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An LP-based proof for the non-existence of a pair of orthogonal Latin squares of order 6
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.
... These can roughly be divided into two types: those building up a design parallel class by parallel class and those proceeding point by point. In Section 3 we use an algorithm of the latter type to classify the 2- (28,4,1) RBIBDs and show that the list of seven known resolutions of 2-(28, 4, 1) designs is complete. These objects are also known as the resolutions of unitals on 28 points. ...
... For each such code C, we first generate all words compatible with C. For a post (j, v), denote by f (j, v) the number of times (j, v) is flagged by the words in C. Denote by N (j, v) the number of words compatible with C that flag (j, v). Next, we find a pair of posts of the form (j, v 1 ...
... Throwing out an idea, perhaps linear programming could be useful in proving nonexistence; cf. [1]. ...
Approaches for classifying resolvable balanced incomplete block designs (RBIBDs) are surveyed. The main approaches can roughly
be divided into two types: those building up a design parallel class by parallel class and those proceeding point by point.
With an algorithm of the latter type — and by refining ideas dating back to 1917 and the doctoral thesis by Pieter Mulder
— it is shown that the list of seven known resolutions of 2-(28, 4, 1) designs is complete; these objects are also known as
the resolutions of unitals on 28 points.
... 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: ...
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 ...
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. ...
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). ...
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]. ...
The quadratic assignment problem (QAP) is one of the hardest combinatorial optimization tasks. It has many real-world applications such as airport gate assignment, and hospital layout problem. Designing enhanced optimization methodologies for the QAP is an active research area. In this paper, we present an integrated firefly algorithm (FA) based on mutually orthogonal Latin squares (MOLS), named as FA-MOLS, to solve the QAP. In the optimization process, the FA-MOLS employs three improvements, namely the MOLS strategy, opposition-based learning scheme, and repeated 2-exchange mutation to maintain the balance between exploitation and exploration abilities. By these improvements, it is intended to avoid the trapping in local optima and improve the convergence speed. The performance of the proposed FA-MOLS is validated on different test instances from the literature. Additionally, two real-world QAPs are simulated and investigated, the office buildings of a single company and the layout of hospital departments. The comprehensive experimental simulations and the nonparametric Wilcoxon's test affirm that the proposed FA-MOLS can provide a highly competitive performance compared with other algorithms from the literature. Therefore, it is concluded that FA-MOLS is an efficient and reliable algorithm for solving industrial quadratic assignment tasks.
The quadratic assignment problem (QAP) is one of the hardest combinatorial optimization tasks. It has many real-world applications such as airport gate assignment, and hospital layout problem. Designing enhanced optimization methodologies for the QAP is an active research area. In this paper, we present an integrated firefly algorithm (FA) based on mutually orthogonal Latin squares (MOLS), named as FA-MOLS, to solve the QAP. In the optimization process, the FA-MOLS employs three improvements, namely the MOLS strategy, opposition-based learning scheme, and repeated 2-exchange mutation to maintain the balance between exploitation and exploration abilities. By these improvements, it is intended to avoid the trapping in local optima and improve the convergence speed. The performance of the proposed FA-MOLS is validated on different test instances from the literature. Additionally, two real-world QAPs are simulated and investigated, the office buildings of a single company and the layout of hospital departments. The comprehensive experimental simulations and the nonparametric Wilcoxon’s test affirm that the proposed FA-MOLS can provide a highly competitive performance compared with other algorithms from the literature. Therefore, it is concluded that FA-MOLS is an efficient and reliable algorithm for solving industrial quadratic assignment tasks.
Latin Squares and Their Applications Second edition offers a long-awaited update and reissue of this seminal account of the subject. The revision retains foundational, original material from the frequently-cited 1974 volume but is completely updated throughout. As with the earlier version, the author hopes to take the reader �from the beginnings of the subject to the frontiers of research�. By omitting a few topics which are no longer of current interest, the book expands upon active and emerging areas. Also, the present state of knowledge regarding the 73 then-unsolved problems given at the end of the first edition is discussed and commented upon. In addition, a number of new unsolved problems are proposed. Using an engaging narrative style, this book provides thorough coverage of most parts of the subject, one of the oldest of all discrete mathematical structures and still one of the most relevant. However, in consequence of the huge expansion of the subject in the past 40 years, some topics have had to be omitted in order to keep the book of a reasonable length.
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.
A pair of orthogonal Latin squares of order six do not exist.
If
is the prime power decomposition of an integer v , and we define the arithmetic function n(v) by
then it is known, MacNeish (10) and Mann (11), that there exists a set of at least n(v) mutually orthogonal Latin squares (m.o.l.s.) of order v . We shall denote by N(v) the maximum possible number of mutually orthogonal Latin squares of order v . Then the Mann-MacNeish theorem can be stated as
MacNeish conjectured that the actual value of N(v) is n(v).
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.
By identifying all latin squares of order n with certain n2-element subsets of an n3-element ground set En a clutter Bn is obtained, which induces an independence system (En, In) in a natural way. Starting from Ryser's conditions for the completion of latin rectangles (cf. Mirsky [15]) we present special classes of circuits of (En, In) and extend Ryser's conditions slightly.Latin squares of order n correspond to the solutions of the planar 3-dimensional assignment problem and, in view of its solution via linear programming techniques, we present some first classes of facet-defining inequalities for P(In) resp. P(Bn), the convex hull of all those 0–1 vectors, which correspond to members of In resp. Bn.
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
List of solutions of the social golfer problem
- W Harvey
W. Harvey, List of solutions of the social golfer problem, http://www-icparc.doc.ic.ac.uk/ ∼ wh/golf/#B-upper.
Le probl eme des 36 oociers
- G Tarry
G. Tarry, Le probl eme des 36 oociers, C.R. Assoc. France Av. Sci. 29 (Part 2) (1900) 170–203.
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)
- C P Gomes
- D B Shmoys
C.P. Gomes, D.B. Shmoys, 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), Le Croisic, France 2002, pp. 291–305.