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 \(n\times n\) 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 \(p\in \{1,2,3\}\), 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.