Abstract
In this paper we address a metaheuristic for an combinatorial optimization problem. For any given graph \(\mathcal {G}=(V,E)\) (where the nodes represent items and edges correlations), we want to find the clique \(\mathcal {C} \subseteq V\) such that the number of links shared between \(\mathcal {C}\) and \(V - \mathcal {C}\) is maximized. This problem is known in the literature as the Max Cut-Clique (MCC).
The contributions of this paper are three-fold. First, the complexity of the MCC is established, and we offer bounds for the MCC using elementary graph theory. Second, an exact Integer Linear Programming (ILP) formulation for the MCC is offered. Third, a full GRASP/VND methodology enriched with a Tabu Search is here developed, where the main ingredients are novel local searches and a Restricted Candidate List that trades greediness for randomization in a multi-start fashion. A dynamic Tabu list considers a bounding technique based on the previous analysis.
Finally, a fair comparison between our hybrid algorithm and the globally optimum solution using the ILP formulation confirms that the globally optimum solution is found by our heuristic for graphs with hundreds of nodes, but more efficiently in terms of time and memory requirements.
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Notes
- 1.
The dataset can be found in the URL http://steinlib.zib.de/steinlib.php.
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Acknowledgements
This work is partially supported by Project 395 CSIC I+D Sistemas Binarios Estocásticos Dinámicos.
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Bourel, M., Canale, E., Robledo, F., Romero, P., Stábile, L. (2019). Complexity and Heuristics for the Max Cut-Clique Problem. In: Sifaleras, A., Salhi, S., Brimberg, J. (eds) Variable Neighborhood Search. ICVNS 2018. Lecture Notes in Computer Science(), vol 11328. Springer, Cham. https://doi.org/10.1007/978-3-030-15843-9_3
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