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# Searching for Mutually Orthogonal Latin Squares via integer and constraint programming

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## Abstract

This paper applies algorithms integrating Integer Programming (IP) and Constraint Programming (CP) to the Mutually Orthogonal Latin Squares (MOLS) problem. We investigate the behaviour of these algorithms against traditional IP and CP schemes. Computational results are obtained with respect to various aspects of the algorithms, using instances of the 2 MOLS and 3 MOLS problems. The benefits of integrating IP with CP on this feasibility problem are clearly exhibited, especially in large problem instances.

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... Second, we develop an improved symmetry breaking method that removes more symmetry from the search space than the "domain reduction" symmetry breaking method used in previous searches [1], [2]. We show that our symmetry breaking method reduces the amount of symmetry present in the 2 MOLS(n) search by an exponential factor in n when compared with domain reduction symmetry breaking. ...
... However, the relaxation solutions provided by the IP solver-despite being relatively expensive to compute-help provide a "global" perspective of the search space. In particular, Appa et al. present IP and CP models to search for MOLS and develop sophisticated techniques for combining solvers in ways that exploit each solver's strengths [1], [2]. ...
... We encode the latin and orthogonality constraints as six sets of n 2 equalities grouped by which subscripts of x ijkl are fixed: x ijkl = 1 ∀k, l orthogonality of X and Y This model can be extended to search for k MOLS(n) using n k+2 variables and k+2 2 n 2 constraints, but this easily exceeds our ability to solve for all but the smallest values of k. Instead, in order to search for triples of MOLS we employ an alternative method described by Appa et al. [1] that relies on using a separate solver to fill in the entries of a third square Z. Assuming the entries of Z have been fixed in advance, the 3 MOLS(n) problem may then be represented as an extension of the above 2 MOLS(n) model with the following additional constraints: ...
Preprint
In this paper we provide results on using integer programming (IP) and constraint programming (CP) to search for sets of mutually orthogonal latin squares (MOLS). Both programming paradigms have previously successfully been used to search for MOLS, but solvers for IP and CP solvers have significantly improved in recent years and data on how modern IP and CP solvers perform on the MOLS problem is lacking. Using state-of-the-art solvers as black boxes we were able to quickly find pairs of MOLS (or prove their nonexistence) in all orders up to ten. Moreover, we improve the effectiveness of the solvers by formulating an extended symmetry breaking method as well as an improvement to the straightforward CP encoding. We also analyze the effectiveness of using CP and IP solvers to search for triples of MOLS, compare our timings to those which have been previously published, and estimate the running time of using this approach to resolve the longstanding open problem of determining the existence of a triple of MOLS of order ten.
... Appa, Magos, and Mourtos [1] and Appa, Mourtos, and Magos [2] compared integer programming and constraint programming in determining whether an OA(q 2 , 4, q, 2) exists for 3 ≤ q ≤ 12 after removing symmetries by introducing constraints. They observed that solving LP relaxations on the nodes of a search tree becomes more beneficial at improving solution times as the number of variables grows. ...
... Such an improvement was developed by Margot [15,16,17,18] and used in [14] to improve the lower bound from 65 to 71 for the 6 matches instance of the famous football pool problem. Next, we describe this algorithm by using the same notation as in [14] adapted for ILP (1). ...
Article
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Classifying orthogonal arrays (OAs) is a well-known important class of problems that asks for finding all non-isomorphic, non-negative integer solutions to a class of systems of constraints. Solved instances are scarce. We develop two new methods based on finding all non-isomorphic solutions of two novel integer linear programming formulations for classifying all non-isomorphic OA(N, k, s, t) given a set of all non-isomorphic OA(N, k − 1, s, t). We also establish the concept of orthogonal design equivalence (OD-equivalence) of OA(N, k, 2, t) to reduce the number of integer linear programs (ILPs) all of whose non-isomorphic solutions need to be enumerated by our methods. For each ILP, we determine the largest group of permutations that can be exploited with the branch-and-bound (B&B) with isomorphism pruning algorithm of Margot [Discrete Optim. 4 (2007), 40–62] without losing isomorphism classes of OA(N, k, 2, t). Our contributions bring the classifications of all non-isomorphic OA(160, k, 2, 4) for k = 9, 10 and OA(176, k, 2, 4) for k = 5, 6, 7, 8, 9, 10 within computational reach. These are the smallest s = 2, t = 4 cases for which classification results are not available in the literature.
... The decision problem version of the PLSE problem is known as the quasigroup completion (QC) problem in AI, CP and SAT communities (Ansótegui 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 2005;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. ...
Article
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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 PLS 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 $$(p,\infty )$$-neighborhood search algorithm that finds an improved solution or concludes that no such solution exists for $$p\in \{1,2,3\}$$. The running time of the algorithm is $$O(n^{p+1})$$. We then propose a novel swap operation, Trellis-swap, which is a generalization of (p, q)-swap with $$p\le 2$$. The proposed Trellis-neighborhood search algorithm runs in $$O(n^{3.5})$$ time. The iterated local search (ILS) algorithm with Trellis-neighborhood is more likely to deliver a high-quality solution than not only ILSs with $$(p,\infty )$$-neighborhood but also state-of-the-art optimization solvers such as IBM ILOG CPLEX and LocalSolver.
... The decision problem version of the PLSE problem is known as the quasigroup completion (QC ) problem in AI, CP and SAT communities. The QC problem has been one of the most frequently used benchmark problems in these areas [28] and various variant problems are studied intensively, e.g., Sudoku [8,9,22,23,26,27] and mutually orthogonal Latin squares [4,25,29]. Our local search may be helpful for those who develop exact solvers for the QC problem since, once a good heuristic algorithm is established, a good lower estimate of the optimal solution size would be delivered quickly. ...
Conference Paper
Full-text available
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.
... The use of mathematical programming for the construction of designs is not new: Lee (2000) shows examples of using semidefinite programming for generating E-,D-and A-optimal designs; Jones and Nediak (2005) use non-linear programming to create D-optimal designs; Appa et al. (2006) combine integer programming and constraint programming to find mutually orthogonal Latin squares; van Dam et al. (2009) and van Dam et al. (2010) use MIP to, respectively, find bounds and create Maximin Latin Hypercube Designs; non-linear mixed integer programming was used to construct balanced incomplete block designs by Yokoyaa and Yam- ada (2011); and, as previously mentioned, Hernandez (2008) creates nearly orthogonal Latin hypercubes by using mixed integer programming. However, to the best of our knowledge, the transformation of the correlation function we use to linearize our MIP is unique, as is our ability to construct new orthogonal designs, and improve existing orthogonal arrays, to obtain efficient, space-filling, balanced designs for discrete-valued factors. ...
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Analysts faced with conducting experiments involving quantitative factors have a variety of potential designs in their portfolio. However, in many experimental settings involving discrete-valued factors (particularly if the factors do not all have the same number of levels), none of these designs are suitable.In this paper, we present a mixed integer programming (MIP) method that is suitable for constructing orthogonal designs, or improving existing orthogonal arrays, for experiments involving quantitative factors with limited numbers of levels of interest. Our formulation makes use of a novel linearization of the correlation calculation.The orthogonal designs we construct do not satisfy the definition of an orthogonal array, so we do not advocate their use for qualitative factors. However, they do allow analysts to study, without sacrificing balance or orthogonality, a greater number of quantitative factors than it is possible to do with orthogonal arrays which have the same number of runs.
... A thorough review regarding assignment problems and their numerous applications can be found in [10,11], where references for exact algorithms, complexity and approximability results are also provided. A more recent computational study for some axial problems using Branch & Bound methods, including related references and applications, can be found in [12], while computational work on the (3, 1)AP n , (3, 2)AP n and (4, 2)AP n appears in [1,13,9] and [14], respectively. A special case of (3, 1)AP n is examined in [15]. ...
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Some Examples The Logic of Propositions The Logic of Discrete Variables The Logic of 0-1 Inequalities Cardinality Clauses Classical Boolean Methods Logic-Based Modeling Logic-Based Branch and Bound Constraint Generation Domain Reduction Constraint Programming Continuous Relaxations Decomposition Methods Branching Rules Relaxation Duality Inference Duality Search Strategies Logic-Based Benders Decomposition Nonserial Dynamic Programming Discrete Relaxations References Index.
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We describe a method for symmetry breaking during search (SBDS) in constraint programming. It has the great advantage of not interfering with heuristic choices. It guarantees to return a unique solution from each set of symmetrically equivalent ones, which is the one found first by the variable and value ordering heuristics. We describe an implementation of SBDS in ILOG Solver, and applications to low autocorrelation binary sequences and the n-queens problem. We discuss how SBDS can be applied when there are too many symmetries to reason with individually, and give applications in graph colouring and Ramsey theory.
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Primal-dual Interior-point Methods. SIAM, Philadelphia Solving open quasigroup problems by propositional reasoning
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Clique facets of the orthogonal Latin squares polytope, CDAM Research Reports Series, LSE-CDAM-2001-04, London School of Economics Available from: <http:// www.cdam.lse.ac Polyhedral results for assignment problems Integrating con-straint and integer programming for the OLS problem
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European Journal of Operational Research 173 (2006) 519–530 www.elsevier.com/locate/ejor References Appa, G., Magos, D., Mourtos, I., Janssen, J.C.M., 2001. Clique facets of the orthogonal Latin squares polytope, CDAM Research Reports Series, LSE-CDAM-2001-04, London School of Economics. Available from: <http:// www.cdam.lse.ac.uk/Reports/Files/cdam-2001-04.pdf>. Appa, G., Magos, D., Mourtos, I., 2002a. Polyhedral results for assignment problems, CDAM Research Reports Series, LSE-CDAM-2002-01, London School of Economics. Available from: <http://www.cdam.lse.ac.uk/Reports/ Files/cdam-2002-01.pdf>. Appa, G., Mourtos, I., Magos, D., 2002b. Integrating con-straint and integer programming for the OLS problem. In: Van Hentenryck, P. (Ed.), Proceedings of CP2002, LNCS 2470. Springer, pp. 17–32.