Discrete & Computational Geometry

, Volume 20, Issue 4, pp 463-476

Geometric Lower Bounds for Parametric Matroid Optimization

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

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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(nr 1/3 ) 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(mn 1/2 ) for arbitrary matroids and O(mn 1/2 / log * n) for uniform matroids were also known.