The All-or-Nothing Flow Problem in Directed Graphs with Symmetric Demand Pairs

Abstract

We study the approximability of the All-or-Nothing multicommodity flow problem in directed graphs with symmetric demand pairs (SymANF). The input consists of a directed graph G = (V, E) and a collection of (unordered) pairs of nodes \({\mathcal{M}} = \{s_1t_1, s_2t_2, \dots, s_kt_k\}\). A subset \({\mathcal{M}}'\) of the pairs is routable if there is a feasible multicommodity flow in G such that, for each pair \(s_it_i \in{\mathcal{M}}'\), the amount of flow from s i to t i is at least one and the amount of flow from t i to s i is at least one. The goal is to find a maximum cardinality subset of the given pairs that can be routed. Our main result is a poly-logarithmic approximation with constant congestion for SymANF. We obtain this result by extending the well-linked decomposition framework of [6] to the directed graph setting with symmetric demand pairs. We point out the importance of studying routing problems in this setting and the relevance of our result to future work.