EURO Journal on Computational Optimization

, Volume 3, Issue 1, pp 1–30 | Cite as

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

  • Luis Gouveia
  • Pedro MartinsEmail author
Original Paper


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.


Maximum edge-weight clique problem Clique’s edge neighborhood Integer formulations Sparse graphs 

Mathematics Subject Classification

90C10 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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Copyright information

© EURO - The Association of European Operational Research Societies 2014

Authors and Affiliations

  1. 1.DEIO, Faculty of SciencesUniversity of LisbonLisbonPortugal
  2. 2.Faculty of Sciences, Operations Research Center (CIO)University of LisbonLisbonPortugal
  3. 3.ISCAC, Polytechnic Institute of Coimbra, Quinta Agrícola, BencantaCoimbraPortugal

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