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 #(m#(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. 1 Introduction In this paper we study connections between combinatorial geometry and matroid optimization theory, as represented by the following problem. Parametric matroid optimization. Given a matroid for which the elements have weights that vary as a linear function of a parameter t , what is the sequence of minimum weight bases over the range of values o...

In this work we present a random pseudo-polynomial algorithm for the problem of finding a base of specified value in a weighted represented matroid, subject to parity conditions. We also describe a specialized version of the algorithm suitable for finding a base of specified value in the intersection of two matroids. This result generalizes an existing pseudo-polynomial algorithm for computing exact arborescences in weighted graphs. Another (simpler) specialized version of our algorithms is also presented for computing perfect matchings of specified value in weighted graphs.

We present a weighted version of Megiddo's multidimensional search technique and use it to obtain faster algorithms for certain convex optimization problems in R d , for fixed d. This leads to speed-ups by a factor of log d n for applications such as solving the Lagrangian duals of matroidal knapsack problems and of constrained optimum subgraph problems on graphs of bounded tree-width.

Two applications of sparsification to parametric computing are given. The first is a fast algorithm for enumerating all distinct minimum spanning trees in a graph whose edge weights vary linearly with a parameter. The second is an asymptotically optimal algorithm for the minimum ratio spanning tree problem, as well as other search problems, on dense graphs. 1 Introduction In the parametric minimum spanning tree problem, one is given an n-node, m-edge undirected graph G where each edge e has a linear weight function w e (#)=a e +#b e . Let Z(#) denote the weight of the minimum spanning tree relative to the weights w e (#). It can be shown that Z(#) is a piecewise linear concave function of # [Gus80]; the points at which the slope of Z changes are called breakpoints. We shall present two results regarding parametric minimum spanning trees. First, we show that Z(#) can be constructed in O(min{nm log n, TMST (2n, n) # Department of Computer Science, Iowa State University, Ames, IA...

A linear-time algorithm for the minimum-ratio spanning tree problem on planar graphs is presented. The algorithm is based on a new planar minimum spanning tree algorithm. The approach extends to other parametric minimum spanning tree problems on planar graphs and to other families of graphs having small separators. 1 Introduction Suppose we are given an undirected graph G where each edge e has two weights a e and b e ; the b e 's are assumed to be either all negative or all positive. The minimum ratio spanning tree problem (MRST) [Cha77] is to find a spanning tree T of G such that the ratio P e2T a e = P e2T b e is minimized. One application of MRST arises in the design of communication networks. The number a e represents the cost of building link e, while b e represents the time required to build that link. The goal is to find a tree that minimizes the ratio of total cost over construction time. Other applications of MRST are given elsewhere [CMV89, Meg83]. The main result of thi...