A BRKGA-based matheuristic for the maximum quasi-clique problem with an exact local search strategy

Abstract

Given a graph $G=(V,E)$ and a threshold $\gamma \in (0,1]$, the maximum cardinality quasi-clique problem consists in finding a maximum cardinality subset $C^*$ of the vertices in $V$ such that the density of the graph induced in $G$ by $C^*$ is greater than or equal to the threshold $\gamma$. This problem has a number of applications in data mining, e.g. in social networks or phone call graphs. We propose a matheuristic for solving the maximum cardinality quasi-clique problem, based on the hybridization of a biased random-key genetic algorithm (BRKGA) with an exact local search strategy. The newly proposed approach is compared with a pure biased random-key genetic algorithm, which was the best heuristic in the literature at the time of writing. Computational results show that the hybrid BRKGA outperforms the pure BRKGA.

NuQClq: An Effective Local Search Algorithm for Maximum Quasi-Clique Problem

“…Another scoring function was used in some versions of BRKGA (Pinto et al 2015(Pinto et al , 2018(Pinto et al , 2019. For example, the BRKGA-IG * algorithm (2018) works as follows: during each iteration, starting from a candidate solution S, it first removes some vertices from S according to the d S value, and then adds some vertices based on the diff information.…”

“…We evaluate NuQClq on a broad range of classic benchmarks as well as sparse instances, and compare it with three state-of-the-art heuristic algorithms. Since previous works use different instances, we select all used instances from (Pinto et al 2019;Djeddi, Haddadene, and Belacel 2019;Zhou, Benlic, and Wu 2020). To be specific, we consider 289 instances, which are mainly divided into two parts: (1) 187 classic instances from DIMACS benchmark (Johnson 1993) 1 and BHOSLIB benchmark (Xu et al 2007) 2 ; (2) 102 sparse instances whose density is from 0.00014% to 3.869% from Florida Sparse Matrix Collection (Davis and Hu 2011) 3 and Stanford Large Network Dataset Collection (Rossi and Ahmed 2015) 4 .…”

“…Oliveira et al (2013) presented a restart iterative greedy algorithm named RIG * . Afterwards, some different versions of biased random-key genetic algorithm (BRKGA) for the MQCP were proposed (Pinto et al 2015(Pinto et al , 2018(Pinto et al , 2019. Among these versions, BRKGA-LSQClique…”

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

“…The experimental results show that the algorithm can find the best known quasi-clique of real-world networks in a reasonable time, and performs better than the existing methods on the DIMACS and BHOSLIB benchmarks. Then, based on the hybrid of a biased random-key genetic algorithm and an exact local search strategy, Pinto et al proposed a method to solve the maximal cardinality quasi-clique problem [10]. Very recently, Chen et al have devised an efficient local search algorithm, called NuQClq, and proposed two novel strategies incorporated into their algorithm [11].…”

“…Recently, algorithms for finding the maximum quasi-cliques have been well studied. A series of algorithms have been proposed including exact algorithms and heuristic methods [4][5][6][7][8][9][10][11]. As an early work, Pajouh et al proposed a branch-and-bound algorithm for the maximum γ -quasi-clique problem [4].…”