Variable neighborhood search for partitioning sparse biological networks into the maximum edge-weighted $k$-plexes

TL;DR

This paper introduces a variable neighborhood search algorithm for partitioning biological networks into maximum edge-weighted $k$-plexes, effectively solving the Max-EkPP problem and outperforming existing methods on real and benchmark datasets.

Contribution

The paper presents a novel VNS algorithm for Max-EkPP that improves solution quality and computational efficiency over existing approaches, including large-scale networks.

Findings

Successfully finds all known optimal solutions on benchmark instances.

Achieves better or equal solutions compared to previous methods.

Effectively handles large-scale biological network data.

Abstract

In a network, a $k$-plex represents a subset of $n$ vertices where the degree of each vertex in the subnetwork induced by this subset is at least $n-k$. The maximum edge-weight $k$-plex partitioning problem (Max-EkPP) is to find the $k$-plex partitioning in edge-weighted network, such that the sum of edge weights is maximal. The Max-EkPP has an important role in discovering new information in large sparse biological networks. We propose a variable neighborhood search (VNS) algorithm for solving Max-EkPP. The VNS implements a local search based on the 1-swap first improvement strategy and the objective function that takes into account the degree of every vertex in each partition. The objective function favors feasible solutions, also enabling a gradual increase in terms of objective function value when moving from slightly infeasible to barely feasible solutions. A comprehensive experimental computation is performed on real metabolic networks and other benchmark instances from literature. Comparing to the integer linear programming method from literature, our approach succeeds to find all known optimal solutions. For all other instances, the VNS either reaches previous best known solution or improves it. The proposed VNS is also tested on a large-scaled dataset which was not previously considered in literature.