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: (nr1/3 ) for a general n-element matroid with rank r, and (mα(n)) for the special case of parametric graph minimum spanning trees. The only previous lower bound was (n logr) for uniform matroids; upper bounds of O(mn1/2 ) for arbitrary matroids and O(mn1/2 / log∗ n) for uniform matroids were also known.

• Type: Conference
• Year: 1995
• Where: STOC
• Authors: David Eppstein
• Keywords: Algorithms, Lower Bound, Matroids, STOC 1995, Uniform Matroids