Label cover instances with large girth and the hardness of approximating basic k-spanner

Michael Dinitz, Guy Kortsarz, Ran Raz

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

6 Scopus citations

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(log1-εn)/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 ⊈ BPTIME(2polylog(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 (log1-εn)/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.

Original languageEnglish (US)
Title of host publicationAutomata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Proceedings
Pages290-301
Number of pages12
EditionPART 1
DOIs
StatePublished - Dec 1 2012
Externally publishedYes
Event39th International Colloquium on Automata, Languages, and Programming, ICALP 2012 - Warwick, United Kingdom
Duration: Jul 9 2012Jul 13 2012

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
NumberPART 1
Volume7391 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other39th International Colloquium on Automata, Languages, and Programming, ICALP 2012
CountryUnited Kingdom
CityWarwick
Period7/9/127/13/12

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

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    Dinitz, M., Kortsarz, G., & Raz, R. (2012). Label cover instances with large girth and the hardness of approximating basic k-spanner. In Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Proceedings (PART 1 ed., pp. 290-301). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7391 LNCS, No. PART 1). https://doi.org/10.1007/978-3-642-31594-7_25