Vertex Sparsifiers: New Results from Old Techniques

Matthias Englert; Anupam Gupta; Robert Krauthgamer; Harald Raecke; Inbal Talgam; Kunal Talwar

doi:10.1137/130908440

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arXiv · 1006.4586

Vertex Sparsifiers: New Results from Old Techniques

Matthias Englert·Anupam Gupta·Robert Krauthgamer·Harald Raecke·Inbal Talgam·Kunal Talwar

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Abstract

Given a capacitated graph $G = (V,E)$ and a set of terminals $K \subseteq V$, 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 $O(\log k/\log \log k)$, where $k = |K|$, (b) a convex combination of trees over the terminals $K$ that maintains congestion up to a factor of $O(\log k)$, and (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 improved bounds immediately imply improved approximation guarantees for several terminal-based cut and ordering problems.

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BibTeX RIS

Matthias Englert, Anupam Gupta, Robert Krauthgamer, Harald Raecke, Inbal Talgam, Kunal Talwar. 2010-06-23. Vertex Sparsifiers: New Results from Old Techniques. https://doi.org/10.1137/130908440

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