Searching for Mutually Orthogonal Latin Squares via integer and constraint programming

Appa, Gautam; Magos, D.; Mourtos, Ioannis

doi:10.1016/j.ejor.2005.01.048

Cited by 11 publications

(21 citation statements)

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“…The decision problem version of the PLSE problem is known as the quasigroup completion ( QC) problem in AI, CP and SAT communities (Ans6tegui et al, 2004;'Gomes and Selman, 1997;Gomes and Shmoys, 2002). The QC problem has been one of the most frequently used benchmark problems in these areas and variant problems are studied intensively, e.g., Sudoku (Crawford et al, 2008(Crawford et al, , 2009Lambert et al, 2006;Lewis, 2007;Simonis;Soto et al, 2013), mutually orthogonal Latin squares (Appa et al, 2006a;Ma and Zhang, 2013;Vieira Jr. et al, 2011), and spatially balanced Latin squares (Gomes et al, 2004a;Le Bras et al, 2012;Smith et al, 2005). Our local search may be helpful for those who develop exact solvers for the QC problem since the local search itself or metaheuristic algorithms employing it would deliver a good initial solution or a tight lower estimate of the optimal solution size quickly.…”

Section: Introductionmentioning

confidence: 99%

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

Haraguchi

2016

J Heuristics

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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.

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Section: Introductionmentioning

confidence: 99%

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

Haraguchi

2016

J Heuristics

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“…For MOLS problems, Appa et al [8,9] proposed a specific method for symmetry breaking. Essentially it fixes some variables' values (or value domains) before the search begins.…”

Section: Symmetry Breakingmentioning

confidence: 99%

“…Moreover, for the first column of the second latin square, the value domains are reduced. The configuration is demonstrated in Figure 2 in [9].…”

Section: Symmetry Breakingmentioning

confidence: 99%

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Finding orthogonal latin squares using finite model searching tools

Zhang

2011

Sci. China Inf. Sci.

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An important class of problems in combinatorics is to find orthogonal latin squares with certain properties. Computer search is a promising approach for solving such problems. But generally its worstcase complexity is high. This paper describes how to use a general-purpose model searching program to find orthogonal latin squares. New techniques for problem representation and symmetry breaking are proposed to increase search efficiency.

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“…The decision problem version of the PLSE problem is known as the quasigroup completion (QC ) problem in AI, CP and SAT communities [3,17,18,38]. The QC problem has been one of the most frequently used benchmark problems in these areas and variant problems are studied intensively, e.g., Sudoku [9,10,29,31,34,36], mutually orthogonal Latin squares [4,33,39], and spatially balanced Latin squares [19,30,35]. Our local search may be helpful for those who develop exact solvers for the QC problem since the local search itself or metaheuristic algorithms employing it would deliver a good initial solution or a tight lower estimate of the optimal solution size quickly.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Local Search for Partial Latin Square Extension Problem

Haraguchi

2015

Integration of AI and OR Techniques in Constraint Programming

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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, Trellisswap, which is a generalization of (1, q)-swap and (2, q)-swap. Our Trellisneighborhood 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.

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Searching for Mutually Orthogonal Latin Squares via integer and constraint programming

Cited by 11 publications

References 10 publications

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

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

Finding orthogonal latin squares using finite model searching tools

An Efficient Local Search for Partial Latin Square Extension Problem

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