Disjoint Paths and Unsplittable Flow (Version 2.7)

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 focus mostly on 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 [70, 71, 139, 141, 120, 132, 119, 88]. We sporadically mention some results on fixed-parameter tractability, but this is not an aspect this survey covers at any depth. An instance of edge-disjoint (vertex-disjoint) paths consists of a graphG = (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. In the decision problem, one asks whether there is a a set of edge-disjoint (vertex-disjoint) paths P1, P2, . . . , Pk, where Pi is an si-ti path, i = 1, . . . , k. The graph G can be either directed or undirected. Typically, a terminal may appear in more than one pair in T . For vertex-disjoint paths one requires that the terminal pairs are mutually disjoint. We abbreviate the edge-disjoint paths problem by Edp and vertex-disjoint paths by Vdp. 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 four basic problem versions. The following polynomial-time reductions exist among them. Any undirected problem can be reduced to its directed counterpart by replacing an undirected edge with an appropriate gadget; both reductions maintain planarity. See [124] and [141, Chapter 70] for details. An edge-disjoint problem can be reduced to its vertex-disjoint counterpart by replacing G with its line graph (or digraph as the case may be). Directed vertex-disjoint paths reduce to directed edge-disjoint paths by replacing every vertex with a pair of new vertices connected by an edge. There is no known reduction from a directed to an undirected problem. These