An Efficient Local Search for Partial Latin Square Extension Problem

TL;DR

This paper introduces an efficient local search algorithm for the NP-hard Partial Latin Square Extension problem, utilizing (p,q)-swap neighborhoods and a novel Trellis-swap operation to improve solution quality.

Contribution

It proposes a new local search framework with (p,q)-swap neighborhoods and Trellis-swap, enhancing solution efficiency for the PLS extension problem.

Findings

The algorithm finds improved solutions in O(n^{p+1}) time for p=1,2,3.

Trellis-swap generalizes existing swaps and operates in O(n^{3.5}) time.

Prototype iterated local search outperforms state-of-the-art solvers.

Abstract

A partial Latin square (PLS) is a partial assignment of n symbols to an nxn 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.