Extensions and limits to vertex sparsification
Author(s)Moitra, Ankur; Leighton, Frank Thomson
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Suppose we are given a graph G = (V, E) and a set of terminals K ⊂ V. We consider the problem of constructing a graph H = (K, E[subscript H]) that approximately preserves the congestion of every multicommodity flow with endpoints supported in K. We refer to such a graph as a flow sparsifier. We prove that there exist flow sparsifiers that simultaneously preserve the congestion of all multicommodity flows within an O(log k / log log k)-factor where |K| = k. This bound improves to O(1) if G excludes any fixed minor. This is a strengthening of previous results, which consider the problem of finding a graph H = (K, E[subscript H]) (a cut sparsifier) that approximately preserves the value of minimum cuts separating any partition of the terminals. Indirectly our result also allows us to give a construction for better quality cut sparsifiers (and flow sparsifiers). Thereby, we immediately improve all approximation ratios derived using vertex sparsification in . We also prove an Ω(log log k) lower bound for how well a flow sparsifier can simultaneously approximate the congestion of every multicommodity flow in the original graph. The proof of this theorem relies on a technique (which we refer to as oblivious dual certifcates) for proving super-constant congestion lower bounds against many multicommodity flows at once. Our result implies that approximation algorithms for multicommodity flow-type problems designed by a black box reduction to a "uniform" case on k nodes (see  for examples) must incur a super-constant cost in the approximation ratio.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Proceedings of the 42nd ACM symposium on Theory of computing (STOC '10)
Association for Computing Machinery (ACM)
F. Thomson Leighton and Ankur Moitra. 2010. Extensions and limits to vertex sparsification. In Proceedings of the 42nd ACM symposium on Theory of computing (STOC '10). ACM, New York, NY, USA, 47-56.
Author's final manuscript