Tree metrics and edge-disjoint S-paths

Given an undirected graph G = (V,E) with a terminal set S ⊆ V , a weight function μ : ( S 2 ) → Z+ on terminal pairs, and an edge-cost a : E → Z+, the μweighted minimum-cost edge-disjoint S-paths problem (μ-CEDP) is to maximize ∑ P∈P μ(sP , tP )−a(P ) over all edge-disjoint sets P of S-paths, where sP , tP denote the ends of P and a(P ) is the sum of edge-cost a(e) over edges e in P . Our main result is a complete characterization of terminal weights μ for which μCEDP is tractable and admits a combinatorial min-max theorem. We prove that if μ is a tree metric, then μ-CEDP is solvable in polynomial time and has a combinatorial min-max formula, which extends Mader’s edge-disjoint S-paths theorem and its minimum-cost generalization by Karzanov. Our min-max theorem includes the dual half-integrality, which was earlier conjectured by Karzanov for a special case. We also prove that μ-EDP, which is μ-CEDP with a = 0, is NP-hard if μ is not a truncated tree metric, where a truncated tree metric is a weight function represented as pairwise distances between balls in a tree. On the other hand, μ-CEDP for a truncated tree metric μ reduces to μ′-CEDP for a tree metric μ′. Thus our result is best possible unless P = NP. As an application, we get a good approximation algorithm for μ-EDP with “near” tree metric μ by utilizing results from the theory of low-distortion embedding.