An efficient local search for partial Latin square extension problem

Abstract

A partial Latin square (PLS) is a partial assignment of n symbols to an n × n grid such that, in each row and in each column, each symbol appears at most once. The partial Latin square extension problem is an NP-hard problem that asks for a largest extension of a given PLS. In this paper we propose an efficient local search for this problem. We focus on the local search such that the neighborhood is defined by (p, q)-swap, i.e., removing exactly p symbols and then assigning symbols to at most q empty cells. For p ∈ {1, 2, 3}, our neighborhood search algorithm finds an improved solution or concludes that no such solution exists in O(n p+1) time. We also propose a novel swap operation, Trellis-swap, which is a generalization of (1, q)-swap and (2, q)-swap. Our Trellis-neighborhood search algorithm takes O(n 3.5) time to do the same thing. Using these neighborhood search algorithms, we design a prototype iterated local search algorithm and show its effectiveness in comparison with state-of-the-art optimization solvers such as IBM ILOG CPLEX and LocalSolver.