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Mathematical Programming

, Volume 109, Issue 2–3, pp 613–624 | Cite as

Reduction of symmetric semidefinite programs using the regular \(\ast\)-representation

  • Etienne de Klerk
  • Dmitrii V. Pasechnik
  • Alexander Schrijver
FULL LENGTH PAPER

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 a method of de Klerk et al. that gives a semidefinite programming lower bound to the crossing number of complete bipartite graphs. It implies that cr(K 8,n ) ≥ 2.9299n 2 − 6n, cr(K 9,n ) ≥ 3.8676n 2 − 8n, and (for any m ≥ 9)
$$\lim_{n\to\infty}\frac{{\rm cr}(K_{m,n})}{Z(m,n)}\geq 0.8594\frac{m}{m-1},$$
where Z(m,n) is the Zarankiewicz number \(\lfloor\frac{1}{4}(m-1)^2\rfloor\lfloor\frac{1}{4}(n-1)^2\rfloor\), which is the conjectured value of cr(K m,n ). Here the best factor previously known was 0.8303 instead of 0.8594.

Keywords

Crossing numbers Complete bipartite graph Semidefinite programming Regular \(*\)-representations 

Mathematics Subject Classification (2000)

90C22 20B40 05C10 

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Etienne de Klerk
    • 1
  • Dmitrii V. Pasechnik
    • 1
  • Alexander Schrijver
    • 2
  1. 1.Department of Econometrics and Operations Research, Faculty of Economics and Business AdministrationTilburg UniversityLE TilburgThe Netherlands
  2. 2.CWI and University of Amsterdam. CWIAmsterdamThe Netherlands

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