Vertex Sparsifiers: New Results from Old Techniques
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 flowsparsifier 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 flowsparsifier $H$ 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 0extension problem, the first one in which the preimages of each terminal are connected in $G$. Moreover, this result extends to minorclosed families of graphs. Our improved bounds immediately imply improved approximation guarantees for several terminalbased cut and ordering problems.
 Publication:

arXiv eprints
 Pub Date:
 June 2010
 arXiv:
 arXiv:1006.4586
 Bibcode:
 2010arXiv1006.4586E
 Keywords:

 Computer Science  Data Structures and Algorithms
 EPrint:
 An extended abstract appears in the 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), 2010. Final version to appear in SIAM J. Computing