Eranda Çela

Linear Assignment Problems and Extensions

Assignment problems deal with the question how to assign n items (e.g. jobs) to n machines (or workers) in the best possible way. They consist of two components: the assignment as underlying combinatorial structure and an objective function modeling the ”best way”.

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.

A dual framework for lower bounds of the quadratic assignment problem based on linearization

Abstract.A dual framework allowing the comparison of various bounds for the quadratic assignment problem (QAP) based on linearization, e.g. the bounds of Adams and Johnson, Carraresi and Malucelli, and Hahn and Grant, is presented. We discuss the differences of these bounds and propose a new and more general bounding procedure based on the dual of the linearization of Adams and Johnson. The new procedure has been applied to problems of dimension up to

2-medians in trees with pos/neg weights

n=72

Hamiltonian cycles in circulant digraphs with two stripes

, and the computational results indicate that the new bound competes well with existing linearization bounds and yields a good trade off between computation time and bound quality.

Heuristics for biquadratic assignment problems and their computational comparison

Abstract This paper deals with facility location problems with pos/neg weights in trees. We consider two different objective functions which model two different ways to handle obnoxious facilities. If we minimize the overall sum of the minimum weighted distances of the vertices from the facilities, the optimal solution has nice combinatorial properties, e.g., vertex optimality. For the pos/neg 2-median problem on a network with n vertices, these properties can be exploited to derive an O(n2) algorithm for trees, an O (n log n) algorithm for stars and a linear algorithm for paths. For the p-median problem with pos/neg weights on a path we give an O(pn2) algorithm. If we minimize the overall sum of the weighted minimum distances of the vertices from the facilities, we can show that there exists a finite set of O(n3) points in the tree which contains the locations of facilities in an optimal solution. This leads to an O(n3) algorithm for finding the optimum 2-medians in a tree. The complexity can be reduced to O(n2), if the medians are restricted to vertices or if the tree is a path.

The Wiener maximum quadratic assignment problem

Abstract The circulant traveling salesman problem ( CTSP ) is the problem of finding a minimum weight Hamiltonian cycle in a weighted graph with circulant distance matrix. The computational complexity of this problem is not known. In fact, even the complexity of deciding Hamiltonicity of the underlying graph is unknown. This paper provides a characterization of Hamiltonian digraphs with circulant distance matrix containing only two nonzero stripes . The corresponding conditions can be checked in polynomial time. Secondly, we show that all Hamiltonian cycles of a circulant 2-digraph are periodic. Based on these two results, a method for enumerating all Hamiltonian cycles in such digraphs is described. Moreover, two simple algorithms are derived for solving the sum and bottleneck versions of CTSP for circulant distance matrices with two nonzero stripes.

The Quadratic Assignment Problem with a Monotone Anti-Monge and a Symmetric Toeplitz Matrix: Easy and Hard Cases

Abstract The biquadratic assignment problem (BiQAP) is a generalization of the quadratic assignment problem (QAP). As for any hard optimization problem also for BiQAP, a reasonable effort to cope with the problem is trying to derive heuristics which solve it suboptimally and which, possibly, yield a good trade off between the solution quality and the time and memory requirements. In this paper we describe several heuristics for BiQAPs, in particular pair exchange algorithms (improvement methods) and variants of simulated annealing and taboo search. We implement these heuristics as C codes and analyze their performances.

Another well-solvable case of the QAP: Maximizing the job completion time variance

We investigate a special case of the maximum quadratic assignment problem where one matrix is a product matrix and the other matrix is the distance matrix of a one-dimensional point set. We show that this special case, which we call the Wiener maximum quadratic assignment problem, is NP-hard in the ordinary sense and solvable in pseudo-polynomial time. Our approach also yields a polynomial time solution for the following problem from chemical graph theory: find a tree that maximizes the Wiener index among all trees with a prescribed degree sequence. This settles an open problem from the literature.

Polynomially solvable cases of the constant rank unconstrained quadratic 0-1 programming problem

The Anti-Monge-Toeplitz QAP (AMT-QAP) is a restricted version of the Quadratic Assignment Problem (QAP), with a monotone Anti-Monge matrix and a symmetric Toeplitz matrix. The following problems can be modeled via the AMT-QAP: (P1) The “Turbine Problem”, i. e. the assignment of given masses to the vertices of a regular polygon such that the distance of the gravity center of the resulting system to the polygons center is minimized. (P2) The Traveling Salesman Problem on symmetric Monge matrices. (P3) The linear arrangement of records with given access probabilities in order to minimize the average access time.