Excluded minors, network decomposition, and multicommodity flow
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Improved bounds for the max-flow min-multicut ratio for planar and Kr,r-free graphs
Information Processing Letters
Cut problems and their application to divide-and-conquer
Approximation algorithms for NP-hard problems
Approximations for the disjoint paths problem in high-diameter planar networks
Journal of Computer and System Sciences
Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms
Journal of the ACM (JACM)
Graph minors. XVI. excluding a non-planar graph
Journal of Combinatorial Theory Series B
Cuts, Trees and ℓ1-Embeddings of Graphs*
Combinatorica
Edge-Disjoint Paths in Planar Graphs
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Multicommodity flow, well-linked terminals, and routing problems
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
A Note on Multiflows and Treewidth
Algorithmica
Edge-Disjoint Paths in Planar Graphs with Constant Congestion
SIAM Journal on Computing
Genus and the geometry of the cut graph
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Maximum Edge-Disjoint Paths in Planar Graphs with Congestion 2
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Routing in undirected graphs with constant congestion
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
A Polylogarithmic Approximation Algorithm for Edge-Disjoint Paths with Congestion 2
FOCS '12 Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
Flow-cut gaps for integer and fractional multiflows
Journal of Combinatorial Theory Series B
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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 $\Omega(\sqrt{n})$ even for planar graphs [11] due to a simple topological obstruction and a major focus, following earlier work [14], has been understanding the gap if some constant congestion is allowed. In planar graphs the integrality gap is O(1) with congestion 2 [19,5]. In general graphs, recent work has shown the gap to be O(polylog(n)) [8,9] with congestion 2. Moreover, the gap is Ω(logΩ(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.