A constructive algorithm for partial Latin square extension problem that solves hardest instances effectively

Abstract

A partial Latin square (PLS) is a partial assignment of n symbols to an n× gridsuchthat,ineachrowandineachcolumn,eachsymbolappearsatmost once. The partial Latin square extension (PLSE) problem asks to find such a PLS that is a maximum extension of a given PLS. Recently Haraguchi et al. proposed a heuristic algorithm for the PLSE problem. In this paper, we present its effectiveness especially for the “hardest” instances. We show by empirical studies that, when n is large to some extent, the instances such that symbols are given in 60-70% of the n2 cells are the hardest. For such instances, the algorithm delivers a better solution quickly than IBM ILOG CPLEX, a state-of-the-art optimization solver, that is given a longer time limit. It also outperforms surrogate constraint based heuristics that are originally developed for the maximum independent set problem.