How Unbalanced Must a Latin Rectangle Be?

Zhengyu Li, Vijay Ganesh

We view a $k\times n$ Latin rectangle as $n$ points in the grid $[n]^k$. Each symbol gives one point, whose coordinates are the columns it occupies. The rectangle is balanced when these points are pairwise equidistant in the $\ell_1$ metric. Its imbalance measures the distance from this ideal. We study the minimum imbalance from three sides. Constructively, we use a simulated annealing that moves symbols within rows and only penalises column clashes. Against the reference table of Voboril et al. (2025), it improves 33 of the 51 non-square values for $8\le n\le 13$, by up to 39.4%, and matches the rest. A cheap invariant, the row group, certifies that none of the 33 new rectangles is isotopic to rows of a group table, so the earlier column-swapping local search cannot reach any of them. Exactly, a computer-assisted branch and bound proves that the minimum imbalance of a $3\times 9$ rectangle is $56$. Díaz et al. found this value, but its optimality was open. A sharper pruning bound, derived from the identity $I=2\sum(d-t)^+$, reduces the search tree by a factor of $2.4$ to $3.3$ on the instances with $k\ge3$ that we solve. Asymptotically, the minimum imbalance is $Θ(n^3)$ for…