Abstract
Consider partitions of a given set A of n distinct points in general position in ℝd into parts where each pair of parts can be separated by a hyperplane that contains a given set of points E. We consider the problem of counting and generating all such partitions (correcting a classic 1967 result of Harding about the number of such partitions into two parts). Applications of the result to partition problems are presented.
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References
Alon N, Onn S (1999) Separable partitions. Discrete Appl Math 91:39–51
Barnes ER, Hoffman AJ, Rothblum UG (1992) Optimal partitions having disjoint convex and conic hulls. Math Program 54:69–86
Buck RC (1943) Partition of space. Am Math Mon 50:541–544
Capoyleas V, Rote G, Woeginger G (1991) Geometric clusterings. J Algorithms 12:341–356
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
Chakravarty AK, Orlin JB, Rothblum UG (1985) Consecutive optimizors for a partitioning problem with applications to optimal inventory groupings for joint replenishment. Oper Res 33:820–834
Edelsbrunner H (1987) Algorithms in combinatorial geometry. Springer, Berlin
Gal S, Klots B (1995) Optimal partitioning which maximizes the sum of the weighted averages. Oper Res 43:500–508
Garey MR, Hwang FK, Johnson DS (1977) Algorithms for a set partitioning problem arising in the design of multipurpose units. IEEE Trans Comput 26:321–328
Golany B, Hwang FK, Rothblum UG (2008) Sphere-separable partitions of multi-parameter elements. Discrete Appl Math 156:838–845
Harding EF (1967) The number of partitions of a set of n points in k dimensions induced by hyperplanes. Proc Edinb Math Soc 15:285–289
Hwang FK, Mallows CL (1995) Enumerating consecutive and nested partitions. J Comb Theory, Ser A 70:323–333
Hwang FK, Rothblum UG (2010) Partitions: Optimality and clustering. World Scientific, Singapore
Hwang FK, Onn S, Rothblum UG (1999) A polynomial time algorithm for shaped partition problems. SIAM J Optim 10:70–81
Onn S, Schulman LJ (2001) The vector partition problem for convex objective functions. Math Oper Res 26:583–590
Pfersky U, Rudolf R, Woeginger G (1994) Some geometric clustering problems. Nord J Comput 1:246–263
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F.K. Hwang is retired. E-mail: fkhwang@gmail.com.
Research of U.G. Rothblum was supported in part by an ISF grant, by the Bernstein Research Fund (administered by the VPR at the Technion) and by the Fund for the Promotion of Research at the Technion.
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Hwang, F.K., Rothblum, U.G. On the number of separable partitions. J Comb Optim 21, 423–433 (2011). https://doi.org/10.1007/s10878-009-9263-4
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DOI: https://doi.org/10.1007/s10878-009-9263-4