Parametric polymatroid optimization and its geometric applications

Naoki Katoh; Hisao Tamaki; Takeshi Tokuyama

Parametric polymatroid optimization and its geometric applications

Naoki Katoh, Hisao Tamaki, Takeshi Tokuyama

Research output: Contribution to conference › Paper › peer-review

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(k^1/3n) 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 language	English
Pages	517-526
Number of pages	10
Publication status	Published - 1999 Jan 1
Externally published	Yes
Event	Proceedings of the 1999 10th Annual ACM-SIAM Symposium on Discrete Algorithms - Baltimore, MD, USA Duration: 1999 Jan 17 → 1999 Jan 19

Other

Other	Proceedings of the 1999 10th Annual ACM-SIAM Symposium on Discrete Algorithms
City	Baltimore, MD, USA
Period	99/1/17 → 99/1/19

ASJC Scopus subject areas

Chemical Health and Safety
Software
Safety, Risk, Reliability and Quality
Discrete Mathematics and Combinatorics

Cite this

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AU - Katoh, Naoki

AU - Tamaki, Hisao

AU - Tokuyama, Takeshi

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N2 - 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.

AB - 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.

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T2 - Proceedings of the 1999 10th Annual ACM-SIAM Symposium on Discrete Algorithms

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Parametric polymatroid optimization and its geometric applications

Abstract

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ASJC Scopus subject areas

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