Solving the maximum edge-weight clique problem in sparse graphs with compact formulations

Abstract

This paper studies the behavior of compact formulations for solving the maximum edge-weight clique (MEWC) problem in sparse graphs. The MEWC problem has long been discussed in the literature, but mostly addressing complete graphs, with or without a cardinality constraint on the clique. Yet, several real-world applications are defined on sparse graphs, where the missing edges are due to some threshold process or because they are not even supposed to be in the graph, at all. Such situations often arise in cell’s metabolic networks, where the amount of metabolites shared among reactions is an important issue to understand the cell’s prevalent elements. We propose new node-discretized formulations for the problem, which are more compact than other models known from the literature. Computational experiments on benchmark and real-world instances are conducted for discussing and comparing the models. These tests indicate that the node-discretized formulations are more efficient for solving large size sparse graphs. Additionally, we also address a new variant of the MEWC problem where the objective to be maximized includes the neighboring edges of the clique.