Geometric lower bounds for parametric matroid optimization

David Eppstein
1995 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing - STOC '95  
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: (nr 1/3 ) 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 log r ) for uniform
more » ... matroids; upper bounds of O(mn 1/2 ) for arbitrary matroids and O(mn 1/2 / log * n) for uniform matroids were also known.
doi:10.1145/225058.225284 dblp:conf/stoc/Eppstein95 fatcat:vmdbwosnh5alfgf2el4idhgryq