Solving the Multiobjective Quasi-Clique Problem

TL;DR

The paper tackles the Multiobjective Quasi-Clique (MOQC) problem, proposing to maximize density and vertex count simultaneously rather than fixing one parameter a priori. It establishes theoretical links between MOQC, MQC, DKS, and MOS, and develops three exact solution strategies (Baseline, Two-phase, Three-phase) that exploit ε-constraint decompositions, dichotomic weighted-sum searches, and vertex-degree-based local searches. Empirical results on real sparse graphs show that the Three-phase approach yields strong performance in running time and effectively generates new efficient quasi-cliques, while the mapping from MOS to MOQC provides a practical route to obtain the MOQC nondominated frontier. The work advances a rigorous multiobjective framework for quasi-cliques with clear pathways for future work, including connectedness guarantees and additional objectives such as connectivity.

Abstract

Given a simple undirected graph $G$, a quasi-clique is a subgraph of $G$ whose density is at least $γ$ $(0 < γ\leq 1)$. 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 $\varepsilon$-constraint scalarization and introduce a Two-phase strategy, which applies a dichotomic search based on weighted sum scalarization in the first phase and an $\varepsilon$-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 $\varepsilon$-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.

Figures (2)

• Figure 1: Illustration of the (weakly) nondominated sets of problems MOQC and MOS, respectively, for the graph $G$ shown in (a).
• Figure 2: Weakly-nondominated points in $\mathcal{\widehat{Z}}^V_G$ identified by the Three-phase strategy for Harward500. $(a)$ Entire weakly-nondominated set. $(b)$ Subset of points ranging from $(210, -21)$ to $(821, -95)$, and $(c)$ from $(1945, -410)$ to $(2043, -500)$, highlighting extreme-supported points (solid circles), and weakly-nondominated points identified by minD and maxD (open circles) and by $\varepsilon$-constraint (crossed circles).

Theorems & Definitions (31)

• Definition 1: MQC problem
• Definition 2: DKS problem
• Definition 3: MOQC problem
• Definition 4: MOS problem
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof