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.
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This research is partially supported by ROC National Science grant NSC 92-2115-M-009-014.
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Chang, F.H., Chen, H.B., Guo, J.Y. et al. One-dimensional optimal bounded-shape partitions for Schur convex sum objective functions. J Comb Optim 11, 321–339 (2006). https://doi.org/10.1007/s10878-006-7911-5
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DOI: https://doi.org/10.1007/s10878-006-7911-5