Approximate duality of multicommodity multiroute flows and cuts: single source case

Given an integer h, a graph G = (V,E) with arbitrary positive edge capacities and k pairs of vertices (s1, t1), (s2, t2), . . . , (sk, tk), called terminals, an h-route cut is a set F ⊆ E of edges such that after the removal of the edges in F no pair si−ti is connected by h edge-disjoint paths (i.e., the connectivity of every si− ti pair is at most h− 1 in (V,E\F )). The h-route cut is a natural generalization of the classical cut problem for multicommodity flows (take h = 1). The main result of this paper is an O(h2(h + log k))approximation algorithm for the minimum h-route cut problem in the case that s1 = s2 = · · · = sk, called the single source case. As a corollary of it we obtain an approximate duality theorem for multiroute multicommodity flows and cuts with a single source. This partially answers an open question posted in several previous papers dealing with cuts for multicommodity multiroute problems.