Solving the Multiobjective Quasi-Clique Problem

Given a simple undirected graph G, a quasi-clique is a subgraph of G whose density is at least γ (0 < γ≤ 1). Finding a maximum quasi-clique has been addressed from two different perspectives: i) maximizing vertex cardinality for a given edge density; and ii) maximizing edge density for a given vertex cardinality. However, when no a priori preference information about cardinality and density is available, a more natural approach is to consider the problem from a multiobjective perspective. We introduce the Multiobjective Quasi-clique Problem (MOQC), which aims to find a quasi-clique by simultaneously maximizing both vertex cardinality and edge density. To efficiently address this problem, we explore the relationship among MOQC, its single-objective counterpart problems, and a biobjective optimization problem, along with several properties of the MOQC problem and quasi-cliques. We propose a baseline approach using ε-constraint scalarization and introduce a Two-phase strategy, which applies a dichotomic search based on weighted sum scalarization in the first phase and an ε-constraint methodology in the second phase. Additionally, we present a Three-phase strategy that combines the dichotomic search used in Two-phase with a vertex-degree-based local search employing novel sufficient conditions to assess quasi-clique efficiency, followed by an ε-constraint in a final stage. Experimental results on real-world sparse graphs indicate that the integrated use of dichotomic search and local search, together with mechanisms to assess quasi-clique efficiency, makes the Three-phase strategy an effective approach for solving the MOQC problem in terms of running time and ability to produce new efficient quasi-cliques.