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

Authors:
Etienne de Klerk;Dmitrii V. Pasechnik;Alexander Schrijver
Affiliations:
Tilburg University, Department of Econometrics and Operations Research, Faculty of Economics and Business Administration, P.O. Box 90153, 5000, LE Tilburg, The Netherlands;Tilburg University, Department of Econometrics and Operations Research, Faculty of Economics and Business Administration, P.O. Box 90153, 5000, LE Tilburg, The Netherlands;CWI and University of Amsterdam. CWI, P.O. Box 90153, Kruislaan 413, 1098, Amsterdam, SJ, The Netherlands
Venue:
Mathematical Programming: Series A and B
Year:
2007

Citing 0
Cited 11

On the Lovász δ-number of almost regular graphs with application to Erdős-Rényi graphs

European Journal of Combinatorics
Expressing combinatorial problems by systems of polynomial equations and hilbert's nullstellensatz

Combinatorics, Probability and Computing
On crossing numbers of geometric proximity graphs

Computational Geometry: Theory and Applications
An iterative scheme for valid polynomial inequality generation in binary polynomial programming

IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
Think co(mpletely)positive ! Matrix properties, examples and a clustered bibliography on copositive optimization

Journal of Global Optimization
The 2-page crossing number of Kn

Proceedings of the twenty-eighth annual symposium on Computational geometry
A branch-and-cut approach to the crossing number problem

Discrete Optimization
Using symmetry to optimize over the sherali-adams relaxation

ISCO'12 Proceedings of the Second international conference on Combinatorial Optimization
Zarankiewicz's Conjecture is finite for each fixed m

Journal of Combinatorial Theory Series B
Exploiting Symmetries in SDP-Relaxations for Polynomial Optimization

Mathematics of Operations Research
Book drawings of complete bipartite graphs

Discrete Applied Mathematics

Quantified Score

Hi-index	0.00

Visualization

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.