Efficient Algorithms for Finding Maximum and Maximal Cliques and Their Applications

Tomita, Eiji

doi:10.1007/978-3-319-53925-6_1

Cited by 28 publications

(16 citation statements)

References 39 publications

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“…(2) Exact Algorithms On the other hand, designing exact algorithms has also been extensively studied in view of the importance of the problem, e.g., in [9,33,34,38,39,42,45,48,49,51,53]. All the exact algorithms follow the branch-&bound framework.…”

Section: Existing Algorithmsmentioning

confidence: 99%

“…In addition, upper bounds are computed for branches such that a branch is pruned if its upper bound is not better than the currently found largest clique. In the literature, various branching and bounding techniques have been developed [33,34,48,49,51].…”

Section: Existing Algorithmsmentioning

confidence: 99%

“…-solvers for MCC-Dense (i.e., MCC over Dense graphs) [9,33,34,38,48,49,51,53] that specifically handle dense graphs by relying on the adjacency matrix graph representation and explicitly storing non-neigbors; -solvers for MCC-Sparse (i.e., MCC over Sparse graphs) [39,42,45] that handle large sparse graphs by using the adjacency list graph representation.…”

Section: Mcc-sparse Vs Mcc-densementioning

confidence: 99%

“…The existing MCC-Dense solvers cannot be directly applied to process large sparse graphs with millions of vertices due to the memory explosion of the adjacency matrix graph representation. 1 In the literature, advanced branching and bound-ing techniques-such as graph recoloring-based bounding [48,49] and degeneracy-based branching [33,34]-have been developed for MCC-Dense. However, the existing MCC-Sparse solvers use only simple branching and bounding techniques-such as standard graph coloring-based bounding and highest color-based branching-but resort to advanced programming techniques such as multi-core [42] and bit-parallel [45].…”

Section: Mcc-sparse Vs Mcc-densementioning

confidence: 99%

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Efficient maximum clique computation and enumeration over large sparse graphs

Chang

2020

The VLDB Journal

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This paper studies the problem of maximum clique computation (MCC) over sparse graphs, as large real-world graphs are usually sparse. In the literature, the problem of MCC over sparse graphs has been studied separately and less extensively than its dense counterpart-MCC over dense graphs-and advanced algorithmic techniques that are developed for MCC over dense graphs have not been utilized in the existing MCC solvers for sparse graphs. In this paper, we design an algorithm MC-BRB for sparse graphs which transforms an instance of MCC over a large sparse graph G to instances of k-clique finding (KCF) over dense subgraphs of G, each of which can be computed by the existing MCC solvers for dense graphs. To further improve the efficiency, we then develop a new branch-reduce-&-bound framework for KCF over dense graphs by proposing light-weight reducing techniques and leveraging the advanced branching and bounding techniques that are used in the existing MCC solvers for dense graphs. In addition, we also design an ego-centric algorithm MC-EGO for heuristically computing a near-maximum clique in near-linear time, and we extend our MC-BRB algorithm to enumerate all maximum cliques. Finally, we parallelize our algorithms to exploit multiple CPU cores. We conduct extensive empirical studies on large real graphs and demonstrate the efficiency and effectiveness of our techniques. KeywordsMaximum clique computation • Maximum clique enumeration • Large sparse graph • Branch-bound-and-reduce • Reducing techniques Footnote 1 continued |V | 2 ) memory consumption is still inevitable; note that the state-of-theart MCC-Dense solver MoMC [34] also materializes the complement graph of the input graph, i.e., explicitly stores the non-neighbors of each vertex, for efficient processing.

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Section: Existing Algorithmsmentioning

confidence: 99%

Section: Existing Algorithmsmentioning

confidence: 99%

Section: Mcc-sparse Vs Mcc-densementioning

confidence: 99%

Section: Mcc-sparse Vs Mcc-densementioning

confidence: 99%

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Efficient maximum clique computation and enumeration over large sparse graphs

Chang

2020

The VLDB Journal

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“…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%

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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Computing Maximum Independent Sets over Large Sparse Graphs

Alsahafy

Chang

2019

Web Information Systems Engineering – WISE 2019

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Efficient Algorithms for Finding Maximum and Maximal Cliques and Their Applications

Cited by 28 publications

References 39 publications

Efficient maximum clique computation and enumeration over large sparse graphs

Efficient maximum clique computation and enumeration over large sparse graphs

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

Computing Maximum Independent Sets over Large Sparse Graphs

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