 and ( s + d ) / 2 > k − 1 , there exists a uniquely satisfiable d-regular ( k , s + 1 ) -CNF formula. Moreover, for k ≥ 7 , we have that u ( k , d ) ≤ f ( k , d ) + 1 .

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“…Calabro, Impagliazzo, and Paturi [2] proved that for both classes satisfiability can be checked in runtime O(γ n ) with γ < 2, and asymptotically related the bounds of k-SAT, SAT with m/n ≤ ∆, and SAT with at most d occurrences per variable to one another, essentially showing that the two latter bounds behave as the former one with k = Θ(log ∆) and k = Θ(log d), respectively. For small values of d, Wahlström [7,6] has given a stronger bound for formulas with at most d occurrences per variable of order O(1.1279 (d−2)n ). The main contribution of this paper is extending the boundaries of the classes of CNF formulas for which satisfiability can be checked faster than in runtime O(2 n ).…”

Section: Introductionmentioning

confidence: 99%

“…The main contribution of this paper is extending the boundaries of the classes of CNF formulas for which satisfiability can be checked faster than in runtime O(2 n ). In particular, we continue the line of research of [7], where the total number of occurrences (positive and negative) of each variable in a CNF formula F was restricted to at most d. Now, for each variable x of F we call the literal ∈ {x, ¬x} with less occurrences in F minor and its negation major. We then only restrict the number of minor literals in F to be at most d, that is, the total number of literal occurrences per variable in F remains unrestricted.…”

Section: Introductionmentioning

confidence: 99%

Solving SAT for CNF Formulas with a One-Sided Restriction on Variable Occurrences

Johannsen

Razgon

Wahlström

2009

Lecture Notes in Computer Science

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Abstract. In this paper we consider the class of boolean formulas in Conjunctive Normal Form (CNF) where for each variable all but at most d occurrences are either positive or negative. This class is a generalization of the class of CNF formulas with at most d occurrences (positive and negative) of each variable which was studied in . Applying complement search [Purdom, 1984], we show that for every d there exists a constantsuch that satisfiability of a CNF formula on n variables can be checked in runtime O(γ To the best of our knowledge, for the considered classes of formulas there are no previous non-trivial upper bounds on the complexity of satisfiability checking.

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Faster Exact Solving of SAT Formulae with a Low Number of Occurrences per Variable

Cited by 11 publications

References 14 publications

A Fast Algorithm for SAT in Terms of Formula Length

A Fast Algorithm for SAT in Terms of Formula Length

Uniquely Satisfiable d-Regular (k,s)-SAT Instances

Solving SAT for CNF Formulas with a One-Sided Restriction on Variable Occurrences

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