Discrete & Computational Geometry

, Volume 20, Issue 4, pp 463–476

Geometric Lower Bounds for Parametric Matroid Optimization

  • D. Eppstein

DOI: 10.1007/PL00009396

Cite this article as:
Eppstein, D. Discrete Comput Geom (1998) 20: 463. doi:10.1007/PL00009396

Abstract.

We relate the sequence of minimum bases of a matroid with linearly varying weights to three problems from combinatorial geometry: k -sets, lower envelopes of line segments, and convex polygons in line arrangements. Using these relations we show new lower bounds on the number of base changes in such sequences: Ω(nr1/3) for a general n -element matroid with rank r , and Ω(mα(n)) for the special case of parametric graph minimum spanning trees. The only previous lower bound was Ω(n log r) for uniform matroids; upper bounds of O(mn1/2) for arbitrary matroids and O(mn1/2/ log* n) for uniform matroids were also known.

Copyright information

© 1998 Springer-Verlag New York Inc.

Authors and Affiliations

  • D. Eppstein
    • 1
  1. 1.Department of Information and Computer Science, University of California, Irvine, CA 92717, USA http://www.ics.uci.edu/~eppstein/ eppstein@ics.uci.eduUS

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