Englert, Matthias, Gupta, A., Krauthgamer, Robert, Räcke, Harald, Talgam-Cohen, Inbal and Talwar, Kunal (2010) Vertex sparsifiers : new results from old techniques. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. Lecture Notes in Computer Science (6302). Springer Verlag, pp. 152-165. ISBN 9783642153686

Abstract

Given a capacitated graph G = (V,E) and a set of terminals K ⊆ V, how should we produce a graph H only on the terminals K so that every (multicommodity) flow between the terminals in G could be supported in H with low congestion, and vice versa? (Such a graph H is called a flow-sparsifier for G.) What if we want H to be a “simple” graph? What if we allow H to be a convex combination of simple graphs? Improving on results of Moitra [FOCS 2009] and Leighton and Moitra [STOC 2010], we give efficient algorithms for constructing: (a) a flow-sparsifier H that maintains congestion up to a factor of ${\smash{O(\frac{\log k}{\log \log k})}}$, where k = |K|. (b) a convex combination of trees over the terminals K that maintains congestion up to a factor of O(logk). (c) for a planar graph G, a convex combination of planar graphs that maintains congestion up to a constant factor. This requires us to give a new algorithm for the 0-extension problem, the first one in which the preimages of each terminal are connected in G. Moreover, this result extends to minor-closed families of graphs.Our bounds immediately imply improved approximation guarantees for several terminal-based cut and ordering problems.

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