Mixed Integer Programming for Searching Maximum Quasi-Bicliques

Abstract

This paper is related to the problem of finding the maximal quasi-bicliques in a bipartite graph (bigraph). A quasi-biclique in the bigraph is its "almost" complete subgraph. The relaxation of completeness can be understood variously; here, we assume that the subgraph is a γ-quasi-biclique if it lacks a certain number of edges to form a biclique such that its density is at least γ ∈ (0, 1]. For a bigraph and fixed γ, the problem of searching for the maximal quasi-biclique consists of finding a subset of vertices of the bigraph such that the induced subgraph is a quasi-biclique and its size is maximal for a given graph. Several models based on Mixed Integer Programming (MIP) to search for a quasi-biclique are proposed and tested for working efficiency. An alternative model inspired by biclustering is formulated and tested; this model simultaneously maximizes both the size of the quasi-biclique and its density, using the least-square criterion similar to the one exploited by triclustering T B .

Cohesive subgraph mining in large graphs

“…Recently, many studies have been conducted on finding cohesive subgraphs on bipartite graphs, including bicliques [16,17,28,[41][42][43][44], (𝛼, 𝛽)-cores [45], quasi-bicliques [26,[46][47][48],…”

“…and ( 2) 𝛾-quasi-biclique [47,53] is a bipartite graph 𝐺 (𝐿, 𝑅, 𝐸) that can miss at most ). Existing works of quasi-bicliques focus on finding subgraphs with a certain density and degree [54,55].…”

“…The maximum quasi-biclique search problem aims to find a 𝛾-quasi-biclique or 𝛿-quasibiclique 𝐻 * such that 𝑉 (𝐻 * ) is maximized, which is a NP-hard problem [26,48]. [47,48] use mixed integer programming to find a maximum 𝛾-quasi-biclique exactly, which cannot handle large datasets. [26,46] propose greedy algorithms to find an approximate maximum 𝛿-quasi-biclique, which cannot be adapted to find the MaxBP exactly.…”

“…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.…”

“…Therefore, it is hard to adapt these algorithms to improve existing methods for finding all MQCs. Second, QC has also been introduced to bipartite graphs [93,94], temporal graphs [95,96] and directed graphs [40]. Specifically, authors in [93,94] define quasi-biclique which is a counterpart of QC in bipartite graphs.…”