Metric extension operators, vertex sparsifiers and Lipschitz extendability

Konstantin Makarychev*, Yury Makarychev

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contribution

31 Scopus citations

Abstract

We study vertex cut and flow sparsifiers that were recently introduced by Moitra [23], and Leighton and Moitra [18]. We improve and generalize their results. We give a new polynomial-time algorithm for constructing O(log k= log log k) cut and flow sparsifiers, matching the best known existential upper bound on the quality of a sparsi-fier, and improving the previous algorithmic upper bound of O(log2 k= log log k). We show that flow sparsifiers can be obtained from linear operators approximating minimum metric extensions. We introduce the notion of (linear) metric extension operators, prove that they exist, and give an exact polynomialtime algorithm for finding optimal operators. We then establish a direct connection between flow and cut sparsifiers and Lipschitz extendability of maps in Banach spaces, a notion studied in functional analysis since 1950s. Using this connection, we obtain a lower bound of Ω(√log k/log log k) for flow sparsifiers and a lower bound of Ω(√log k/log log k) for cut sparsifiers. We show that if a certain open question posed by Ball in 1992 has a positive answer, then there exist Õ(√log k) cut sparsifiers. On the other hand, any lower bound on cut sparsifiers better than Ω(√log k) would imply a negative answer to this question.

Original languageEnglish (US)
Title of host publicationProceedings - 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS 2010
Pages255-264
Number of pages10
DOIs
StatePublished - Dec 1 2010
Event2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS 2010 - Las Vegas, NV, United States
Duration: Oct 23 2010Oct 26 2010

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428

Other

Other2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS 2010
CountryUnited States
CityLas Vegas, NV
Period10/23/1010/26/10

ASJC Scopus subject areas

  • Computer Science(all)

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