On the maximum quasi-clique problem

Pattillo, Jeffrey; Veremyev, Alexander; Butenko, Sergiy; Boginski, Vladimir

doi:10.1016/j.dam.2012.07.019

Cited by 98 publications

(95 citation statements)

References 16 publications

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“…Let S be an average s-plex of size k in a graph G. Then, there is some γ ≤ 1 such that G[S] is a γ-quasi-clique and γ · (k − 1) ≥ k − s. Now, since the γ-quasi-clique property is quasi-hereditary [8,49], there is a vertex v ∈ S such that S \ {v} a γ-quasi-clique. That is, G[S \ {v}] has average degree at least γ · (k − 2).…”

Section: Propositionmentioning

confidence: 99%

“…In fact, the property is hereditary only for the two extreme cases γ = 1 (when only cliques fulfill the property) and γ = 0 (when every vertex set fulfills the property). The property is, however, quasi-hereditary for every γ ∈ [0, 1]: removing a vertex of minimum degree from a γ-quasi-clique results in another γ-quasi-clique [8,49]. As for the other clique relaxations, we are interested in finding large γ-quasi-cliques.…”

Section: Definitionmentioning

confidence: 99%

“…Input: An undirected graph G and an integer k. The γ-QUASI-CLIQUE problem is NP-hard for all γ ∈ (0, 1] [49]. Since the γ-quasi-clique property is quasi-hereditary, one may solve γ-QUASI-CLIQUE by deciding whether G has a γ-quasi-clique of size exactly k. Consequently, γ-QUASI-CLIQUE can be directly reduced to DENSE-k-SUBGRAPH, where we aim to find a size-k set S ⊆ V such that G[S] has a maximum number of edges [50].…”

Section: γ-Quasi-cliquementioning

confidence: 99%

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Multivariate Algorithmics for Finding Cohesive Subnetworks

Komusiewicz

2016

Algorithms

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Community detection is an important task in the analysis of biological, social or technical networks. We survey different models of cohesive graphs, commonly referred to as clique relaxations, that are used in the detection of network communities. For each clique relaxation, we give an overview of basic model properties and of the complexity of the problem of finding large cohesive subgraphs under this model. Since this problem is usually NP-hard, we focus on combinatorial fixed-parameter algorithms exploiting typical structural properties of input networks.

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Section: Propositionmentioning

confidence: 99%

Section: Definitionmentioning

confidence: 99%

Section: γ-Quasi-cliquementioning

confidence: 99%

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Multivariate Algorithmics for Finding Cohesive Subnetworks

Komusiewicz

2016

Algorithms

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“…Specifically, a λ-quasi-clique is defined as a connected subgraph in which the ratio between the degree of each node and the highest possible degree is at least λ, where λ ∈ (0, 1]. There has been some prior work on finding quasi-cliques from graphs [5,16,17]. However, none of them can handle the quasi-clique search problem, which is not studied before, and has many applications in the real world (see below).…”

Section: Introductionmentioning

confidence: 99%

“…Since the maximum clique problem is NP-Hard [15] and no polynomial time algorithm can approximate it within a factor of n 1− ( > 0) [8], it is not surprising that finding the maximum quasi-clique is also NPHard and not approximable in polynomial time. Existing studies on quasi-clique maximization (without any query) are mainly based on local search [9,5,16], in which a solution moves to the best neighboring solution iteratively, updated node-by-node. However, none of the existing approaches are designed for quasi-clique search and can efficiently handle the QMQ problem.…”

Section: Introductionmentioning

confidence: 99%

Query-Driven Maximum Quasi-Clique Search

Lee¹,

Lakshmanan²

2016

Proceedings of the 2016 SIAM International Conference on Data Mining

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Quasi-cliques are an elegant way to model dense subgraphs, with each node adjacent to at least a fraction λ ∈ (0, 1] of other nodes in the subgraph. In this paper, we focus on a new graph mining problem, the query-driven maximum quasi-clique (QMQ) search, which aims to find the largest λ-quasi-clique containing a given query node set S. This problem has many applications and is proved to be NP-Hard and inapproximable. To solve the problem efficiently in practice, we propose the notion of core tree to organize dense subgraphs recursively, which reduces the search space and effectively helps find the solution within a few tree traversals. To optimize a solution to a better solution, we introduce three refinement operations: Add, Remove and Swap. We propose two iterative maximization algorithms, DIM and SUM, to approach QMQ by deterministic and stochastic means respectively. With extensive experiments on three real datasets, we demonstrate that our algorithms significantly outperform several baselines in running time and the quality.

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On maximum degree‐based ‐quasi‐clique problem: Complexity and exact approaches

et al. 2017

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We consider the problem of finding a degree‐based γ‐quasi‐clique of maximum cardinality in a given graph for some fixed γ∈(0,1]. A degree‐based γ‐quasi‐clique (often referred to as simply a quasi‐clique) is a subgraph, where the degree of each vertex is at least γ times the maximum possible degree of a vertex in the subgraph. A degree‐based γ‐quasi‐clique is a relative clique relaxation model, where the case of γ=1 corresponds to the well‐known concept of a clique. In this article, we first prove that the problem is NP‐hard for any fixed γ∈(0,1], which addresses one of the open questions in the literature. More importantly, we also develop new exact solution methods for solving the problem and demonstrate their advantages and limitations in extensive computational experiments with both random and real‐world networks. Finally, we outline promising directions of future research including possible functional generalizations of the considered clique relaxation model. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 71(2), 136–152 2018

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On the maximum quasi-clique problem

Abstract: 2016-12-23T18:52:10

Cited by 98 publications

References 16 publications

Multivariate Algorithmics for Finding Cohesive Subnetworks

Multivariate Algorithmics for Finding Cohesive Subnetworks

Query-Driven Maximum Quasi-Clique Search

On maximum degree‐based ‐quasi‐clique problem: Complexity and exact approaches

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