Bounds for the crossing number of the complete bipartite graph

Computes the bounds $\alpha_m$ and $\beta_m$ for the crossing number of $K_{n,m}$ described in the paper

New lower bounds on crossing numbers of Km,n from permutation modules and semidefinite programming, Daniel Brosch and Sven Polak, ArXiv:2206.02755, 2022.

To compute the bounds:

• SolveSDP_Alpha.jl contains a function solveAlpha(m) which computes $\alpha_m$ with Convex.jl and rounds the solution to a feasible rational solution. The same file also provides a more efficient function solveAlphaExternal(m) which manually generates the SDPA file, avoiding Convex.jl and JuMP.jl, and verifies the solution.
• SolveSDP_Beta_Iterative.jl contains a function solveBetaIterative(m) which computes $\beta_m$ and rounds the solution to a feasible rational solution.
• VisualizeSolutions.jl computes $\beta_m$ for $m\in{5,7,9,11,13}$ and plots the coefficients of the eigenvector of the rank 1 solutions.

Quick test of solveAlpha

Assume Julia installed; run julia in terminal, in the main package directory (with readme.md).

• hit ] to get pkg> prompt
• type activate . (and hit return)
• type initialize
• hit backspace to get back to julia> prompt
• type include("src/SolveSDP_Alpha.jl")
• type solveAlpha(5); after a little while, it will give a rational close to 1.947213595... as the answer.
• etc... (e.g., type solveAlpha(7) to get 4.359315494807...)