Vertex Sparsifiers and Abstract Rounding Algorithms
Moses Charikar, Tom Leighton, Shi Li, Ankur Moitra

TL;DR
This paper investigates the limitations of vertex sparsifiers in approximating all minimum cuts in a graph, establishes new lower bounds, and introduces algorithms for constructing near-optimal sparsifiers and routing schemes.
Contribution
It provides the first super-constant lower bounds for cut-sparsifiers, and offers new algorithms for vertex sparsifier construction and routing with optimal guarantees.
Findings
Lower bound of ( log^{1/4} k) for cut-sparsifiers
Optimal O( log k)-competitive Steiner oblivious routing schemes
Efficient construction of vertex-sparsifiers matching existential bounds
Abstract
The notion of vertex sparsification is introduced in \cite{M}, where it was shown that for any graph and a subset of terminals , there is a polynomial time algorithm to construct a graph on just the terminal set so that simultaneously for all cuts , the value of the minimum cut in separating from is approximately the same as the value of the corresponding cut in . We give the first super-constant lower bounds for how well a cut-sparsifier can simultaneously approximate all minimum cuts in . We prove a lower bound of -- this is polynomially-related to the known upper bound of . This is an exponential improvement on the bound given in \cite{LM} which in fact was for a stronger vertex sparsification guarantee, and did not apply to cut…
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Taxonomy
TopicsComplexity and Algorithms in Graphs · Advanced Graph Theory Research · Optimization and Search Problems
