A linear programming reformulation of the standard quadratic optimization problem
Journal of Global Optimization
A GAP package for computation with coherent configurations
ICMS'10 Proceedings of the Third international congress conference on Mathematical software
Journal of Global Optimization
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
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It has been long conjectured that the crossing number $\Cr(K_{m,n})$ of the complete bipartite graph $K_{m,n}$ equals the Zarankiewicz number $Z(m,n):= \floor{\frac{m-1}{2}} \floor{\frac{m}{2}} \floor{\frac{n-1}{2}} \floor{\frac{n}{2}}$. Another longstanding conjecture states that the crossing number $\Cr(K_n)$ of the complete graph $K_n$ equals $Z(n):=\frac{1}{4}\smallfloor{\frac{n}{2}} \smallfloor{\frac{n-1}{2}} \smallfloor{\frac{n-2}{2}}\smallfloor{\frac{n-3}{2}}$. In this paper we show the following improved bounds on the asymptotic ratios of these crossing numbers and their conjectured values: \begin{itemize} \item[(i)] for each fixed $m\ge 9$, $\lim_{n\to\infty} \Cr(K_{m,n})/Z(m,n) \ge 0.83m/(m-1)$; \item[(ii)] $\lim_{n\to\infty} \Cr(K_{n,n})/Z(n,n) \ge 0.83$; and \item[(iii)] $\lim_{n\to\infty} \Cr(K_{n})/Z(n) \ge 0.83$. \end{itemize} The previous best known lower bounds were $0.8m/(m-1), 0.8$, and $0.8$, respectively. These improved bounds are obtained as a consequence of the new bound $\Cr(\ksn) \ge 2.1796n^2 - 4.5n$. To obtain this improved lower bound for $\Cr(\ksn)$, we use some elementary topological facts on drawings of $K_{2,7}$ to set up a quadratic program on $6!$ variables whose minimum $p$ satisfies $\Cr(\ksn) \ge (p/2)n^2 - 4.5n$, and then use state-of-the-art quadratic optimization techniques combined with a bit of invariant theory of permutation groups to show that $p \ge 4.3593$.