Computing autotopism groups of partial Latin rectangles: a pilot study

TL;DR

This study compares various algorithms for computing autotopism groups of partial Latin rectangles, highlighting their effectiveness depending on the size and structure of the rectangles, and aims to guide practical software development.

Contribution

It provides an experimental comparison of six algorithmic families and entry invariants for autotopism group computation of partial Latin rectangles, informing future software design.

Findings

Few entries imply many symmetries best identified mathematically.

Intermediate entries benefit from quick invariants to reduce computation.

Large autotopism groups in Latin squares hinder computational performance.

Abstract

Computing the autotopism group of a partial Latin rectangle can be performed in a variety of ways. This pilot study has two aims: (a) to compare these methods experimentally, and (b) to identify the design goals one should have in mind for developing practical software. To this end, we compare six families of algorithms (two backtracking methods and four graph automorphism methods), with and without the use of entry invariants, on two test suites. We consider two entry invariants: one determined by the frequencies of row, column, and symbol representatives, and one determined by $2 \times 2$ submatrices. We find: (a) with very few entries, many symmetries often exist, and these should be identified mathematically rather than computationally, (b) with an intermediate number of entries, a quick-to-compute entry invariant was effective at reducing the need for computation, (c) with an almost-full partial Latin rectangle, more sophisticated entry invariants are needed, and (d) the performance for (full) Latin squares is significantly poorer than other partial Latin rectangles of comparable size, obstructed by the existence of Latin squares with large (possibly transitive) autotopism groups.