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# Reduction of symmetric semidefinite programs using the regular representation.

 dc.contributor.author Klerk, Etienne de. dc.contributor.author Pasechnik, Dmitrii V. dc.contributor.author Schrijver, Alexander. dc.date.accessioned 2012-03-09T00:44:11Z dc.date.available 2012-03-09T00:44:11Z dc.date.copyright 2006 dc.date.issued 2012-03-09 dc.identifier.citation Klerk, E. d., Pasechnik, D. V. & Schrijver, A. (2006). Reduction of symmetric semidefinite programs using the regular representation. Mathematical Programming, 109, 613-624. dc.identifier.uri http://hdl.handle.net/10220/7625 dc.description.abstract We consider semidefinite programming problems on which a permutation group is acting.We describe a general technique to reduce the size of such problems, exploiting the symmetry. The technique is based on a low-order matrix ∗-representation of the commutant (centralizer ring) of the matrix algebra generated by the permutation matrices.We apply it to extending amethod of de Klerk et al. that gives a semidefinite programming lower bound to the crossing number of complete bipartite graphs. It implies that cr(K8,n) ≥ 2.9299n2−6n, cr(K9,n) ≥ 3.8676n2 − 8n, and (for any m ≥ 9) lim n→∞ cr(Km,n)/Z(m, n) ≥ 0.8594 m/m − 1, where Z(m,n) is the Zarankiewicz number [1/4(m-1)2][1/4(n-1)2], which is the conjectured value of cr(K m,n ). Here the best factor previously known was 0.8303 instead of 0.8594. dc.format.extent 11 p. dc.language.iso en dc.relation.ispartofseries Mathematical programming dc.rights © 2006 Springer-Verlag. This is the author created version of a work that has been peer reviewed and accepted for publication by Mathematical Programming, Springer-Verlag. It incorporates referee’s comments but changes resulting from the publishing process, such as copyediting, structural formatting, may not be reflected in this document. The published version is available at: [DOI: http://dx.doi.org/10.1007/s10107-006-0039-7 ]. dc.subject DRNTU::Science::Mathematics. dc.title Reduction of symmetric semidefinite programs using the regular representation. dc.type Journal Article dc.contributor.school School of Physical and Mathematical Sciences dc.identifier.doi http://dx.doi.org/10.1007/s10107-006-0039-7 dc.description.version Accepted version

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