Canonical labelling of Latin squares in average-case polynomial time

TL;DR

This paper proves that most Latin squares lack large proper subsquares and introduces an average-case polynomial-time algorithm for canonical labelling, aiding isomorphism testing for various combinatorial structures.

Contribution

The paper establishes a probabilistic bound on large subsquares in random Latin squares and develops an efficient average-case algorithm for their canonical labelling.

Findings

Most Latin squares have no large proper subsquares.

The canonical labelling algorithm runs in polynomial average time.

Applicable to isomorphism problems in multiple combinatorial objects.

Abstract

A Latin square of order $n$ is an $n\times n$ matrix in which each row and column contains each of $n$ symbols exactly once. For $\epsilon>0$, we show that with high probability a uniformly random Latin square of order $n$ has no proper subsquare of order larger than $n^{1/2}\log^{1/2+\epsilon}n$. Using this fact we present a canonical labelling algorithm for Latin squares of order $n$ that runs in average time bounded by a polynomial in $n$. The algorithm can be used to solve isomorphism problems for many combinatorial objects that can be encoded using Latin squares, including quasigroups, Steiner triple systems, Mendelsohn triple systems, $1$-factorisations, nets, affine planes and projective planes.