Solving the maximum edge-weight clique problem in sparse graphs with compact formulations
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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.
KeywordsMaximum edge-weight clique problem Clique’s edge neighborhood Integer formulations Sparse graphs
Mathematics Subject Classification90C10 90C35 90C90
The authors would like to thank the referees for their comments and suggestions which led to a significantly improved version of the paper. Thanks are also due to the Editor for the suggested observations. This work has been partially supported by the Portuguese National Funding by FCT (project PEst-OE/MAT/UI0152).
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