There is a planar graph almost as good as the complete graph
SCG '86 Proceedings of the second annual symposium on Computational geometry
On the structure of minimum-weight k-connected spanning networks
SIAM Journal on Discrete Mathematics
Approximating s-t minimum cuts in Õ(n2) time
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Minimum 0-extensions of graph metrics
European Journal of Combinatorics
Approximation algorithms for the 0-extension problem
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
An improved approximation algorithm for the 0-extension problem
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Graph sparsification by effective resistances
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Degree Bounded Network Design with Metric Costs
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Proceedings of the forty-first annual ACM symposium on Theory of computing
Graph partitioning using single commodity flows
Journal of the ACM (JACM)
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Vertex sparsifiers: new results from old techniques
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Optimal lower bounds for universal and differentially private steiner trees and TSPs
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
On vertex sparsifiers with Steiner nodes
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Routing in undirected graphs with constant congestion
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
On mimicking networks representing minimum terminal cuts
Information Processing Letters
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Suppose we are given a graph G = (V, E) and a set of terminals K ⊂ V. We consider the problem of constructing a graph H = (K, EH) that approximately preserves the congestion of every multicommodity flow with endpoints supported in K. We refer to such a graph as a flow sparsifier. We prove that there exist flow sparsifiers that simultaneously preserve the congestion of all multicommodity flows within an O(log k / log log k)-factor where |K| = k. This bound improves to O(1) if G excludes any fixed minor. This is a strengthening of previous results, which consider the problem of finding a graph H = (K, EH) (a cut sparsifier) that approximately preserves the value of minimum cuts separating any partition of the terminals. Indirectly our result also allows us to give a construction for better quality cut sparsifiers (and flow sparsifiers). Thereby, we immediately improve all approximation ratios derived using vertex sparsification in [14]. We also prove an Ω(log log k) lower bound for how well a flow sparsifier can simultaneously approximate the congestion of every multicommodity flow in the original graph. The proof of this theorem relies on a technique (which we refer to as oblivious dual certifcates) for proving super-constant congestion lower bounds against many multicommodity flows at once. Our result implies that approximation algorithms for multicommodity flow-type problems designed by a black box reduction to a "uniform" case on k nodes (see [14] for examples) must incur a super-constant cost in the approximation ratio.