Maximum Edge-Disjoint Paths in k-Sums of Graphs

Abstract

We consider the approximability of the maximum edge-disjoint paths problem (MEDP) in undirected graphs, and in particular, the integrality gap of the natural multicommodity flow based relaxation for it. The integrality gap is known to be Ω( √ n) even for planar graphs [14] due to a simple topological obstruction and a major focus, following earlier work [17], has been understanding the gap if some constant congestion is allowed. In planar graphs the integrality gap is O(1) with congestion 2 [23, 7]. In general graphs, recent work has shown the gap to be polylog(n) [10, 11] with congestion 2. Moreover, the gap is log Ω(c) n in general graphs with congestion c for any constant c ≥ 1 [1]. It is natural to ask for which classes of graphs does a constant-factor constant-congestion property hold. It is easy to deduce that for given constant bounds on the approximation and congestion, the class of "nice" graphs is minor-closed. Is the converse true? Does every proper minor-closed family of graphs exhibit a constant factor, constant congestion bound relative to the LP relaxation? We conjecture that the answer is yes. One stumbling block has been that such bounds were not known for bounded treewidth graphs (or even treewidth 3). In this paper we give a polytime algorithm which takes a fractional routing solution in a graph of bounded treewidth and is able to integrally route a constant fraction of the LP solution's value. Note that we do not incur any edge congestion. Previously this was not known even for series parallel graphs which have treewidth 2. The algorithm is based on a more general argument that applies to k-sums of graphs in some graph family, as long as the graph family has a constant factor, constant congestion bound. We then use this to show that such bounds hold for the class of k-sums of bounded genus graphs.

New algorithms for maximum disjoint paths based on tree-likeness

“…3.1]; see also Proposition 3.3 of Chekuri et al [14]. As a corollary of Theorem 2, we immediately obtain the following proposition about the integrality gap of MaxEDP LP.…”

“…Recently, Ene et al [21] showed that MaxEDP admits an O(w 3 )-approximation algorithm on graphs of treewidth at most w. Theirs is the best known approximation ratio in terms of w, improving on an earlier O(w · 3 w )-approximation algorithm due to Chekuri et al [15]. This shows that the problem seems more amenable on "tree-like" graphs.…”

“…A well-known integrality gap instance due to Garg et al [26] shows that this relaxation has an integrality gap of Ω( √ n), and this is the main obstacle for improving the O( √ n)-approximation ratio in general graphs. This led Chekuri et al [15] to study the approximability of MaxEDP with respect to the treewidth of the underlying graph. In particular, they pose the following conjecture:…”

Polynomial Bounds for the Grid-Minor Theorem

“…The decomposition W i already guarantees that S i has the α BW -bandwidth property. If S i is not a good router, then it must be impossible to send large amounts of flow from S i to T in G. In this case, using known techniques (see appendix of [CNS13]), we can show that we can delete an edge from G[S i ], while preserving the 1-well-linkedness of the terminals 6 , contradicting the minimality of G. Therefore, if W i contains at least one large cluster, then it contains a good router. Assume now that all clusters in W i are small.…”

Excluded Grid Theorem

“…that the paths in P(C) are internally disjoint from C ∪ T . The following lemma can be seen as a variation of the Deletable Edge Lemma of Chekuri, Khanna and Shepherd [5] (the proof of their original lemma can be found in [9]), though it is somewhat simpler. The proof is omitted from this extended abstract.…”