Parametric polymatroid optimization and its geometric applications

Naoki Katoh, Hisao Tamaki, Takeshi Tokuyama

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)

Abstract

We give an optimal bound on the number of transitions of the minimum weight base of an integer valued parametric polymatroid. This generalizes and unifies Tamal Dey's O(k1/3n) upper bound on the number of k-sets (and the complexity of the k-level of a straight-line arrangement), David Eppstein's lower bound on the number of transitions of the minimum weight base of a parametric matroid, and also the Θ(kn) bound on the complexity of the at-most-k level (the union of i-levels for i = 1, 2,..., k) of a straight-line arrangement. As applications, we improve Welzl's upper bound on the sum of the complexities of multiple levels, and apply this bound to the number of different equal-sized-bucketings of a planar point set with parallel partition lines. We also consider an application to a special parametric transportation problem.

Original languageEnglish
Pages (from-to)429-443
Number of pages15
JournalInternational Journal of Computational Geometry and Applications
Volume12
Issue number5
DOIs
Publication statusPublished - 2002 Oct 1

Keywords

  • Combinatorial geometry
  • K-level
  • Matroid

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics

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