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

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 nodesM = {s1t1, s2t2, . . . , sktk}. A subsetM′ of the pairs is routable if there is a feasible multicommodity flow in G such that, for each pair siti ∈ M′, the amount of flow from si to ti is at least one and the amount of flow from ti to si 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 [9] 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. ∗Department of Computer Science, University of Illinois at Urbana-Champaign. [email protected]. Supported in part by NSF grants CCF-1016684 and CCF-1319376. Part of this work was done while the author was supported by TTI Chicago on a sabbatical visit in Fall 2012. †Center for Computational Intractability, Princeton University and Department of Computer Science and DIMAP, University of [email protected] Supported in part by NSF grants CCF-1016684 and CCF-0844872. Part of this work was done while the author was an intern at TTI Chicago.