Geometric lower bounds for parametric matroid optimization
David Eppstein · 1995
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: Q(nr1i3) for a general n-element matroid with rank r, and Q(rmr(n)) for the special case of parametric graph minimum spanning trees.The only previous lower bound was fl(n log r) for uniform matroids; upper bounds of 0(rmz1i2) for arbitrary matroids and 0(mn112/ log* n) for uniform matroids were also known.