New lower bounds on crossing numbers of $K_{m,n}$ from semidefinite programming

TL;DR

This paper develops advanced semidefinite programming techniques leveraging symmetry to derive improved lower bounds on the crossing numbers of complete bipartite graphs, extending previous methods and providing concrete bounds for specific cases.

Contribution

It introduces a novel symmetry-based decomposition method for semidefinite programming, leading to tighter bounds on crossing numbers of $K_{m,n}$ graphs.

Findings

Improved bounds for $ ext{cr}(K_{10,n})$, $ ext{cr}(K_{11,n})$, $ ext{cr}(K_{12,n})$, and $ ext{cr}(K_{13,n})$.

A new relaxation of the SDP bound requiring only one small positive semidefinite matrix block.

Full block-diagonalization of the matrix algebra exploiting problem symmetry.

Abstract

In this paper, we use semidefinite programming and representation theory to compute new lower bounds on the crossing number of the complete bipartite graph $K_{m,n}$, extending a method from de Klerk et al. [SIAM J. Discrete Math. 20 (2006), 189--202] and the subsequent reduction by De Klerk, Pasechnik and Schrijver [Math. Prog. Ser. A and B, 109 (2007) 613--624]. We exploit the full symmetry of the problem using a novel decomposition technique. This results in a full block-diagonalization of the underlying matrix algebra, which we use to improve bounds on several concrete instances. Our results imply that $\text{cr}(K_{10,n}) \geq 4.87057 n^2 - 10n$, $\text{cr}(K_{11,n}) \geq 5.99939 n^2-12.5n$, $\text{cr}(K_{12,n}) \geq 7.25579 n^2 - 15n$, $\text{cr}(K_{13,n}) \geq 8.65675 n^2-18n$ for all $n$. The latter three bounds are computed using a new and well-performing relaxation of the original semidefinite programming bound. This new relaxation is obtained by only requiring one small matrix block to be positive semidefinite.