Abstract
A partition problem in one-dimensional space is to seek a partition of a set of numbers that maximizes a given objective function. In some partition problems, the partition size, i.e., the number of nonempty parts in a partition, is fixed; while in others, the size can vary arbitrarily. We call the former the size-partition problem and the latter the open-partition problem. In general, it is much harder to solve open problems since the objective functions depend on size. In this paper, we propose a new approach by allowing empty parts and transform the open problem into a size problem allowing empty parts, called a relaxed-size problem. While the sortability theory has been established in the literature as a powerful tool to attack size partition problems, we develop the sortability theory for relaxed-size problems as a medium to solve open problems.
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References
Anily S, Federgruen A (1991) Structured partitioning problems. Oper Res 39:130–149
Boros E, Hammer PL (1989) On clustering problems with connected optima in Euclidean spaces. Discrete Math 75:81–88
Chakravarty AK, Orlin JB, Rothblum UG (1982) A partitioning problem with additive objective with an application to optimal inventory grouping for joint replenishment. Oper Res 30:1018–1022
Chang GJ, Chen FL, Huang LL, Hwang FK, Nuan ST, Rothblum UG, Sun IF, Wang JW, Yeh HG (1999) Sortabilities of partition properties. J Comb Optim 2:413–427
Gal S, Klots B (1995) Optimal partitioning which maximizes the sum of the weighted averages. Oper Res 43:500
Hwang FK (1981) Optimal partitions. J Optim Theory Appl 34:1–10
Hwang FK, Rothblum UG (2010) Optimality and clustering. Series on appl math, vol 1. World Scientific, Singapore
Hwang FK, Rothblum UG, Yao YC (1996) Localizing combinatorial properties of partitions. Discrete Math 160:1–23
Hwang FK, Sun J, Yao EY (1985) Optimal set partitioning. SIAM J Algebra Discrete Math 6:163–170
Kreweras G (1972) Sur les partitions non croisées d’un cycle. Discrete Math 1:333–350
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Chang, H., Hwang, F.K. & Rothblum, U.G. A new approach to solve open-partition problems. J Comb Optim 23, 61–78 (2012). https://doi.org/10.1007/s10878-010-9341-7
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DOI: https://doi.org/10.1007/s10878-010-9341-7