Efficient maximum clique computation and enumeration over large sparse graphs

Chang, Lijun

doi:10.1007/s00778-020-00602-z

Cited by 21 publications

(19 citation statements)

References 52 publications

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“…Definition 3.2: (Maximal Balanced Clique) Given a signed network G = (V, E + , E − ), a maximal balanced clique C is a maximal subgraph of G that satisfies the following constraints: [26], [27], our MBCS problem is also NP-Hard.…”

Section: Problem Statementmentioning

confidence: 99%

“…Maximum clique search problem is a fundamental and hot research topic in graph analysis. In the literature, numerous studies have been conducted, such as maximum clique search [26], [27], maximum quasiclique search [28], maximum bi-clique search [29], k*-partite clique with maximum edges [30], clique with maximum edge/vertex weight on weighted graph [31], [32]. Motivated by this, we aim to devise a maximum balanced clique search algorithm to find out the balanced clique with maximum vertex size, which can scale to large-scale real signed networks (with more than 100 million edges).…”

Section: Introductionmentioning

confidence: 99%

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Balanced Clique Computation in Signed Networks: Concepts and Algorithms

Chen¹,

Yuan²,

Lin³

et al. 2022

Preprint

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Clique is one of the most fundamental models for cohesive subgraph mining in network analysis. Existing clique model mainly focuses on unsigned networks. However, in real world, many applications are modeled as signed networks with positive and negative edges. As the signed networks hold their own properties different from the unsigned networks, the existing clique model is inapplicable for the signed networks. Motivated by this, we propose the balanced clique model that considers the most fundamental and dominant theory, structural balance theory, for signed networks. Following the balanced clique model, we study the maximal balanced clique enumeration problem (MBCE) which computes all the maximal balanced cliques in a given signed network. Moreover, in some applications, users prefer a unique and representative balanced clique with maximum size rather than all balanced cliques. Thus, we also study the maximum balanced clique search problem (MBCS) which computes the balanced clique with maximum size. We show that MBCE problem and MBCS problem are both NP-Hard. For the MBCE problem, a straightforward solution is to treat the signed network as two unsigned networks and leverage the off-the-shelf techniques for unsigned networks. However, such a solution is inefficient for large signed networks. To address this problem, in this paper, we first propose a new maximal balanced clique enumeration algorithm by exploiting the unique properties of signed networks. Based on the new proposed algorithm, we devise two optimization strategies to further improve the efficiency of the enumeration. For the MBCS problem, we first propose a baseline solution. To overcome the huge search space problem of the baseline solution, we propose a new search framework based on search space partition. To further improve the efficiency of the new framework, we propose multiple optimization strategies regarding to redundant search branches and invalid candidates. We conduct extensive experiments on large real datasets. The experimental results demonstrate the efficiency, effectiveness and scalability of our proposed algorithms for MBCE problem and MBCS problem.

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Section: Problem Statementmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Balanced Clique Computation in Signed Networks: Concepts and Algorithms

Chen¹,

Yuan²,

Lin³

et al. 2022

Preprint

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“…Chang [40,41] notes that even though a lot of real-world networks are usually sparse, MC has been more extensively studied for dense instances. Thus, the authors propose a branch-and-reduce algorithm that leverages the existing work on MC for dense instances by transforming an instance of MC over a sparse graph to instances of k-clique finding (KCF) over dense subgraphs.…”

Section: Finding and Enumerating Maximum Cliquesmentioning

confidence: 99%

Recent Advances in Practical Data Reduction

Abu-Khzam¹,

Lamm²,

Mnich³

et al. 2020

Preprint

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Over the last two decades, significant advances have been made in the design and analysis of fixedparameter algorithms for a wide variety of graph-theoretic problems. This has resulted in an algorithmic toolbox that is by now well-established. However, these theoretical algorithmic ideas have received very little attention from the practical perspective. We survey recent trends in data reduction engineering results for selected problems. Moreover, we describe concrete techniques that may be useful for future implementations in the area and give open problems and research questions.2 Recent Advances for NP-Hard Problems Maximum Independent Set and Minimum Vertex CoverGiven an undirected graph G = (V, E), the goal of the maximum independent set (MIS) problem is to compute a set of vertices I ⊆ V such that (1) no two vertices in I are adjacent to one another, (2) the set I has maximum cardinality among all such sets. The complement of an independent set I is called a vertex cover V \ I. The MIS problem and the complementary problem of finding a minimum vertex cover (MVC) are well-studied NP-hard optimization problems [75] that attract both researchers and practitioners alike. Furthermore, there is no polynomial time approximation algorithm for MIS that can provide an O(n 1−ε ) guarantee for any constant ε > 0, unless P=NP [174]. Finally, MIS is W [1]-hard [58] when parameterized by solution size k. This makes it unlikely that the problem is fixed-parameter tractable in k [58]. On the other hand, MVC is fixed-parameter tractable in solution size k [58]. Exact ApproachesIn recent years, the bridge between theoretically efficient algorithms and their practical applicability has been significantly reduced. In particular, the branch-and-reduce paradigm, i.e., branching algorithms that use a wide variety of reduction rules, have been (1) shown to achieve theoretical running times that are among the best for both MIS and MVC [71,170], and (2) are able to solve large real-world networks in practice [5]. However, most often the approaches used in practice only use a small subset of the reduction rules that have been proposed to achieve good theoretical running times.Abu-Khzam et al.[4] introduced and analyzed the crown reduction rule (and the usage of data reduction rules in this context in practice). Even though the crown rule is not as powerful as the linear programming (LP)-based rule [135] when considering the worst-case size of the resulting kernel, they experimentally verified that it often performs as well as the LP-based rule and is significantly faster in many cases. Furthermore, they show that the LP-based rule is most useful for fairly sparse graphs and should be avoided for dense graphs, as it yields little to no reduction in size.Later, Akiba and Iwata [5] were the first to show the practicality of the branch-and-reduce paradigm for MVC (and MIS) compared to other state-of-the-art approaches like branch-and-bound and branch-and-cut. Their algorithm uses a wide spectrum of reduction rules that form the foundation...

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“…The set S(u i ) are used as search area to find the maximum clique including u i in [36]- [38]. If the vertices are sorted in the reverse degeneracy ordering computed by the Core-matrix algorithm, then N K(u i ) = S(u i ).…”

Section: Basic Conceptsmentioning

confidence: 99%

NK-MaxClique and MMCQ: Tow New Exact Branch and Bound Algorithms for the Maximum Clique Problem

Mohammadi

Kadivar

2020

IEEE Access

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The maximum clique problem (MCP) is a fundamental problem in combinatorial optimization which finds important applications in real-word. This paper describes two new efficient branch-and-bound maximum clique algorithms NK-MaxClique and MMCQ, designed for solving MCP. We define some pruning conditions based on core numbers and vertex ordering to efficiently remove many of the search space. With respect to this ordering, the algorithms consider the vertices respectively to find the corresponding maximum clique in subproblems. Simulation results demonstrate that the algorithms outperform the previous wellknown algorithms for many instances when applied to DIMACS benchmark and random graphs.

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Efficient maximum clique computation and enumeration over large sparse graphs

Cited by 21 publications

References 52 publications

Balanced Clique Computation in Signed Networks: Concepts and Algorithms

Balanced Clique Computation in Signed Networks: Concepts and Algorithms

Recent Advances in Practical Data Reduction

NK-MaxClique and MMCQ: Tow New Exact Branch and Bound Algorithms for the Maximum Clique Problem

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