Counting Problems Associated With Steiner Trees In Graphs

Abstract

This paper considers counting problems associated with K-spanning and K-disconnecting sets for a specified terminal set K in an undirected graph G. In particular, we consider the problems of computing the number of Steiner trees and minK-cuts for G, as well as K-spanning and K-disconnecting sets of cardinality close to the minimum values. Among other things, these numbers are critical to the efficient approximation of K-connected reliability measures in stochastic networks. Although the counting problems considered in this paper are NP-hard in general, a large number of methods for finding shortest paths, min cuts, and Steiner trees in graphs can be extended to efficiently countK-spanning and K-disconnecting sets in important special cases.