Subnetwork Constraints for Tighter Upper Bounds and Exact Solution of the Clique Partitioning Problem

TL;DR

This paper introduces a novel subnetwork constraint method to compute tighter upper bounds and efficiently solve the NP-hard clique partitioning problem in weighted graphs, improving convergence speed.

Contribution

The paper presents a new subnetwork constraint approach combined with branch-and-bound for exact solutions, outperforming existing methods in speed and bound tightness.

Findings

Tighter upper bounds achieved on diverse graph instances

Significant convergence speed improvements observed

Effective in both random and real-world networks

Abstract

We consider a variant of the clustering problem for a complete weighted graph. The aim is to partition the nodes into clusters maximizing the sum of the edge weights within the clusters. This problem is known as the clique partitioning problem, being NP-hard in the general case of having edge weights of different signs. We propose a new method of estimating an upper bound of the objective function that we combine with the classical branch-and-bound technique to find the exact solution. We evaluate our approach on a broad range of random graphs and real-world networks. The proposed approach provided tighter upper bounds and achieved significant convergence speed improvements compared to known alternative methods.