An improved local search algorithm for 3-SAT

Brueggemann, Tobias; Kern, Walter

doi:10.1016/j.endm.2004.03.021

Cited by 12 publications

(13 citation statements)

References 3 publications

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“…Exact exponential-time algorithms with improved running times have been designed for various other important NP-complete problems. For example, Dantsin et al [DGH + 02] pushed the trivialÕ(2 n ) bound for the three satisfiability problem down toÕ(1.481 n ), which was further improved toÕ(1.473 n ) by Brueggemann and Kern [BK04]. Schöning [Sch02], Hofmeister et al [HSSW02] and Paturi et al [PPSZ98] proposed even better randomized algorithms for the satisfiability problem.…”

Section: Introductionmentioning

confidence: 99%

An Exact 2.9416 n Algorithm for the Three Domatic Number Problem

Riege

Rothe

2005

Mathematical Foundations of Computer Science 2005

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The three domatic number problem asks whether a given undirected graph can be partitioned into at least three dominating sets, i.e., sets whose closed neighborhood equals the vertex set of the graph. Since this problem is NP-complete, no polynomial-time algorithm is known for it. The naive deterministic algorithm for this problem runs in time 3n , up to polynomial factors. In this paper, we design an exact deterministic algorithm for this problem running in time 2.9416 n . Thus, our algorithm can handle problem instances of larger size than the naive algorithm in the same amount of time. We also present another deterministic and a randomized algorithm for this problem that both have an even better performance for graphs with small maximum degree.

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Section: Introductionmentioning

confidence: 99%

An Exact 2.9416 n Algorithm for the Three Domatic Number Problem

Riege

Rothe

2005

Mathematical Foundations of Computer Science 2005

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“…where the β j are given from observation and the β j,γ are approximated according to (3). The goal is then to determine…”

Section: The Garnier/kallel-approachmentioning

confidence: 99%

“…For the calculation of βj,γ according to (3) we implemented the following procedure, which actually approximates βj,γ since we employ an approximation of the Γ-function. We recall that in (3) the (unknown) N is substituted by M/r, where M is selected as described in Section IV-A and r is a variable in our calculations.…”

Section: B Approximation Of H γmentioning

confidence: 99%

“…A first summary was presented in [8] along with an empirical analysis of run-time distributions for various local search-based methods such as WalkSAT [22]. Improvements on run-time estimations for k-SAT problems as well as for CNFs with unconstrained clause lengths are reported in [3], [4], [6], [14], [17], [18], which are partly based on randomised local search methods. Significant progress has been achieved in the analysis of phase transitions since this effect was reported in [11], [23].…”

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confidence: 99%

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Combinatorial Landscape Analysis for k-SAT Instances

Albrecht¹,

Lane²,

Steinhöfel³

2008

2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence)

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Abstract-Over the past ten years, methods from statistical physics have provided a deeper inside into the average complexity of hard combinatorial problems, culminating in a rigorous proof for the asymptotic behaviour of the k-SAT phase transition threshold by Achlioptas and Peres in 2004. On the other hand, when dealing with individual instances of hard problems, gathering information about specific properties of instances in a pre-processing phase might be helpful for an appropriate adjustment of local search-based procedures. In the present paper, we address both issues in the context of landscapes induced by k-SAT instances: Firstly, we utilize a sampling method devised by Garnier and Kallel in 2002 for approximations of the number of local maxima in landscapes generated by individual k-SAT instances and a simple neighbourhood relation. The objective function is given by the number of satisfied clauses. Secondly, we outline a method for obtaining upper bounds for the average number of local maxima in k-SAT instances which indicates some kind of phase transition for the neighbourhood-specific ratio m/n = Θ(2 k /k).

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“…A first summary was presented in [14] along with an empirical analysis of run-time distributions for various local search-based methods such as WalkSAT [30]. Improvements on run-time estimations for k-SAT problems as well as for CNFs with unconstrained clause lengths are reported by a number of authors [1,7,9,11,21,25,26], where the results are partly based on randomised local search methods. Significant progress has been achieved in the analysis of phase transitions since this effect was reported in [18] and [31].…”

Section: Introductionmentioning

confidence: 99%

Analysis of Local Search Landscapes for k-SAT Instances

2010

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Abstract. Stochastic local search is a successful technique in diverse areas of combinatorial optimisation and is predominantly applied to hard problems. When dealing with individual instances of hard problems, gathering information about specific properties of instances in a preprocessing phase is helpful for an appropriate parameter adjustment of local search-based procedures. In the present paper, we address parameter estimations in the context of landscapes induced by k-SAT instances: at first, we utilise a sampling method devised by Garnier and Kallel in 2002 for approximations of the number of local maxima in landscapes generated by individual k-SAT instances and a simple neighbourhood relation. The objective function is given by the number of satisfied clauses. The procedure provides good approximations of the actual number of local maxima, with a deviation typically around 10%. Secondly, we provide a method for obtaining upper bounds for the average number of local maxima in k-SAT instances. The method allows us to obtain the upper bound 2 n−O( √ n/k) for the average number of local maxima, if m is in the region of 2 k ·n/k.Mathematics Subject Classification (2010). Primary 68R99; Secondary 90C27.

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An improved local search algorithm for 3-SAT

Abstract: We slightly improve the pruning technique presented in Dantsin et. al. (2002) to obtain an O * (1.473 n ) algorithm for 3-SAT.

Cited by 12 publications

References 3 publications

An Exact 2.9416 n Algorithm for the Three Domatic Number Problem

An Exact 2.9416 n Algorithm for the Three Domatic Number Problem

Combinatorial Landscape Analysis for k-SAT Instances

Analysis of Local Search Landscapes for k-SAT Instances

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