  Given a simple undirected graph $G$, a quasi-clique is a subgraph of $G$
whose density is at least $\gamma$ $(0 < \gamma \leq 1)$. 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.
