Linear-shaped partition problems

We establish the polynomial-time solvability of a class of vector partition problems with linear objectives subject to restrictions on the number of elements in each part. c © 2000 Published by Elsevier Science B.V. All rights reserved. 1. Shaped partition problems The shaped partition problem concerns the partitioning of n vectors A; : : : ; A in d-space into p parts so as to maximize an objective function which is convex on the sum of vectors in each part subject to arbitrary constraints on the number of elements in each part. This class of problems has applications in diverse elds that include circuit layout, clustering, inventory, scheduling and reliability (see [2,3,5,9] and references therein) as well as important recent applications to symbolic computation [11]. In its outmost generality, the shaped partition problem instantly captures NP-hard problems hence is intractable [8]. The purpose of this article is to exhibit polynomial-time solvability for a broad class of shaped partition prob∗ Corresponding author. Fax: +972-4 823 5194. E-mail addresses: [email protected] (F.K. Hwang), [email protected] (S. Onn), [email protected] (U.G. Rothblum) 1 Research supported in part by a grant from the Israel Science Foundation (ISF), by a VPR grant at the Technion, and by the Fund for the Promotion of Research at the Technion. lems with linear objectives. To de ne the problem formally, describe our results and raise some remaining questions, we next introduce some notations. Let Q and N denote, respectively, the rational numbers and nonnegative integers. All vectors are columns by default. The vectors of all-ones and all-zeros, of dimension that is clear from the context, are denoted by 1 and 0, respectively. A p-partition of the set [n]:={1; : : : ; n} is an ordered collection =( 1; : : : ; p) of pairwise disjoint (possibly empty) sets whose union is [n]. The shape of is the tuple | |:=(| 1|; : : : ; | p|) of nonnegative integers which describes the number of elements in each part of . Let N p n :={ ∈N: 1 T = n} denote the set of all p-shapes of n. The rst ingredient of the problem data is a subset ⊆Nn of admissible shapes. The feasible solutions to the problem are then all partitions of [n] of admissible shape | | ∈ . The second ingredient of the problem data is a d × n matrix A whose jth column A represents d numerical attributes associated with the jth element of the partitioned set [n]. With each p-partition of [n] we associate the following d × p matrix whose kth column represents the total 0167-6377/00/$ see front matter c © 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0167 -6377(99)00069 -3 160 F.K. Hwang et al. / Operations Research Letters 26 (2000) 159–163 attribute vector of the kth part,