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Label Cover Instances with Large Girth and the Hardness of Approximating Basic k-Spanner

  • Michael Dinitz
  • Guy Kortsarz
  • Ran Raz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)

Abstract

We study the well-known Label Cover problem under the additional requirement that problem instances have large girth. We show that if the girth is some k, the problem is roughly \(2^{(\log^{1-\epsilon} n)/k}\) hard to approximate for all constant ε > 0. A similar theorem was claimed by Elkin and Peleg [ICALP 2000] as part of an attempt to prove hardness for the basic k-spanner problem, but their proof was later found to have a fundamental error. Thus we give both the first non-trivial lower bound for the problem of Label Cover with large girth as well as the first full proof of strong hardness for the basic k-spanner problem, which is both the simplest problem in graph spanners and one of the few for which super-logarithmic hardness was not known. Assuming \(NP \not\subseteq BPTIME(2^{polylog(n)})\), we show (roughly) that for every k ≥ 3 and every constant ε > 0 it is hard to approximate the basic k-spanner problem within a factor better than \(2^{(\log^{1-\epsilon} n) / k}\). This improves over the previous best lower bound of only Ω(logn)/k from [17]. Our main technique is subsampling the edges of 2-query PCPs, which allows us to reduce the degree of a PCP to be essentially equal to the soundness desired. This turns out to be enough to basically guarantee large girth.

Keywords

Small Cycle Parallel Repetition Checkable Proof Direct Spanner Large Girth 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Michael Dinitz
    • 1
  • Guy Kortsarz
    • 2
  • Ran Raz
    • 1
  1. 1.Weizmann Institute of ScienceIsrael
  2. 2.Department of Computer ScienceRutgersCamdenUSA

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