Vertex Sparsifiers: New Results from Old Techniques

Matthias Englert, Anupam Gupta, Robert Krauthgamer, Harald Räcke, Inbal Talgam-Cohen, Kunal Talwar
2014 SIAM journal on computing (Print)  
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
more » ... t algorithms for constructing: (a) a flow-sparsifier H that maintains congestion up to a factor of O( k/ k), where k = |K|, (b) a convex combination of trees over the terminals K that maintains congestion up to a factor of O( 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.
doi:10.1137/130908440 fatcat:p5ykt26u2fhvzb3ryge6llj6pm