A Sparse Corner-Difference MILP for Minimum Square Tiling

Abstract

We study the problem of tiling an $N\times N$ square with axis-aligned squares whose side lengths are integers strictly less than $N$, with the objective of minimizing the number of tiles. The standard placement-based exact-cover MILP contains $Θ(N^5)$ placement-to-cell nonzero coefficients and is also affected by the dihedral symmetry of the square domain. We introduce a Corner-Difference MILP (CD-MILP) that represents each selected square by at most four signed corner coefficients. Two-dimensional prefix reconstruction shows that the resulting constraints are equivalent to the original cell-cover equations, while reducing the coverage-related nonzero count to $Θ(N^3)$. We also introduce lightweight corner-ordering constraints that retain at least one representative from every $D_4$ orbit, although ties may leave residual symmetry. Computational experiments on nine prime-size instances compare the baseline, CD-MILP, the baseline with corner ordering, and their combination using Gurobi and COPT. Under the stated experimental settings, the combined formulation solves eight instances with Gurobi and seven with COPT, compared with four and five, respectively, for the baseline. Its average presolved matrixs contain approximately $10^5$ nonzero coefficients on average, compared with approximately $10^7$ for the baseline.