Geometric Lower Bounds for Parametric Matroid Optimization

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. Received November 12, 1996, and in revised form February 19, 1997.