Mining Largest Maximal Quasi-Cliques

Abstract

Quasi-cliques are dense incomplete subgraphs of a graph that generalize the notion of cliques. Enumerating quasi-cliques from a graph is a robust way to detect densely connected structures with applications in bioinformatics and social network analysis. However, enumerating quasi-cliques in a graph is a challenging problem, even harder than the problem of enumerating cliques. We consider the enumeration of top- k degree-based quasi-cliques and make the following contributions: (1) we show that even the problem of detecting whether a given quasi-clique is maximal (i.e., not contained within another quasi-clique) is NP-hard. (2) We present a novel heuristic algorithm K ernel QC to enumerate the k largest quasi-cliques in a graph. Our method is based on identifying kernels of extremely dense subgraphs within a graph, followed by growing subgraphs around these kernels, to arrive at quasi-cliques with the required densities. (3) Experimental results show that our algorithm accurately enumerates quasi-cliques from a graph, is much faster than current state-of-the-art methods for quasi-clique enumeration (often more than three orders of magnitude faster), and can scale to larger graphs than current methods.

MIP formulations for induced graph optimization problems: a tutorial

“…Several approaches have been proposed in the literature for detecting maximum and maximal quasi‐cliques. These include heuristics and metaheuristics (Abello et al., 2002; Tsourakakis et al., 2013; Pinto et al., 2018, 2021; Zhou et al., 2020; Peng et al., 2021; Sanei‐Mehri et al., 2021), integer programming (Pattillo et al., 2013; Veremyev et al., 2016; Marinelli et al., 2021), and backtracking (Ribeiro and Riveaux, 2019).…”

A local search algorithm with hybrid strategies for the maximum weighted quasi‐clique problem

“…Cai et al proposed a new MWCP algorithm with a semi-exact approach [18]. Although many studies focused on γ -quasicliques and weighted cliques [19][20][21][22][23], there are rare works to discuss the quasi-clique structures in the weighted graphs. To study the γ -quasiclique identification algorithms in weighted graphs, in this paper, we introduce the definition of the vertex-weighted quasi-clique and the maximum weighted γ -quasi-clique problem and then propose a local search algorithm with hybrid vertex selection strategies to solve the problem.…”

Cohesive subgraph mining in large graphs

“…There are many variants of QC mining which consider various problem settings [33,76,[90][91][92], different types of graphs [40,[93][94][95][96], and different definitions of QC [75][76][77]. In the sequel, we review these studies.…”

“…They also adopt a BB framework while developing some pruning rules based on the query set. Some other studies aim to find the (top-k) largest QC 𝐺 [𝐻] such that |𝐻| is maximized [33,92].…”