In this paper we propose an algorithm for enumerating diagonal Latin squares of small order. It relies on specific properties of diagonal Latin squares to employ symmetry breaking techniques, and on several heuristic optimizations and bit arithmetic techniques to make use of computational power of state-of-the-art CPUs. Using this approach we enumerated diagonal Latin squares of order at most 9, and vertically symmetric diagonal Latin squares of order at most 10.
Authors are not aware of algorithms developed specifically for enumeration of diagonal Latin squares. Papers @cite_8 @cite_12 @cite_6 describe the approaches that led to enumeration of Latin squares of orders 9, 10 and 11. The corresponding algorithms heavily rely on the ability to permute rows and columns to construct equivalence classes and evaluate their properties. To the best of our knowledge the results of these papers are not applicable to diagonal Latin squares because the vast majority of row-column permutations break diagonal property. Also, diagonal Latin squares form relatively small equivalence classes.
@cite_6 @cite_12 @cite_8
2069392999 1729440882
We (1) determine the number of Latin rectangles with 11 columns and each possible number of rows, including the Latin squares of order 11, (2) answer some questions of Alter by showing that the number of reduced Latin squares of order n is divisible by f! where f is a particular integer close to ( 1 2 n, ) (3) provide a formula for the number of Latin squares in terms of permanents of (+1, −1)-matrices, (4) find the extremal values for the number of 1-factorisations of k-regular bipartite graphs on 2n vertices whenever 1 ≤ k ≤ n ≤ 11, (5) show that the proportion of Latin squares with a non-trivial symmetry group tends quickly to zero as the order increases. We show the existence of a large family of representations supported by the orbit closure of the determinant. However, the validity of our result is based on the validity of the celebrated ‘Latin square conjecture’ due to Alon and Tarsi or, more precisely, on the validity of an equivalent ‘column Latin square conjecture’ due to Huang and Rota. While there has been significant progress on algorithmic aspects of the Lovasz Local Lemma (LLL) in recent years, a noteworthy exception is when the LLL is used in the context of random permutations: the "lopsided" version of the LLL is usually at play here, and we do not yet have subexponential-time algorithms. We resolve this by developing a randomized polynomial-time algorithm for such applications. A noteworthy application is for Latin Transversals: the best-known general result here (, improving on Erdos and Spencer), states that any n x n matrix in which each entry appears at most (27 256)n times, has a Latin transversal. We present the first polynomial-time algorithm to construct such a transversal. Our approach also yields RNC algorithms: for Latin transversals, as well as the first efficient ones for the strong chromatic number and (special cases of) acyclic edge-coloring. Comon's (1994) well-known scheme for independent component analysis (ICA) is based on the maximal diagonalization, in a least-squares sense, of a higher-order cumulant tensor. In a previous paper, we proved that for fourth-order cumulants, the computation of an elementary Jacobi rotation is equivalent to the computation of the best rank-1 approximation of a fourth-order tensor. In this paper, we show that for third-order tensors, the computation of an elementary Jacobi rotation is again equivalent to a best rank-1 approximation; however, here, it is a matrix that has to be approximated. This favorable computational load makes it attractive to do "something third-order-like" for fourth-order cumulant tensors as well. We show that simultaneous optimal diagonalization of "third-order tensor slices" of the fourth-order cumulant is a suitable strategy. This "simultaneous third-order tensor diagonalization" approach (STOTD) is similar in spirit to the efficient JADE-algorithm.