A matroid algorithm and its application to the efficient solution of two optimization problems on graphs

We address the problem of finding a minimum weight baseB of a matroid when, in addition, each element of the matroid is colored with one ofm colors and there are upper and lower bound restrictions on the number of elements ofB with colori, fori = 1, 2,,m. This problem is a special case of matroid intersection. We present an algorithm that exploits the special structure, and we apply it to two optimization problems on graphs. When applied to the weighted bipartite matching problem, our algorithm has complexity O(|EV|+|V| ²log|V|). HereV denotes the node set of the underlying bipartite graph, andE denotes its edge set. The second application is defined on a general connected graphG = (V,E) whose edges have a weight and a color. One seeks a minimum weight spanning tree with upper and lower bound restrictions on the number of edges with colori in the tree, for eachi. Our algorithm for this problem has complexity O(|EV|+m ² |V|+ m|V| ²). A special case of this constrained spanning tree problem occurs whenV ^* is a set of pairwise nonadjacent nodes ofG. One must find a minimum weight spanning tree with upper and lower bound restrictions on the degree of each node ofV ^*. Then the complexity of our algorithm is O(|VE|+|V ^* V| ²). Finally, we discuss a new relaxation of the traveling salesman problem.This report was supported in part by NSF grant ECS 8601660.