Donald Bren School of Information and Computer Sciences

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: for a general n-element matroid with rank r, and omega(ma(n)) for the special case of parametric graph minimum spanning trees. The only previous lower bound was omega(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.