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.

This work is partially supported by JSPS KAKENHI Grant Number 25870661.