Tree metrics and edge-disjoint $S$-paths

Abstract

Given an undirected graph $G=(V,E)$ with a terminal set $S \subseteq V$, a weight function on terminal pairs, and an edge-cost $a: E \rightarrow \mathbf{Z}_+$, the $\mu$-weighted minimum-cost edge-disjoint $S$-paths problem ($\mu$-CEDP) is to maximize $\sum \nolimits _{P \in \mathcal{P}} \mu (s_P,t_P) - a(P)$ over all edge-disjoint sets $\mathcal{P}$ of $S$-paths, where $s_P,t_P$ 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 $\mu$ for which $\mu$-CEDP is tractable and admits a combinatorial min–max theorem. We prove that if $\mu$ is a tree metric, then $\mu$-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 $\mu$-EDP, which is $\mu$-CEDP with $a = 0$, is NP-hard if $\mu$ 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, $\mu$-CEDP for a truncated tree metric $\mu$ reduces to $\mu '$-CEDP for a tree metric $\mu '$. Thus our result is best possible unless P = NP. As an application, we obtain a good approximation algorithm for $\mu$-EDP with “near” tree metric $\mu$ by utilizing results from the theory of low-distortion embedding.