Edge-Disjoint Paths and Unsplittable Flow

Finding disjoint paths in graphs is a problem that has attracted considerable attention from at least three perspectives: graph theory, VLSI design and network routing/flow. The corresponding literature is extensive. In this chapter we limit ourselves mostly to results on offline approximation algorithms for problems on general graphs as influenced from the network flow perspective. Surveys examining the underlying graph theory, combinatorial problems in VLSI, and disjoint paths on special graph classes can be found in [35, 36, 86, 88, 76, 83, 75, 51]. An instance of disjoint paths consists of a (directed or undirected) graph G = (V, E) and a multiset T = {(si, ti) : si ∈ V, ti ∈ V, i = 1, . . . , k} of k source-sink pairs. Any source or sink is called a terminal. An element of T is also called a commodity. One seeks a set of edge(or vertex-)disjoint paths P1, P2, . . . , Pk, where Pi is an si−ti path, i = 1, . . . , k. In the case of vertex-disjoint paths we are interested in paths that are internally disjoint, i.e., a terminal may appear in more than one pair in T . We abbreviate the edge-disjoint paths problem by Edp. The notation introduced will be used throughout the chapter to refer to an input instance. We will also denote |V | by n and |E| by m for the corresponding graph. Based on whether G is directed or undirected and the edgeor vertex-disjointness condition one obtains