Combining MaxSAT Reasoning and Incremental Upper Bound for the Maximum Clique Problem

Li, Chu-Min; Fang, Zhen; Xu, Ke

doi:10.1109/ictai.2013.143

Cited by 51 publications

(40 citation statements)

References 16 publications

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“…and possibly with shortcuts taken for certain vertices [48], and possibly followed by a repair step to improve the colouring [53], or stronger bounding rules based upon MaxSAT inference [25,26,46]). Such colourings will not give us the required property that vertices in remaining ∩ connected come last (so they are selected first by the reverse branching order).…”

Section: Ensuring Connectedness In a Clique-based Approachmentioning

confidence: 99%

Clique and Constraint Models for Maximum Common (Connected) Subgraph Problems

McCreesh¹,

Ndiaye²,

Prosser³

et al. 2016

Lecture Notes in Computer Science

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Abstract. The maximum common subgraph problem is to find the largest subgraph common to two given graphs. This problem can be solved either by constraint-based search, or by reduction to the maximum clique problem. We evaluate these two models using modern algorithms, and see that the best choice depends mainly upon whether the graphs have labelled edges. We also study a variant of this problem where the subgraph is required to be connected. We introduce a filtering algorithm for this property and show that it may be combined with a restricted branching technique for the constraint-based approach. We show how to implement a similar branching technique in clique-inspired algorithms. Finally, we experimentally compare approaches for the connected version, and see again that the best choice depends on whether graphs have labels.

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Section: Ensuring Connectedness In a Clique-based Approachmentioning

confidence: 99%

Clique and Constraint Models for Maximum Common (Connected) Subgraph Problems

McCreesh¹,

Ndiaye²,

Prosser³

et al. 2016

Lecture Notes in Computer Science

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show abstract

“…One type is heuristic algorithms mainly including stochastic local search (e.g., Pullan & Hoos, 2006;Cai, Su, & Sattar, 2011;Cai, Su, Luo, & Sattar, 2013;Fang, Chu, Qiao, Feng, & Xu, 2014a). Another is exact algorithms including branch-and-bound (BnB) search (e.g., Östergård, 2002;Régin, 2003;Tomita & Seki, 2003;Konc & Janezic, 2007;Li & Quan, 2010b;Tomita & Kameda, 2007;Li, Fang, & Xu, 2013). Heuristic algorithms are able to solve large-scale instances but cannot guarantee the optimality of their solutions.…”

Section: Introductionmentioning

confidence: 99%

“…The excellent performance of Max-CLQ shows that MaxSAT reasoning technologies allows to compute a tight upper bound for MC within reasonable time. IncMaxCLQ (Li et al, 2013) combines an incremental upper bound and MaxSAT reasoning to compute a tight upper bound more efficiently. In addition to the independent set partition and MaxSAT reasoning, other approaches, such as the graph matching (Régin, 2003), are also used in the upper bounding for MC.…”

Section: Introductionmentioning

confidence: 99%

An Exact Algorithm Based on MaxSAT Reasoning for the Maximum Weight Clique Problem

Fang¹,

Li²,

Xu³

2016

jair

Self Cite

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Recently, MaxSAT reasoning is shown very effective in computing a tight upper bound for a Maximum Clique (MC) of a (unweighted) graph. In this paper, we apply MaxSAT reasoning to compute a tight upper bound for a Maximum Weight Clique (MWC) of a wighted graph. We first study three usual encodings of MWC into weighted partial MaxSAT dealing with hard clauses, which must be satisfied in all solutions, and soft clauses, which are weighted and can be falsified. The drawbacks of these encodings motivate us to propose an encoding of MWC into a special weighted partial MaxSAT formalism, called LW (Literal-Weighted) encoding and dedicated for upper bounding an MWC, in which both soft clauses and literals in soft clauses are weighted. An optimal solution of the LW MaxSAT instance gives an upper bound for an MWC, instead of an optimal solution for MWC. We then introduce two notions called the Top-k literal failed clause and the Top-k empty clause to extend classical MaxSAT reasoning techniques, as well as two sound transformation rules to transform an LW MaxSAT instance. Successive transformations of an LW MaxSAT instance driven by MaxSAT reasoning give a tight upper bound for the encoded MWC. The approach is implemented in a branch-and-bound algorithm called MWCLQ. Experimental evaluations on the broadly used DIMACS benchmark, BHOSLIB benchmark, random graphs and the benchmark from the winner determination problem show that our approach allows MWCLQ to reduce the search space significantly and to solve MWC instances effectively. Consequently, MWCLQ outperforms state-of-the-art exact algorithms on the vast majority of instances. Moreover, it is surprisingly effective in solving hard and dense instances.

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“…Many branch-and-bound algorithms have been developed for the maximum clique problem [36,37,42], which use vertex reordering and pruning techniques based on approximate graph coloring [42] or MaxSAT [32], and can be further sped up by applying local search to obtain an initial solution of high quality [7].…”

Section: Related Workmentioning

confidence: 99%

Scalable Kernelization for Maximum Independent Sets

Hespe¹,

Schulz²,

Strash³

2018

2018 Proceedings of the Twentieth Workshop on Algorithm Engineering and Experiments (ALENEX)

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The most efficient algorithms for finding maximum independent sets in both theory and practice use reduction rules to obtain a much smaller problem instance called a kernel. The kernel can then be solved quickly using exact or heuristic algorithms-or by repeatedly kernelizing recursively in the branch-and-reduce paradigm. It is of critical importance for these algorithms that kernelization is fast and returns a small kernel. Current algorithms are either slow but produce a small kernel, or fast and give a large kernel. We attempt to accomplish both of these goals simultaneously, by giving an efficient parallel kernelization algorithm based on graph partitioning and parallel bipartite maximum matching.We combine our parallelization techniques with two techniques to accelerate kernelization further: dependency checking that prunes reductions that cannot be applied, and reduction tracking that allows us to stop kernelization when reductions become less fruitful. Our algorithm produces kernels that are orders of magnitude smaller than the fastest kernelization methods, while having a similar execution time. Furthermore, our algorithm is able to compute kernels with size comparable to the smallest known kernels, but up to two orders of magnitude faster than previously possible. Finally, we show that our kernelization algorithm can be used to accelerate existing state-of-the-art heuristic algorithms, allowing us to find larger independent sets faster on large real-world networks and synthetic instances.

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Combining MaxSAT Reasoning and Incremental Upper Bound for the Maximum Clique Problem

Cited by 51 publications

References 16 publications

Clique and Constraint Models for Maximum Common (Connected) Subgraph Problems

Clique and Constraint Models for Maximum Common (Connected) Subgraph Problems

An Exact Algorithm Based on MaxSAT Reasoning for the Maximum Weight Clique Problem

Scalable Kernelization for Maximum Independent Sets

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