A study of the quadratic semi-assignment polytope

We study a polytope which arises from a mixed integer programming formulation of the quadratic semi-assignment problem. We introduce an isomorphic projection and transform the polytope to a tractable full-dimensional polytope. As a result, some basic polyhedral properties, such as the dimension, the affine hull, and the trivial facets, are obtained. Further, we present valid inequalities called cut- and clique-inequalities and give complete characterizations for them to be facet-defining. We also discuss a simultaneous lifting of the clique-type facets. Finally, we show an application of the quadratic semi-assignment problem to hub location problems with some computational experiences.     

A polynomially solvable class of quadratic semi-assignment problems

The Quadratic Semi-Assignment Problem (QSAP) models a large variety of practical applications. In the present note we will consider a particular class of QSAP that can be solved by determining the maximum cost flow on a network. This class of problems arises in schedule synchronization and in transportation.     

Universal number partition problem with divisibility

We examine a version of the Universal Number Partition Problem with a divisibility property referred to as the Universal Shelf Packing Problem (USPP). We show that if a shelf length is a product of powers of two primes the USPP is always partitionable. In the case where the shelf length is a product of three distinct primes we propose an efficient scheme to determine when such a case is not partitionable. We also show that a shelf length that is a product of powers of four or more primes always has at least one partition failure. Our analysis uses elementary number theory, known results related to the linear Diophantine Frobenius problem, and a new result on Frobenius gaps.     

Binomial coefficients and quadratic fields

Let be a real quadratic field with discriminant where is an odd prime. For we determine modulo in terms of a Lucas sequence, the fundamental unit and the class number of .

     

GRASP with path-relinking for the generalized quadratic assignment problem

The generalized quadratic assignment problem (GQAP) is a generalization of the NP-hard quadratic assignment problem (QAP) that allows multiple facilities to be assigned to a single location as long as the capacity of the location allows. The GQAP has numerous applications, including facility design, scheduling, and network design. In this paper, we propose several GRASP with path-relinking heuristics for the GQAP using different construction, local search, and path-relinking procedures. We introduce a novel approximate local search scheme, as well as a new variant of path-relinking that deals with infeasibilities. Extensive experiments on a large set of test instances show that the best of the proposed variants is both effective and efficient.     

Algorithm for cardinality-constrained quadratic optimization

This paper describes an algorithm for cardinality-constrained quadratic optimization problems, which are convex quadratic programming problems with a limit on the number of non-zeros in the optimal solution. In particular, we consider problems of subset selection in regression and portfolio selection in asset management and propose branch-and-bound based algorithms that take advantage of the special structure of these problems. We compare our tailored methods against CPLEX’s quadratic mixed-integer solver and conclude that the proposed algorithms have practical advantages for the special class of problems we consider. The research of D. Bertsimas was partially supported by the Singapore-MIT alliance. The research of R. Shioda was partially supported by the Singapore-MIT alliance, the Discovery Grant from NSERC and a research grant from the Faculty of Mathematics, University of Waterloo.     

The partition problem

In this paper we describe several forms of thek-partition problem and give integer programming formulations of each case. The dimension of the associated polytopes and some basic facets are identified. We also give several valid and facet defining inequalities for each of the polytopes.Partial Support from NSF Grants DMS 8606188 and ECS 8800281 is gratefully acknowledged.     

Multipoint problem for hyperbolic equations with variable coefficients

By using the metric approach, we study the problem of classical well-posedness of a problem with multipoint conditions with respect to time in a tube domain for linear hyperbolic equations of order 2n (n ≥ 1) with coefficients depending onx. We prove metric theorems on lower bounds for small denominators appearing in the course of the solution of the problem.     

A graph partition problem

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The partition bargaining problem

Consider the problem of partitioning n items among d players where the utility of each player for bundles of items is additive; so, player r has utility for item i and the utility of that player for a bundle of items is the sum of the 's over the items i in his/her bundle. Each partition S of the items is then associated with a d-dimensional utility vector V^S whose coordinates are the utilities that the players assign to the bundles they get under S. Also, lotteries over partitions are associated with the corresponding expected utility vectors. We model the problem as a Nash bargaining game over the set of lotteries over partitions and provide methods for computing the corresponding Nash solution, to prescribed accuracy, with effort that is polynomial in n. In particular, we show that points in the pareto-optimal set of the corresponding bargaining set correspond to lotteries over partitions under which each item, with the possible exception of at most d(d-1)/2 items, is assigned in the same way.     

The partition problem for finite abelian groups

By the aid of a slight generalization of the Hales-Jewett theorem [Trans. Amer. Math. Soc.106 (1963), 222–229] we investigate the partition problem for finite Abelian groups. In particular the partition problem for the class of finitely generated free modules over Z_q is solved. By the results of Deuber and Rothschild [“Coll. Math. Soc. János Bolyai 18,” 1976] this yields a complete characterization of those finite Abelian groups with respect to which the class FAB of all finite Abelian groups has the partition property. Especially it turns out that FAB has the partition property with respect to the cyclic group Z_m, m > 1.     

The satisfactory partition problem

The SATISFACTORY PARTITION problem consists in deciding if a given graph has a partition of its vertex set into two nonempty parts such that each vertex has at least as many neighbors in its part as in the other part. This problem was introduced by Gerber and Kobler [Partitioning a graph to satisfy all vertices, Technical report, Swiss Federal Institute of Technology, Lausanne, 1998; Algorithmic approach to the satisfactory graph partitioning problem, European J. Oper. Res. 125 (2000) 283-291] and further studied by other authors but its complexity remained open until now. We prove in this paper that SATISFACTORY PARTITION, as well as a variant where the parts are required to be of the same cardinality, are NP-complete. However, for graphs with maximum degree at most 4 the problem is polynomially solvable. We also study generalizations and variants of this problem where a partition into k nonempty parts (k?3) is requested.     

Nonlocal boundary-value problem for parabolic equations with variable coefficients

We study the boundary-value problem for Petrovskii parabolic equations of arbitrary order with variable coefficients with conditions nonlocal in time. We establish conditions for the existence and uniqueness of a classical solution of this problem and prove metric theorems on lower bounds of small denominators appearing in the construction of a solution of the problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 7, pp. 915–921, July, 1995.This work was partially supported by the Foundation for Fundamental Research of the Ukrainian State Committee on Science and Technology.