The quadratic assignment problem with a monotone anti-Monge and a symmetric Toeplitz matrix: Easy and hard cases

This paper investigates a restricted version of the Quadratic Assignment Problem (QAP), where one of the coefficient matrices is an Anti-Monge matrix with non-decreasing rows and columns and the other coefficient matrix is a symmetric Toeplitz matrix. This restricted version is called the Anti-Monge—Toeplitz QAP. There are three well-known combinatorial problems that can be modeled via the Anti-Monge—Toeplitz QAP: (Pl) The Turbine Problem, i.e. the assignment of given masses to the vertices of a regular polygon such that the distance of the center of gravity of the resulting system to the center of the polygon is minimized. (P2) The Traveling Salesman Problem on symmetric Monge distance matrices. (P3) The arrangement of data records with given access probabilities in a linear storage medium in order to minimize the average access time. We identify conditions on the Toeplitz matrixB that lead to a simple solution for the Anti-Monge—Toeplitz QAP: The optimal permutation can be given in advance without regarding the numerical values of the data. The resulting theorems generalize and unify several known results on problems (P1), (P2), and (P3). We also show that the Turbine Problem is NP-hard and consequently, that the Anti-Monge—Toeplitz QAP is NP-hard in general. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Dedicated to the memory of Gene LawlerThis research has been supported by the Spezialforschungsbereich F 003 Optimierung und Kontrolle, Projektbereich Diskrete Optimierung.     

Improved semidefinite programming bounds for quadratic assignment problems with suitable symmetry

Semidefinite programming (SDP) bounds for the quadratic assignment problem (QAP) were introduced in Zhao et?al. (J Comb Optim 2:71–109, 1998). Empirically, these bounds are often quite good in practice, but computationally demanding, even for relatively small instances. For QAP instances where the data matrices have large automorphism groups, these bounds can be computed more efficiently, as was shown in Klerk and Sotirov (Math Program A, 122(2), 225–246, 2010). Continuing in the same vein, we show how one may obtain stronger bounds for QAP instances where one of the data matrices has a transitive automorphism group. To illustrate our approach, we compute improved lower bounds for several instances from the QAP library QAPLIB.     

A contribution to quadratic assignment problems

We discuss special eases of the quadratic assignment problem (QAP) being polynomially solvable. In particular we give an algebraic condition for the cost; Matrices of a QAP which guarantees that it is equivalent with a linear assignment problem. Based on these results we develop an approximation algorithm for QAPs with non-negative symmetric cost matrices.     

Completely normal matrices

Normal matrices in which all submatrices are normal are said to be completely normal. We characterize this class of matrices, determine the possible inertias of a particular completely normal matrix, and show that real matrices in this class are closed under (general) Schur complementation. We provide explicit formulas for the Moore–Penrose inverse of a completely normal matrix of size at least four. A result on irreducible principally normal matrices is derived as well.     

循环矩阵的两种特性的判定

揭示几类矩阵之间的紧密联系.借助于群的子群的判定以及循环布尔矩阵是本原矩阵的判定方法,得到循环模糊矩阵成为幂等矩阵的充要条件,反循环布尔矩阵成为本原矩阵的充要条件.并给出了循环模糊矩阵成为幂等矩阵的判定方法,反循环布尔矩阵成为本原矩阵的判定方法.     

Williamson matrices up to order 59

A recent result of Schmidt has brought Williamson matrices back into the spotlight. In this article, a new algorithm is introduced to search for hard to find Williamson matrices. We find all nonequivalent Williamson matrices of odd order n up to n = 59. It turns out that there are none for n = 35, 47, 53, 59 and it seems that the Turyn class may be the only infinite class of these matrices.      

A new infinite family of minimally nonideal matrices

Minimally nonideal matrices are a key to understanding when the set covering problem can be solved using linear programming. The complete classification of minimally nonideal matrices is an open problem. One of the most important results on these matrices comes from a theorem of Lehman, which gives a property of the core of a minimally nonideal matrix. Cornuéjols and Novick gave a conjecture on the possible cores of minimally nonideal matrices. This paper disproves their conjecture by constructing a new infinite family of square minimally nonideal matrices. In particular, we show that there exists a minimally nonideal matrix with r ones in each row and column for any r?3.     

Taussky定理推广与应用

本文在Ｃａｓｓｉｎｉ卵形域上推广了Ｔａｕｓｓｋｙ定理。所得结果修正了Ｂｒａｕｅｒ定理,作为应用给出不可约双对角占优矩阵非奇异的充要条件,最后把基本结果推广到分块矩阵上。     

Approximation of high-dimensional kernel matrices by multilevel circulant matrices

Kernels are important in developing a variety of numerical methods, such as approximation, interpolation, neural networks, machine learning and meshless methods for solving engineering problems. A common problem of these kernel-based methods is to calculate inverses of kernel matrices generated by a kernel function and a set of points. Due to the denseness of these matrices, finding their inverses is computationally costly. To overcome this difficulty, we introduce in this paper an approximation of the kernel matrices by appropriate multilevel circulant matrices so that the fast Fourier transform can be applied to reduce the computational cost. Convergence analysis for the proposed approximation is established based on certain decay properties of the kernels.     

Effective partitioning method for computing generalized inverses and their gradients

We extend the algorithm for computing {1}, {1, 3}, {1, 4} inverses and their gradients from [11] to the set of multiple-variable rational and polynomial matrices. An improvement of this extension, appropriate to sparse polynomial matrices with relatively small number of nonzero coefficient matrices as well as in the case when the nonzero coefficient matrices are sparse, is introduced. For that purpose, we exploit two effective structures form [6], which make use of only nonzero addends in polynomial matrices, and define their partial derivatives. Symbolic computational package MATHEMATICA is used in the implementation. Several randomly generated test matrices are tested and the CPU times required by two used effective structures are compared and discussed.     

A new exact discrete linear reformulation of the quadratic assignment problem

The quadratic assignment problem (QAP) is a challenging combinatorial problem. The problem is NP-hard and in addition, it is considered practically intractable to solve large QAP instances, to proven optimality, within reasonable time limits. In this paper we present an attractive mixed integer linear programming (MILP) formulation of the QAP. We first introduce a useful non-linear formulation of the problem and then a method of how to reformulate it to a new exact, compact discrete linear model. This reformulation is efficient for QAP instances with few unique elements in the flow or distance matrices. Finally, we present optimal results, obtained with the discrete linear reformulation, for some previously unsolved instances (with the size n = 32 and 64), from the quadratic assignment problem library, QAPLIB.     

Binary Representation of Normalized Symmetric and Correlation Matrices

With a view toward the correlation matrices, it is shown that the normalized real symmetric matrices are the affine hull of the binary correlation matrices, while the convex hull is a proper subset of the correlation matrices. A number of ways to identify the correlation matrices in the affine hull are discussed.     

Binary Representation of Normalized Symmetric and Correlation Matrices

With a view toward the correlation matrices, it is shown that the normalized real symmetric matrices are the affine hull of the binary correlation matrices, while the convex hull is a proper subset of the correlation matrices. A number of ways to identify the correlation matrices in the affine hull are discussed.     

A Family of Asymmetrical Orthogonal Arrays with Run Sizes 4p^2

Nowadays orthogonal arrays play important roles in statistics, computer science, coding theory and cryptography. The usual difference matrices are essential for the construction of many mixed orthogonal arrays. But there are also many orthogonal arrays, especially mixed-level or asymmetrical which can not be obtained by the usual difference matrices. In order to construct these asymmetrical orthogonal arrays, a class of special matrices, so-called generalized difference matrices, were discovered by Zhang（1989, 1990, 1993） by the orthogonal decompositions of projective matrices. In this article, an interesting equivalent relationship between the orthogonal arrays and the generalized difference matrices is presented. As an application, a family of orthogonal arrays of run sizes 4p2, such as L36（6^13^42^10）, are constructed.     

Symmetric Weighing Matrices Constructed using Group Matrices

A weighing matrix of order n and weight m² is a square matrix M of order n with entries from {-1,0,+1} such that MM^T=m²I where I is the identity matrix of order n. If M is a group matrix constructed using a group of order n, M is called a group weighing matrix. Recently, group weighing matrices were studied intensively, especially when the groups are cyclic and abelian. In this paper, we study the abelian group weighing matrices that are symmetric, i.e.M^T=M. Some new examples are found. Also we obtain a few exponent bounds on abelian groups that admit symmetric group weighing matrices. In particular, we prove that there is no symmetric abelian group weighing matrices of order 2p^r and weight p² where p is a prime and p≥ 5.Communicated by: K.T. Arasu     

等差-等比循环矩阵的逆矩阵求法

本文给出了等差-等此循环矩阵的逆矩阵求法,推广文[5]的结果.     

Set inertia-preserving matrices∗ †

The inertia-preservers of several sets of matrices are identified. The sets include: all real matrices, all complex matrices, triangular matrices, real symmetric matrices and Hermitian matrices.     

On the eigenvalues of double band matrices

We consider matrices containing two diagonal bands of positive entries. We show that all eigenvalues of such matrices are of the form rζ, where r is a nonnegative real number and ζ is a pth root of unity, where p is the period of the matrix, which is computed from the distance between the bands. We also present a problem in the asymptotics of spectra in which such double band matrices are perturbed by banded matrices.     

Set inertia-preserving matrices

The inertia-preservers of several sets of matrices are identified. The sets include: all real matrices, all complex matrices, triangular matrices, real symmetric matrices and Hermitian matrices.     

k-广义Hermite矩阵

本文给出了k-广义(反)Hermite矩阵的概念,研究了它的性质及其与k-广义酉矩阵之间的联系,推广了酉矩阵和(反)Hermite矩阵的相应结果.