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Searching for Mutually Orthogonal Latin Squares via integer and constraint programming
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.
... 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: ...
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). ...
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. ...
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. ...
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. ...
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]. ...
The (k,s)assignment problem sets a unified framework for studying the facial structure of families of assignment polytopes. Through this framework, we derive classes of clique facets for all axial and planar assignment polytopes. For each of these classes, a polynomial-time separation procedure is described. Furthermore, we provide computational experience illustrating the efficiency of these facet-defining inequalities when applied as cutting planes.
A partial Latin square (PLS) is a partial assignment of n symbols to an n× gridsuchthat,ineachrowandineachcolumn,eachsymbolappearsatmost once. The partial Latin square extension (PLSE) problem asks to find such a PLS that is a maximum extension of a given PLS. Recently Haraguchi et al. proposed a heuristic algorithm for the PLSE problem. In this paper, we present its effectiveness especially for the “hardest” instances. We show by empirical studies that, when n is large to some extent, the instances such that symbols are given in 60-70% of the n2 cells are the hardest. For such instances, the algorithm delivers a better solution quickly than IBM ILOG CPLEX, a state-of-the-art optimization solver, that is given a longer time limit. It also outperforms surrogate constraint based heuristics that are originally developed for the maximum independent set problem.
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 worst-case 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. © 2011 Science China Press and Springer-Verlag Berlin Heidelberg.
Many real-life Constraint Satisfaction Problems (CSPs) involve some constraints similar to the alld-ifferent constraints. These constraints are called con-straints of difference. They are defined on a subset of variables by a set of tuples for which the values oc-curing in the same tuple are all different. In this pa-per, a new filtering algorithm for these constraints is presented. It achieves the generalized arc-consistency condition for these non-binary constraints. It is based on matching theory and its complexity is low. In fact, for a constraint defined on a subset of p variables hav-ing domains of cardinality at most d, its space com-plexity is OCpd) and its time complexity is O(p2d2). This filtering algorithm has been successfully used in the system RESYN (Vismara et al. 1992), to solve the subgraph isomorphism problem.
In the first part of the paper, we present a framework for describing basic techniques to improve the representation of a mixed integer programming problem. We elaborate on identification of infeasibility and redundancy, improvement of bounds and coefficients, and fixing of binary variables. In the second part of the paper, we discuss recent extensions to these basic techniques and elaborate on the investigation and possible uses of logical consequences.
INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.
We combine mixed integer linear programming (MILP) and constraint programming (CP) to solve planning and scheduling problems.
Tasks are allocated to facilities using MILP and scheduled using CP, and the two are linked via logic-based Benders decomposition.
Tasks assigned to a facility may run in parallel subject to resource constraints (cumulative scheduling). We solve minimum
cost problems, as well as minimum makespan problems in which all tasks have the same release date and deadline. We obtain
computational speedups of several orders of magnitude relative to the state of the art in both MILP and CP.
We study a hybrid MIP/CP solution approach in which CP is used for detecting infeasibilities and generating cuts within a branch-and-cut algorithm for MIP. Our framework applies to MIP problems augmented by monotone constraints that can be handled by CP. We illustrate our approach on a generic multiple machine scheduling problem, and compare it to other hybrid MIP/CP algorithms.
This paper introduces an Integer Programming model for multidimensional assignment problems and examines the underlying polytopes. It also proposes a certain hierarchy among assignment polytopes. The dimension for classes of multidimensional assignment polytopes is established, unifying and generalising previous results. The framework introduced constitutes the first step towards a polyhedral characterisation for classes of assignment prob-lems. The generic nature of this approach is illustrated by identifying a family of facets for a certain class of multidi-mensional assignment problems, namely "axial" problems.
Since 1782, when Euler addressed the question of existence of a pair of Orthogonal Latin Squares (OLS) by stating his famous conjecture ([8, 9, 13]), these structures have remained an active area of research due to their theoretical properties as well as their applications in a variety of fields. In the current work we consider the polyhedral aspects of OLS. In particular we establish the dimension of the OLS polytope, describe all cliques of the underlying intersection graph and categorize them into three classes. For two of these classes we show that the related inequalities have Chvátal rank two and both are facet defining. For each such class, we give a separation algorithm of the lowest possible complexity, i.e. linear in the number of variables.
In recent years we have seen an increasing interest in combining CSP and LP based techniques for solving hard computational problems. While considerable progress has been made in the integration of these techniques for solving problems that exhibit a mixture of linear and combinatorial constraints, it has been sur-prisingly difficult to successfully integrate LP-based and CSP-based methods in a purely combinatorial setting. We propose a complete randomized backtrack search method for combinatorial problems that tightly couples CSP propagation techniques with randomized LP rounding. Our approach draws on recent results on approximation algorithms with theoretical guarantees, based on LP relaxations and randomized rounding tech-niques, as well on results that provide evidence that the run time distributions of combinatorial search methods are often heavy-tailed. We present experimental re-sults that show that our hybrid CSP/LP backtrack search method outperforms the pure CSP and pure LP strategies on instances of a hard combinatorial problem.
Mixed logical/linear programming (MLLP) is an extension of mixed integer/linear programming (MILP). It can represent the discrete elements of a problem with logical propositions and provides a more natural modeling framework than MILP. It can also have computational advantages, partly because it eliminates integer variables when they serve no purpose, provides alternatives to the traditional continuous relaxation, and applies logic processing algorithms. This paper surveys previous work and attempts to organize ideas associated with MLLP, some old and some new, into a coherent framework. It articulates potential advantages of MLLP's wider choice of modeling and solution options and illustrates some of them with computational experiments.
Symmetries in constraint satisfaction problems (CSPs) are one of the difficulties that practitioners have to deal with. We present in this paper a new method based on the symmetries of decisions taken from the root of the search tree. This method can be seen as an improvement of the SBDD method presented by Focacci and Milano [7] and Fahle, Schamberger and Sellmann [5]. We present a simple formalization of our method for which we prove correctness and completeness results. We show that our method is theoretically more efficient as the size of each no-good is smaller. This theoretical analysis is confirmed by thorough experimental evaluation on highly symmetrical real world problems. We are able to break all symmetries for problems with more than 1078 symmetries.
We combine mixed integer linear programming (MILP) and constraint programming (CP) to solve planning and scheduling problems. Tasks are allocated to facilities using MILP and scheduled using CP, and the two are linked via logic-based Benders decomposition. Tasks assigned to a facility may run in parallel subject to resource constraints (cumulative scheduling). We solve minimum cost problems, as well as minimum makespan problems in which all tasks have the same release date and deadline. We obtain computational speedups of several orders of magnitude relative to the state of the art in both MILP and CP.
Symmetries in constraint satisfaction or combinatorial optimization problems can cause considerable difficulties for exact solvers. One way to overcome the problem is to employ sophisticated models with no or at least less symmetries. However, this often requires a lot of experience from the user who is carrying out the modeling. Moreover, some problems even contain inherent symmetries that cannot be broken by remodeling. We present an approach that detects symmetric choice points during the search. It enables the user to find solutions for complex problems with minimal effort spent on modeling.
. There are many open problems in the study of quasigroups. Recently, automated techniques have been employed to attack these open problems. In this paper, we show how a propositional satisfiability prover is used to solve many open problems in quasigroups. Our success relies on a powerful propositional prover called SATO and a useful technique called the cyclic group construction. We provide detailed solutions to open problems solved by SATO. 1 Introduction In the recent years, there has been considerable renewed interest in the propositional satisfiability problem (SAT). Because the SAT problem is the first known NP-complete problem, it is relatively easy to transform any NP-complete problem in mathematics, computer science and electrical engineering into the SAT problem. The SAT problem is known to be difficult to solve in theory. However, contrary to the common perception that transforming a problem into the SAT problem will not make the problem easier to solve, many problems can ...
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.
The maximal partial spreads of PG(3,4) were recently classified by Leonard Soicher. Each such partial spread (with r lines, say) yields a translation net of order 16 and degree r and hence a set of r−2 mutually orthogonal Latin squares of order 16. We determine which of these nets are transversal-free. In particular, we obtain sets of t MAXMOLS(16) for two previously unknown cases, namely for t=9 and 10.
In this paper, we examine the orthogonal Latin squares (OLS) problem from an integer programming perspective. The OLS problem has a long history and its significance arises from both theoretical aspects and practical applications. The problem is formulated as a four-index assignment problem whose solutions correspond to OLS. This relationship is exploited by various routines (preliminary variable fixing, branching, etc) of the Branch & Cut algorithm we present. Clique, odd-hole and antiweb inequalities implement the ‘Cut’ component of the algorithm. For each cut type a polynomial-time separation algorithm is implemented. Extensive computational analysis examines multiple aspects concerning the design of our algorithm. The results illustrate clearly the improvement achieved over simple Branch & Bound.Journal of the Operational Research Society (2004) 55, 298–307. doi:10.1057/palgrave.jors.2601655
In this paper we address ourselves to identifying facets of the set packing polyhedron, i.e., of the convex hull of integer solutions to the set covering problem with equality constraints and/or constraints of the form . This is done by using the equivalent node-packing problem derived from the intersection graph associated with the problem under consideration. First, we show that the cliques of the intersection graph provide a first set of facets for the polyhedron in question. Second, it is shown that the cycles without chords of odd length of the intersection graph give rise to a further set of facets. A rather strong geometric property of this set of facets is exhibited.
The task of finding global optima to general classes of nonconvex optimization problem is attracting increasing attention. McCormick [4] points out that many such problems can conveniently be expressed in separable form, when they can be tackled by the special methods of Falk and Soland [2] or Soland [6], or by Special Ordered Sets. Special Ordered Sets, introduced by Beale and Tomlin [1], have lived up to their early promise of being useful for a wide range of practical problems. Forrest, Hirst and Tomlin [3] show how they have benefitted from the vast improvements in branch and bound integer programming capabilities over the last few years, as a result of being incorporated in a general mathematical programming system.
Nevertheless, Special Ordered Sets in their original form require that any continuous functions arising in the problem be approximated by piecewise linear functions at the start of the analysis. The motivation for the new work described in this paper is the relaxation of this requirement by allowing automatic interpolation of additional relevant points in the course of the analysis.
This is similar to an interpolation scheme as used in separable programming, but its incorporation in a branch and bound method for global optimization is not entirely straightforward. Two by-products of the work are of interest. One is an improved branching strategy for general special-ordered-set problems. The other is a method for finding a global minimum of a function of a scalar variable in a finite interval, assuming that one can calculate function values and first derivatives, and also bounds on the second derivatives within any subinterval.
The paper describes these methods, their implementation in the UMPIRE system, and preliminary computational experience.
We examine a family of graphs called webs. For integers n ⩾ 2 and , the web W(n, k) has vertices Vn = {1, …, n} and edges {(i, j): j = i+k, …, i+n − k, for iϵVn (sums mod n)}. A characterization is given for the vertex packing polyhedron of W(n, k) to contain a facet, none of whose projections is a facet for the lower dimensional vertex packing polyhedra of proper induced subgraphs of W(n, k). Simple necessary and sufficient conditions are given for W(n, k) to contain W(n′, k′) as an induced subgraph; these conditions are used to show that webs satisfy the Strong Perfect Graph Conjecture. Complements of webs are also studied and it is shown that if both a graph and its complement are webs, then the graph is either an odd hole or its complement.
In the two centuries since Euler first asked about mutually orthogonal latin squares, substantial progress has been made. The biggest breakthroughs came in 1960 with the celebrated theorems of Bose, Shrikhande, and Parker, and in 1974 in the research of Wilson. Current efforts have concentrated on refining these approaches, and finding new applications of the substantial theory opened. This paper provides a detailed list of constructions for MOLS, concentrating on the uses of pairwise balanced designs and transversal designs in recursive constructions as pioneered in the papers of Bose, Shrikhande, and Parker. In addition, several new lower bounds for MOLS are given and an up-to-date table of lower bounds for MOLS is provided.
We consider the problem of Mutually Orthogonal Latin Squares and propose two algorithms which integrate Integer Programming
(IP) and Constraint Programming (CP). Their behaviour is examined and compared to traditional CP and IP algorithms. The results
assess the quality of inference achieved by the CP and IP, mainly in terms of early identification of infeasible subproblems.
It is clearly illustrated that the integration of CP and IP is beneficial and that one hybrid algorithm exhibits the best
performance as the problem size grows. An approach for reducing the search by excluding isomorphic cases is also presented.
We introduce branch-and-infer, a unifying framework for integer linear programming and finite domain constraint programming. We use this framework to compare the two approaches with respect to their modeling and solving capabilities, to introduce symbolic constraint abstractions into integer programming, and to discuss possible combinations of the two approaches.
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.
Dealing with highly symmetric problems in a constraint programming context is an area of growing interest. The Social Golfer Problem is a highly symmetric problem on which many researchers are trying out their algorithms. In this paper we take a closer look at the social golfer problem and some of the techniques which can be used to solve it, focussing in particular on Fahle et. al's Symmetry Breaking via Dominance Detection. 1
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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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