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The semidefinite relaxation of the $$k$$-partition polytope is strong. (English) Zbl 1049.90136
Cook, William J. (ed.) et al., Integer programming and combinatorial optimization. 9th international IPCO conference, Cambridge, MA, USA, May 27–29, 2002. Proceedings. Berlin: Springer (ISBN 3-540-43676-6). Lect. Notes Comput. Sci. 2337, 273-290 (2002).
Summary: 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 $$\sqrt{2} - 1$$ or any of the (exponentially many) facet-defining clique constraints by $$\frac{1}{2}$$ or more.
For the entire collection see [Zbl 0992.00060].

##### MSC:
 90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut 90C22 Semidefinite programming 90B18 Communication networks in operations research
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