TY - JOUR
T1 - Parametric polymatroid optimization and its geometric applications
AU - Katoh, Naoki
AU - Tamaki, Hisao
AU - Tokuyama, Takeshi
PY - 2002/10/1
Y1 - 2002/10/1
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.
KW - Combinatorial geometry
KW - K-level
KW - Matroid
UR - http://www.scopus.com/inward/record.url?scp=0036811613&partnerID=8YFLogxK
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U2 - 10.1142/S0218195902000967
DO - 10.1142/S0218195902000967
M3 - Article
AN - SCOPUS:0036811613
VL - 12
SP - 429
EP - 443
JO - International Journal of Computational Geometry and Applications
JF - International Journal of Computational Geometry and Applications
SN - 0218-1959
IS - 5
ER -