Abstract
This paper proposes an exact algorithm for the Max-Mean dispersion problem (\(Max-Mean DP\)), an NP-Hard combinatorial optimization problem whose aim is to select the subset of a set such that the average distance between elements is maximized. The problem admits a natural non-convex quadratic fractional formulation from which a semidefinite programming (SDP) relaxation can be derived. This relaxation can be tightened by means of a cutting plane algorithm which iteratively adds the most violated triangular inequalities. The proposed approach embeds the SDP relaxation and the cutting plane algorithm into a branch and bound framework to solve \(Max-Mean DP\) instances to optimality. Computational experiments show that the proposed method is able to solve to optimality in reasonable time instances with up to 100 elements, outperforming other alternative approaches.
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Acknowledgments
The authors want to thank Brian Borchers from New Mexico Tech and Christoph Helmberg from Chemnitz University of Technology for their helpful suggestions on the use of their solvers (CSDP and ConicBundle, respectively).
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Garraffa, M., Della Croce, F. & Salassa, F. An exact semidefinite programming approach for the max-mean dispersion problem. J Comb Optim 34, 71–93 (2017). https://doi.org/10.1007/s10878-016-0065-1
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DOI: https://doi.org/10.1007/s10878-016-0065-1