A review on algorithms for maximum clique problems

Wu, Qinghua; Hao, Jin‐Kao

doi:10.1016/j.ejor.2014.09.064

Cited by 255 publications

(172 citation statements)

References 137 publications

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“…We have many approximation algorithms for finding a maximum clique [37], while finding a good approximate solution for the maximum clique problem is considered to be very hard as shown that the maximum clique problem is not polynomial-time approximable within |V| 1− for any unless NP = P [38]. The most important problem is a proper choice of the trade-off between the quality of the approximate solution and the time required to obtain it.…”

Section: An Approximate Solution As An Initial Lower Boundmentioning

confidence: 99%

“…Hence, much effort has been devoted to this problem theoretically and experimentally [4], [21], [37]. In particular, see [37] for a recent progress of algorithms for this problem.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, see [37] for a recent progress of algorithms for this problem. Furthermore, much faster algorithms are in great demand so that much more practical problems can be solved.…”

Section: Introductionmentioning

confidence: 99%

“…Here, finding a maximum clique in a graph has many important practical applications. These applications include coding theory [24], pattern recognition and image processing [9], [20], bioinformatics [5], [18], [33], design of wireless networks [16], and others [4], [36], [37].…”

Section: Introductionmentioning

confidence: 99%

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A Much Faster Algorithm for Finding a Maximum Clique with Computational Experiments

Tomita

Matsuzaki

Nagao

et al. 2017

Journal of Information Processing

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Abstract:We present further improvements to a branch-and-bound maximum-clique-finding algorithm MCS (WALCOM 2010, LNCS 5942, pp.191-203) that was shown to be fast. First, we employ a variant of an efficient approximation algorithm KLS for finding a maximum clique. Second, we make use of appropriate sorting of vertices only near the root of the search tree. Third, we employ a lightened approximate coloring mainly near the leaves of the search tree. A new algorithm obtained from MCS with the above improvements is named k5 MCT. It is shown that k5 MCT is much faster than MCS by extensive computational experiments. In particular, k5 MCT is shown to be faster than MCS for gen400 p0.9 75, gen400 p0.9 65 and gen400 p0.9 55 by over 81,000, 39,000 and 19,000 times, respectively.

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Section: An Approximate Solution As An Initial Lower Boundmentioning

confidence: 99%

“…Hence, much effort has been devoted to this problem theoretically and experimentally [4], [21], [37]. In particular, see [37] for a recent progress of algorithms for this problem.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, see [37] for a recent progress of algorithms for this problem. Furthermore, much faster algorithms are in great demand so that much more practical problems can be solved.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Much Faster Algorithm for Finding a Maximum Clique with Computational Experiments

Tomita

Matsuzaki

Nagao

et al. 2017

Journal of Information Processing

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“…The problem has been thoroughly studied and exact and heuristic methods have been developed. For an overview of the theoretical results and existing methods, see Bomze et al Bomze et al (1999) and Wu and Hao Wu and Hao (2015). In the existing methods, the weights of the vertices are known beforehand, but in our model the costs of the vertices depend on which vertices are in the clique.…”

Section: Cost Of the Verticesmentioning

confidence: 99%

Two decomposition algorithms for solving a minimum weight maximum clique model for the air conflict resolution problem

Lehouillier

Omer

Soumis

et al. 2017

European Journal of Operational Research

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International audienceIn this article, we tackle the conflict resolution problem using a new variant of the minimum-weight maximum-clique model. The problem involves identifying maneuvers that maintain the required separation distance between all pairs of a set of aircraft while minimizing fuel costs. We design a graph in which the vertices correspond to a finite set of maneuvers and the edges connect conflict-free maneuvers. A maximum clique of minimal weight yields a conflict-free situation that involves all the aircraft and minimizes the costs induced. The innovation of the model is its cost structure: the costs of the vertices cannot be determined a priori, since they depend on the vertices in the clique. We formulate the problem as a mixed integer linear program. Since the modeling of the aircraft dynamics and the computation of trajectories is separated from the solution process, the model is flexible. As a consequence, our mathematical framework is valid for any hypotheses. In particular, the aircraft can perform dynamic velocity, heading, and flight-level changes. To solve instances involving a large number of aircraft spread over several flight levels, we introduce two decomposition algorithms. The first is a sequential mixed integer linear optimization procedure that iteratively refines the discretization of the maneuvers to yield a trade-off between computational time and cost. The second is a large neighborhood search heuristic that uses the first procedure as a subroutine. The best solutions for the available set of maneuvers are obtained in less than 10 seconds for instances with up to 250 aircraft randomly allocated to 20 flight levels

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A new parallel tabu search algorithm for the optimization of the maximum vertex weight clique problem

Dülger,

Dökeroğlu

2023

Concurrency and Computation

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SummaryThe efficiency of metaheuristic algorithms depends significantly on the number of fitness value evaluations performed on candidate solutions. In addition to various intelligent techniques used to obtain better results, parallelization of calculations can substantially improve the solutions in cases where the problem is NP‐hard and requires many evaluations. This study proposes a new parallel tabu search method for solving the Maximum Vertex Weight Clique Problem (MVWCP) on the Non‐Uniform Memory Access (NUMA) architectures using the OpenMP parallel programming paradigm. Achieving scalability in the NUMA architectures presents significant challenges due to the high complexity of their memory systems, which can lead to performance loss. However, our proposed Tabu‐NUMA algorithm provides up to speed‐up with 64 cores for ten basic problem instances in DIMACS‐W and BHOSLIB‐W benchmarks. And it improves the performance of the serial Multi Neighborhood Tabu Search (MN/TS) algorithm for 38 problem instances in DIMACS‐W and BHOSLIB‐W benchmarks. We further evaluate our algorithm on larger datasets with thousands of edges and vertices from Network Data Repository benchmark problem instances, and we report significant improvements in terms of speed up. Our results confirm that the Tabu‐NUMA algorithm is among the best recent algorithms for solving MVWCP on the NUMA architectures.

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A review on algorithms for maximum clique problems

Cited by 255 publications

References 137 publications

A Much Faster Algorithm for Finding a Maximum Clique with Computational Experiments

A Much Faster Algorithm for Finding a Maximum Clique with Computational Experiments

Two decomposition algorithms for solving a minimum weight maximum clique model for the air conflict resolution problem

A new parallel tabu search algorithm for the optimization of the maximum vertex weight clique problem

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