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The Semidefinite Relaxation of the k-Partition Polytope Is Strong

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Integer Programming and Combinatorial Optimization (IPCO 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2337))

Abstract

Radio frequency bandwidth has become a very scarce resource. This holds true in particular for the popular mobile communication system GSM. Carefully planning the use of the available frequencies is thus of great importance to GSM network operators. Heuristic optimization methods for this task are known, which produce frequency plans causing only moderate amounts of disturbing interference in many typical situations. In order to thoroughly assess the quality of the plans, however, lower bounds on the unavoidable interference are in demand. The results obtained so far using linear programming and graph theoretic arguments do not suffice. By far the best lower bounds are currently obtained from semidefinite programming. The link between semidefinite programming and the bound on unavoidable interference in frequency planning is the semidefinite relaxation of the graph minimum k-partition problem.

Here, we take first steps to explain the surprising strength of the semidefinite relaxation. This bases on a study of the solution set of the semidefinite relaxation in relation to the circumscribed k-partition polytope. Our focus is on the huge class of hypermetric inequalities, which are valid and in many cases facet-defining for the k-partition polytope. We show that a “slightly shifted version” of the hypermetric inequalities is implicit to the semidefinite relaxation. In particular, no feasible point for the semidefinite relaxation violates any of the facet-defining triangle inequalities for the k-partition polytope by more than √2 - 1 or any of the (exponentially many) facet-defining clique constraints by1/2 or more.

This article is based on results contained in the Ph. D. thesis of the author.

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Author information

Authors and Affiliations

  1. Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB), Takustr. 7, D-14195, Berlin, Germany

    Andreas Eisenblätter

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  1. Andreas Eisenblätter
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Editors and Affiliations

  1. Program in Applied and Computational Mathematics, University of Princeton, Fine Hall, Washington Road, Princeton, NJ, 08544-1000, USA

    William J. Cook

  2. Sloan School of Management, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA

    Andreas S. Schulz

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Eisenblätter, A. (2002). The Semidefinite Relaxation of the k-Partition Polytope Is Strong. In: Cook, W.J., Schulz, A.S. (eds) Integer Programming and Combinatorial Optimization. IPCO 2002. Lecture Notes in Computer Science, vol 2337. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47867-1_20

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  • DOI: https://doi.org/10.1007/3-540-47867-1_20

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  • Print ISBN: 978-3-540-43676-8

  • Online ISBN: 978-3-540-47867-6

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