An efficient iterated local search for the minimum quasi-clique partitioning problem

Abstract

Given a simple undirected graph $G$ and a real constant $\gamma \in (0,1]$, a $\gamma$-quasi-clique is defined as a subset of vertices that induces a subgraph with an edge density of at least $\gamma$. The minimum quasi-clique partitioning problem (MQCPP) seeks to identify the minimum cardinality of $\gamma$-quasi-clique partitions in $G$. This work presents an efficient iterated local search (ILS) method to address MQCPP by using a two-phase local search procedure for local improvement, and a greedy-based perturbation procedure for diversifying the search process. An evaluation function that records the number of intra edges of each quasi-clique is used for neighboring solution evaluation, and a fast incremental evaluation technique is employed to speed up the evaluation. Numerical results on three sets of 321 benchmark instances demonstrate the superior performance of the proposed algorithm compared with state-of-the-art approaches. Specifically, ILS reports 153 (47.7%) new upper bounds and fails to reach the best known solution for only 2 instances. Additional analysis experiments are conducted to evaluate the effects of the key components of ILS, including the two-phase local search procedure, the greedy-based perturbation procedure, and the fast incremental evaluation technique.