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

TL;DR

This work tackles the issue of disconnected solutions in Maximum Quasi-Clique and Densest k-Subgraph problems by introducing two flow-based connectedness constraints, C-STree and C-Flow, that can be integrated into existing MILP formulations. C-STree provides a spanning-tree-based, single-commodity flow characterization applicable to MQC and DKS, while C-Flow offers a specialized flow-based approach for DCKS with a known subgraph size $k$. Extensive experiments on real-world sparse graphs show that C-STree generally delivers the best performance for MCQC, particularly when disconnections are prevalent, and C-Flow is highly competitive for DCKS, especially for larger $k$. The results demonstrate that enforcing connectedness via these constraints yields practical, scalable improvements over prior approaches and broadens applicability in domains requiring cohesive, connected clusters.

Abstract

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