Finding optimal solutions to several gray pattern instances

Abstract

In this paper we compare two new binary linear formulations to a standard quadratic binary program for the gray pattern problem and solved all three by the Gurobi solver. One formulation performed significantly better and obtained seven optimal solutions that were not proven optimal before. It is interesting that the formulation that performed best is based on significantly more variables and constraints.

Appendix

Let $s^{(p)^*}=(s^*(1), s^*(2),\ldots , s^*(p))$ denote the optimal solution (the optimal configuration of p points in the rectangle of size $n=n_1\times n_2$; in our case, $n=64=8\times 8$). Further, let $n_1=n_2,~ p'=k^2\times p$ (without the loss of generality), $k=1, 2,\ldots ,$ and let $s^{(p')^{**}}=(s^{**}(1), s^{**}(2),\ldots , s^{**}(p'))$ correspond to the configuration of $p'$ points in the rectangle (square) of size $k^2\times n=k\times n_1\times k\times n_1$ – such that the rectangle of size $k^2\times n$ is obtained from the rectangle of size n by replicating this rectangle $k^2$ times. Then, there exists a solution denoted by $s^{{(p')}^{***}}$ – such that the following statements hold: $\frac{p'}{p}\times f(s^{{(p)}^*} )\le f\left( s^{(p')^{***}}\right)$, $f\left( s^{(p')^{**}}\right) =\omega \times f\left( s^{(p')^{***}}\right)$; here $\frac{p'}{p}=k^2$, $\omega (\omega \ge 1)$ is an empirical constant close or equal to 1, and $f\left( s^{(p)^*}\right) , f\left( s^{(p')^{**}}\right) , f\left( s^{(p')^{***}}\right)$ denote the objective function values corresponding, respectively, to $s^{(p)^*}, s^{(p')^{**}}, s^{(p')^{***}}.$