NDLI: Extensions and limits to vertex sparsification

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Extensions and limits to vertex sparsification

Content Provider	ACM Digital Library
Author	Moitra, Ankur Leighton, F. Thomson
Abstract	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, $E_{H})$ 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, $E_{H})$ (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 [14]. 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 [14] for examples) must incur a super-constant cost in the approximation ratio.
Starting Page	47
Ending Page	56
Page Count	10
File Format	PDF
ISBN	9781450300506
DOI	10.1145/1806689.1806698
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2010-06-05
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Oblivious reductions Multicommodity flow
Content Type	Text
Resource Type	Article

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