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Journal of Combinatorial Optimization

, Volume 22, Issue 4, pp 563–571 | Cite as

On the vertex characterization of single-shape partition polytopes

  • Yu-Chi Liu
  • Jun-Jie Pan
Article
  • 54 Downloads

Abstract

Given a partition of distinct d-dimensional vectors into p parts, the partition sum of the partition is the sum of vectors in each part. The shape of the partition is a p-tuple of the size of each part. A single-shape partition polytope is the convex hull of partition sums of all partitions that have a prescribed shape. A partition is separable if the convex hull of its parts are pairwise disjoint. The separability of a partition is a necessary condition for the associated partition sum to be a vertex of the single-shape partition polytope. It is also a sufficient condition for d=1 or p=2. However, the sufficiency fails to hold for d≥3 and p≥3. In this paper, we give some geometric sufficient conditions as well as some necessary conditions of vertices in general d and p. Thus, the open case for d=2 and p≥3 is resolved.

Keywords

Sum-partition problem Single-shape partition problem Partition polytope Separable partition Polytope vertex 

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References

  1. Barnes ER, Hoffman AJ, Rothblum UG (1992) On optimal partitions having disjoint convex and conic hulls. Math Program 54:69–86 MathSciNetMATHCrossRefGoogle Scholar
  2. Hwang FK, Rothblum UG (2010) Partitions: optimality and clustering. World Scientific, Singapore MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.School of Biological SciencesMonash UniversityVictoriaAustralia
  2. 2.Department of MathematicsFu Jen Catholic UniversityTaipei CountyTaiwan

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