2 01 4 Vertex Sparsifiers : New Results from Old Techniques ∗

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 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 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); (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. A preliminary version appeared in the Proceedings of the 13th Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2010). Department of Computer Science and DIMAP, University of Warwick, Coventry, UK. Supported by EPSRC grant EP/F043333/1 and DIMAP (the Centre for Discrete Mathematics and its Applications). [email protected]. Computer Science Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA. Research was partly supported by the NSF award CCF-0729022, and an Alfred P. Sloan Fellowship. This research was done when visiting Microsoft Research SVC, La Avenida, Mountain View CA. [email protected]. This work was supported in part by The Israel Science Foundation (grant #452/08), and by a Minerva grant. Weizmann Institute of Science, Rehovot, Israel. [email protected]. Institut für Informatik, Technische Universität München, Munich, Germany. [email protected]. Computer Science Department, Stanford University, Stanford, CA 94350, USA. Microsoft Research Silicon Valley, 1065 La Avenida, Mountain View, CA, USA. [email protected].