One-dimensional optimal bounded-shape partitions for Schur convex sum objective functions

F. H. Chang, H. B. Chen, J. Y. Guo (), F. K. Hwang and Uriel G. Rothblum
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F. H. Chang: National Chiaotung University
H. B. Chen: National Chiaotung University
J. Y. Guo: National Chiaotung University
F. K. Hwang: National Chiaotung University
Uriel G. Rothblum: Technion—Israel Institute of Technology

Journal of Combinatorial Optimization, 2006, vol. 11, issue 3, No 5, 339 pages

Abstract: Abstract Consider the problem of partitioning n nonnegative numbers into p parts, where part i can be assigned n i numbers with n i lying in a given range. The goal is to maximize a Schur convex function F whose ith argument is the sum of numbers assigned to part i. The shape of a partition is the vector consisting of the sizes of its parts, further, a shape (without referring to a particular partition) is a vector of nonnegative integers (n 1,..., n p ) which sum to n. A partition is called size-consecutive if there is a ranking of the parts which is consistent with their sizes, and all elements in a higher-ranked part exceed all elements in the lower-ranked part. We demonstrate that one can restrict attention to size-consecutive partitions with shapes that are nonmajorized, we study these shapes, bound their numbers and develop algorithms to enumerate them. Our study extends the analysis of a previous paper by Hwang and Rothblum which discussed the above problem assuming the existence of a majorizing shape.

Keywords: Optimal partition; Bounded-shape partition; Sum partition; Schur convex function (search for similar items in EconPapers)
Date: 2006
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DOI: 10.1007/s10878-006-7911-5

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