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.

翻译：给定一个简单无向图$G$，拟团是$G$的一个密度至少为$γ$ $(0 < γ\leq 1)$的子图。寻找最大拟团的研究存在两种不同视角：$i)$ 在给定边密度条件下最大化顶点数量；$ii)$ 在给定顶点数量条件下最大化边密度。然而，当缺乏关于顶点数量和密度的先验偏好信息时，更自然的方法是从多目标角度考虑该问题。本文提出多目标拟团问题（MOQC），其目标是通过同时最大化顶点数量和边密度来寻找拟团。为高效求解该问题，我们深入探讨了MOQC与其单目标对应问题、双目标优化问题之间的关联，并分析了MOQC问题及拟团的若干性质。我们提出基于$\varepsilon$-约束标量化的基线方法，并设计了一种两阶段策略：第一阶段采用基于加权和标量化的二分搜索，第二阶段应用$\varepsilon$-约束方法。此外，我们提出三阶段策略：首先采用两阶段策略中的二分搜索，随后结合基于顶点度的局部搜索（运用新提出的充分条件评估拟团效率），最终阶段实施$\varepsilon$-约束方法。在真实世界稀疏图上的实验结果表明，二分搜索与局部搜索的集成使用，结合拟团效率评估机制，使得三阶段策略在运行时间和生成新高效拟团的能力方面，成为求解MOQC问题的有效方法。