Variable Neighborhood Search for Partitioning Sparse Biological Networks into the Maximum Edge-Weighted $k$k-Plexes

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 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 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 and enables a gradual increase of the function's value, when moving from slightly infeasible to barely feasible solutions. Experimental computation is performed on real metabolic networks and other benchmark instances from the literature. Comparing to the previously proposed integer linear programming (ILP), VNS 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-scale dataset not considered up to now.