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

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.