Geometric lower bounds for parametric matroid optimization

Abstract

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 matroids; upper bounds of O(mn 1/2 ) for arbitrary matroids and O(mn 1/2 / log * n) for uniform matroids were also known.

Choosing Subsets with Maximum Weighted Average

“…Thus our original problem, of finding the maximum weighted average among all n − k point sets, can be rephrased as one of searching for the root of F (A). This can also be viewed as a parametric matroid optimization problem [7] in which the matroid in question is uniform. By binary search we can find the position of a root of F within an accuracy of (1 + ) in time O(n log 1/ ).…”

“…(See e.g. [7] for the equivalence between the breakpoints of F and the k-set problem studied in [23].) The worst case time of this algorithm may be larger than we wish, but it gives a simple method that may be useful in practice.…”

Using sparsification for parametric minimum spanning tree problems

“…We will describe our algorithm in geometric terms, following Eppstein [Epp95]. Graph the weight functions of the individual edges in G u as lines w = w e (λ), drawn in a plane with coordinates (λ, w).…”

“…First, we show that Z(λ) can be constructed in O(min{nm log n, T MST (2n, n)· b(m, n)}) time, where T MST (m, n) is the time to compute a minimum spanning tree and b(m, n) is the worst-case number of breakpoints of Z(λ). It is known that b(m, n) = O(m √ n) [Gus80,KaIb83] and b(m, n) = Ω(mα(n)) [Epp95], and that T MST (m, n) = O(m log β(m, n)) [GGST86] (here β(m, n) = min{i : log (i) n ≤ m/n}) 1 . Our algorithm improves on the Eisner-Severance method [EiSe76], which, when applied to parametric minimum spanning trees, takes O(T MST (m, n) · b(m, n)) time.…”

K-Levels of Concave Surfaces

“…Eppstein [9] pointed out that the complexity of k concave chains is asymptotically equivalent to the number of transitions of a parametric matroid of rank k, and his theory can be further extended to parametric polymatroids [12].…”