Geometric lower bounds for parametric matroid optimization

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.