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An LP-based proof for the non-existence of a pair of orthogonal Latin squares of order 6

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Abstract

This paper presents an alternative proof for the non-existence of orthogonal Latin squares of order 6. Our method is algebraic, rather than enumerative, and applies linear programming in order to obtain appropriate dual vectors. The proof is achievable only after extending previously known results for symmetry elimination.

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... As a result of a decade long research, a rich tool-box for hybridization is now available: from the idea of optimization constraints [7,14,17] and associated notions of relaxed or approximated consistency [5,19], reduced-cost filtering [16], to sophisticated problem-dependent techniques based on Bender's decomposition [9], Lagrangian decomposition [6,18,20,21], or column generation [4,11]. Also, specialized hybrid approaches have been developed for special problems like computing orthogonal Latin squares [2] or to solve the social golfer problem [22]. ...
... Conversely, let s(D CSP ) = (y iu | 1 ≤ i ≤ n, u ∈ D i ) denote a solution of the negative BCSP. If y iu = true in s(D CSP ), then for any j, by (2), there exists no value v such that (u, v) ∈ R ij and y jv = true. Since V j must take at least one value, it means that there exists a value v k ∈ D j , with (u, v k ) / ∈ R ij such that y jv k = true. ...
... Our discussion of the two representations for BCSPs in Section 2.1, and in particular the formulation of constraints as logical implications provides the basis to model BCSPs as 0-1 integer linear programs: A logical formula written in conjunctive normal form (CNF) can be easily modeled as a set of inequalities involving 0-1 variables. Using the (1) and (2) in CNF in the following way: ...
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We present a theoretical study on the idea of using mathematical programming relaxations for filtering binary constraint satisfaction problems. We introduce the consistent value polytope and give a linear programming description that is provably tighter than a recently studied formulation. We then provide an experimental study that shows that, despite the theoretical progress, in practice filtering based on mathematical programming relaxations continues to perform worse than standard arc-consistency algorithms for binary constraint satisfaction problems. KeywordsCost-based filtering-hybrid methods-mathematical programming
... Such schemes, integrating the results introduced here with logic-based methods, have been implemented for constructing OLS pairs algorithmically (see [4]). Variants of these implementations can be used to check whether a particular Latin square has an orthogonal mate (see [1] for models related to this problem), whether a partially filled OLS pair can be completed, etc. ...
... 1k(x 1 ;i q ,j ) − 6 1k(x 1 ;1,j ) + 6 i q k(x 1 ;1,j ) − 6 i q k(x 1 ;i q ,j ) ) − ( 2 1l(x 1 ;i q ,1) − 2 1l(x 1 ;1,1) + 2 i q l(x 1 ;1,1) − 2 i q l(x 1 ;i q ,1) 1) ). (4.10) Table 5 Vertices x and x 1 ...
... To prove that the rank is at most 2, we perform a series of operations analogous to those implemented in the proof of Proposition 4.10. With respect to the matrix A, adding the rows (i 0 , j 0 ), (i 0 , k 0 ), (i 0 , l 1 ), each one weighted by 1 2 , rounding down both sides and considering only variables indexed exclusively by elements of the tuples c, s (by omitting the remaining variables), results in the inequality ...
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Since 1782, when Euler addressed the question of existence of a pair of orthogonal Latin squares (OLS) by stating his famous conjecture, these structures have remained an active area of research. In this paper, we examine 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. Two of these classes are shown to induce facet-defining inequalities of Chvátal rank two. For each such class, we provide a polynomial separation algorithm of the lowest possible complexity.
... A similar approach was adopted by Fischer and Yates [3] in 1934 during their enumeration of Latin squares of order 6. Analytical proofs of this result were also given by Yamamoto [7] in 1954 and Stinson [5] in 1984, while Appa et al. [1] established the same result by means of linear programming in 2004. However, the methods of [1,3,6] are difficult to validate without carrying out the necessary numerical computations oneself and the proofs in [5,7] are difficult to follow without extensive prior knowledge of abstract algebra and design theory. ...
... Analytical proofs of this result were also given by Yamamoto [7] in 1954 and Stinson [5] in 1984, while Appa et al. [1] established the same result by means of linear programming in 2004. However, the methods of [1,3,6] are difficult to validate without carrying out the necessary numerical computations oneself and the proofs in [5,7] are difficult to follow without extensive prior knowledge of abstract algebra and design theory. ...
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The non-existence of a pair of mutually orthogonal Latin squares of order six is a well-known result in the theory of combinatorial designs. It was conjectured by Euler in 1782 and was first proved by Tarry in 1900 by means of an exhaustive enumeration of equivalence classes of Latin squares of order six. Various further proofs have since been given, but these proofs generally require extensive prior subject knowledge in order to follow them, or are ‘blind’ proofs in the sense that most of the work is done by computer or by exhaustive enumeration. In this paper we present a graph-theoretic proof of a somewhat weaker result, namely the non-existence of self-orthogonal Latin squares of order six, by introducing the concept of a self-orthogonal Latin square graph. The advantage of this proof is that it is easily verifiable and accessible to discrete mathematicians not intimately familiar with the theory of combinatorial designs. The proof also does not require any significant prior knowledge of graph theory.
... It is not difficult to see that these constraints are of the form X^l^wo^ • (^o? jo) ^ ^/co} = 1 for each qo and for each r. The above model is implicitly used in [3] in the context of an LP-based proof for the infeasibility of 2MOLS for n = 6. ...
... Concerning the case n == 6, an LP-based proof of infeasibility is illustrated in [3]. This proof uses known results about the classification of Latin squares of order 6 into 12 equivalence classes and therefore requires the solution of only 12 LPs of the type obtained from the LP relaxation of (4.13). ...
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In this chapter we present various equivalent formulations or models for the Mutually Orthogonal Latin Squares (MOLS) problem and its generalization. The most interesting feature of the problem is that for some parameters the problem may be infeasible. Our evaluation of different formulations is geared to tackling this feasibility problem. Starting from a Constraint Programming (CP) formulation which emanates naturally from the problem definition, we develop several Integer Programming (IP) formulations. We also discuss a hybrid CP-IP approach in both modelling and algorithmic terms. A non-linear programming formulation and an interesting modelling approach based on the intersection of matroids are also considered.
... Definition 1 [51]. A square matrix L nÂn is denoted as a Latin square if any row and any column L nÂn involve the permutation of the numbers 1, 2, ..., n, where n denotes the order of L [46]. ...
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... For parameters with very few RBIBDs—or even none—on the other hand, the prospects of a breakthrough are much better. Throwing out an idea, perhaps linear programming could be useful in proving nonexistence; cf. [1]. Finally, note that 1-factorizations of complete graphs can be viewed as resolutions of 2-(v, 2, 1) designs. ...
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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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The promise of LP to boost CSP techniques for combinatorial problems, Proceedings of the Fourth International Workshop of Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems (CP-AI-OR’02)
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