Ensuring connectedness for the Maximum Quasi-clique and Densest $k$-subgraph problems

TL;DR

This paper introduces flow-based connectedness constraints for MILP formulations to ensure connected quasi-cliques and densest subgraphs, improving solution relevance for real-world applications.

Contribution

It proposes novel flow-based constraints integrated into MILP models to guarantee connectedness in solutions for MQC and DKS problems.

Findings

Constraints are competitive in solving time.

Enhanced models solve more connected instances.

Results show practical applicability of the approach.

Abstract

Given an undirected graph $G$, a quasi-clique is a subgraph of $G$ whose density is at least $\gamma$ $(0 < \gamma \leq 1)$. Two optimization problems can be defined for quasi-cliques: the Maximum Quasi-Clique (MQC) Problem, which finds a quasi-clique with maximum vertex cardinality, and the Densest $k$-Subgraph (DKS) Problem, which finds the densest subgraph given a fixed cardinality constraint. Most existing approaches to solve both problems often disregard the requirement of connectedness, which may lead to solutions containing isolated components that are meaningless for many real-life applications. To address this issue, we propose two flow-based connectedness constraints to be integrated into known Mixed-Integer Linear Programming (MILP) formulations for either MQC or DKS problems. We compare the performance of MILP formulations enhanced with our connectedness constraints in terms of both running time and number of solved instances against existing approaches that ensure quasi-clique connectedness. Experimental results demonstrate that our constraints are quite competitive, making them valuable for practical applications requiring connectedness.