Iterated local search with Trellis-neighborhood for the partial Latin square extension problem

Abstract

A partial Latin square (PLS) is a partial assignment of n symbols to an n x 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. We consider the local search such that the neighborhood is defined by (p, q)-swap, i.e., the operation of dropping exactly p symbols and then assigning symbols to at most q empty cells. As a fundamental result, we provide an efficient (p, oo)-neighborhood search algorithm that finds an improved solution or concludes that no such solution exists for p E {1, 2, 3}. The running time of the algorithm is O(nP+ 1). We then propose a novel swap operation, Trellisswap, which is a generalization of (p, q)-swap with p ~ 2. The proposed Trellis-neighborhood search algorithm runs in O(n 3 • 5) time. The iterated local search (ILS) algorithm with Trellisneighborhood is more likely to deliver a high-quality solution than not only ILSs with (p, oo)neighborhood but also state-of-the-art optimization solvers such as IBM ILOG CPLEX and LOCALSOLVER.

An Efficient Local Search for the Constrained Symmetric Latin Square Construction Problem

“…We could develop a 3-neighborhood search algorithm that runs in O(n 4 ) time, by extending the 3-neighborhood search algorithm for the PLSE problem [17]. The algorithm takes into account the observation on Itoyanagi et al's 3-neighborhood search in the maximum independent set problem for an ordinary graph [23].…”

“…In [17], for p ∈ {1, 2, 3}, the author proposed a neighborhood search algorithm that runs in O(n p+1 ) time. He also invented a generalization of 2-swap operation, Trellis-swap, and proposed a neighborhood search algorithm that runs in O(n 3.5 ) time.…”

“…As was done for the PLSE problem in [17], we reduce the MaxCSLSC problem to the maximum independent set problem and utilize efficient local search that is developed by Andrade et al [3].…”

“…Furthermore, if we applied their methodology as is, then the time bound of the (p, p+1)-neighborhood search algorithm would be up to O(n p+3 ). The strategy is more or less similar to [17] in which the author developed efficient pneighborhood search algorithms for the PLSE problem. For the MaxCSLSC problem, we need trickier arguments to achieve the time bounds above.…”

“…For "kicking" in 5, we employ the mechanism that the author used for the PLSE problem in [17]. Let us review it briefly.…”