A New Algorithm for Generating All the Maximal Independent Sets

Tsukiyama, Shuji; Ide, Mikio; Ariyoshi, Hiromu; Shirakawa, Isao

doi:10.1137/0206036

Cited by 530 publications

(348 citation statements)

References 7 publications

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“…Because S can be found and G ′ can be constructed in polynomial time, the minimal connected vertex covers of G can be enumerated in time O * (3 n/3 ) using the algorithms of e.g., [26,21] as mentioned in the preliminaries.…”

Section: Theoremmentioning

confidence: 99%

Enumeration and Maximum Number of Minimal Connected Vertex Covers in Graphs

Golovach

Heggernes

Kratsch

2016

Lecture Notes in Computer Science

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Abstract. Connected Vertex Cover is one of the classical problems of computer science, already mentioned in the monograph of Garey and Johnson [16]. Although the optimization and decision variants of finding connected vertex covers of minimum size or weight are well studied, surprisingly there is no work on the enumeration or maximum number of minimal connected vertex covers of a graph. In this paper we show that the maximum number of minimal connected vertex covers of a graph is at most 1.8668 n , and these can be enumerated in time O(1.8668 n ). For graphs of chordality at most 5, we are able to give a better upper bound, and for chordal graphs and distance-hereditary graphs we are able to give tight bounds on the maximum number of minimal connected vertex covers.

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Section: Theoremmentioning

confidence: 99%

Enumeration and Maximum Number of Minimal Connected Vertex Covers in Graphs

Golovach

Heggernes

Kratsch

2016

Lecture Notes in Computer Science

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“…the runtime is proportional to the size of the output. These algorithms stem from the Tsukiyama et al [41] algorithm, which has a running time of O(|V | |E|µ), where µ is the number of maximal cliques. Other output sensitive algorithms include [8,25,21,31], with [31] providing one of the best theoretical guarantees.…”

Section: Related Workmentioning

confidence: 99%

“…While our algorithm maybe more broadly applicable, in this work we focus our implementation on the widely used MapReduce [10,11,17] framework for cluster computing. While MCE is widely studied in the sequential setting [4,5,8,25,13,23,21,31,40,41], there is relatively less work on parallel methods [45,12,38,43,30].…”

Section: Introductionmentioning

confidence: 99%

Mining maximal cliques from a large graph using MapReduce: Tackling highly uneven subproblem sizes

Svendsen

Mukherjee

Tirthapura

2015

Journal of Parallel and Distributed Computing

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We consider Maximal Clique Enumeration (MCE) from a large graph. A maximal clique is perhaps the most fundamental dense substructure in a graph, and MCE is an important tool to discover densely connected subgraphs, with numerous applications to data mining on web graphs, social networks, and biological networks. While effective sequential methods for MCE are known, scalable parallel methods for MCE are still lacking.We present a new parallel algorithm for MCE, Parallel Enumeration of Cliques using Ordering (PECO" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: normal; font-size: 14.4px; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; maxwidth: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">PECO), designed for the MapReduce framework. Unlike previous works, which required a post-processing step to remove duplicate and non-maximal cliques, PECO" role="presentation" style="boxsizing: border-box; display: inline-block; line-height: normal; font-size: 14.4px; word-spacing: normal; wordwrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; minwidth: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">PECOenumerates only maximal cliques with no duplicates. The key technical ingredient is a total ordering of the vertices of the graph which is used in a novel way to achieve a load balanced distribution of work, and to eliminate redundant work among processors. We implemented PECO" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: normal; font-size: 14.4px; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">PECO on Hadoop MapReduce, and our experiments on a cluster show that the algorithm can effectively process a variety of large real-world graphs with millions of vertices and tens of millions of maximal cliques, and scales well with the degree of available parallelism. KeywordsGraph mining, Maximal clique enumeration, Enumeration algorithm, MapReduce, Hadoop, Parallel algorithm, Clique, Load balancing Disciplines Electrical and Computer EngineeringComments This is a manuscript of an article from Svendsen, Michael, Arko Provo Mukherjee, and Srikanta Tirthapura. "Mining maximal cliques from a large graph using mapreduce: Tackling highly uneven subproblem sizes. h i g h l i g h t s• Scalable method for enumerating maximal cliques in a graph using MapReduce.• Effective solution to load balancing.• Experimental evaluation of our solution on large real world graphs.• Outperforms previous MapReduce solutions by orders of magnitude. a r t i c l e i n f o b s t r a c tWe consider Maximal Clique Enumeration (MCE) from a large graph. A maximal clique is perhaps the most fundamental dense substru...

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“…Tsukiyama et al [5] show that we can enumerate a family of maximal independent sets of a general connected graph in O(nmm(G)) time; here n is the number of vertices, m is the number of edges and m(G) is the number of maximal independent sets of a graph G. Ortiz and Villanueva [4] show that m(C(P k )), the number of maximal independent sets of a (usual) caterpillar graph C(P k ) is the same as m(G k ), the number of maximal independent sets of its contraction graph, G k . They give an algorithm to find a family of maximal independent sets of a caterpillar graph in time polynomial in the number of maximal independent sets.…”

Section: Introductionmentioning

confidence: 99%

Maximal independent sets in a generalisation of caterpillar graph

Neethi¹,

Saxena²

2015

J Comb Optim

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A caterpillar graph is a tree which on removal of all its pendant vertices leaves a chordless path. The chordless path is called the backbone of the graph. The edges from the backbone to the pendant vertices are called the hairs of the caterpillar graph. Ortiz and Villanueva (C.Ortiz and M.Villanueva, Discrete Applied Mathematics, 160(3): 259-266, 2012) describe an algorithm, linear in the size of the output, for finding a family of maximal independent sets in a caterpillar graph.In this paper, we propose an algorithm, again linear in the output size, for a generalised caterpillar graph, where at each vertex of the backbone, there can be any number of hairs of length one and at most one hair of length two.

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A New Algorithm for Generating All the Maximal Independent Sets

Cited by 530 publications

References 7 publications

Enumeration and Maximum Number of Minimal Connected Vertex Covers in Graphs

Enumeration and Maximum Number of Minimal Connected Vertex Covers in Graphs

Mining maximal cliques from a large graph using MapReduce: Tackling highly uneven subproblem sizes

Maximal independent sets in a generalisation of caterpillar graph

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