Small-set expansion in shortcode graph and the 2-to-2 conjecture

Boaz Barak, Pravesh K. Kothari, David Steurer

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

2 Scopus citations

Abstract

Dinur, Khot, Kindler, Minzer and Safra (2016) recently showed that the (imperfect completeness variant of) Khot’s 2 to 2 games conjecture follows from a combinatorial hypothesis about the soundness of a certain “Grassmanian agreement tester”. In this work, we show that soundness of Grassmannian agreement tester follows from a conjecture we call the “Shortcode Expansion Hypothesis” characterizing the non-expanding sets of the degree-two Short code graph. We also show the latter conjecture is equivalent to a characterization of the non-expanding sets in the Grassman graph, as hypothesized by a follow-up paper of Dinur et al. (2017). Following our work, Khot, Minzer and Safra (2018) proved the “Shortcode Expansion Hypothesis”. Combining their proof with our result and the reduction of Dinur et al. (2016), completes the proof of the 2 to 2 conjecture with imperfect completeness. We believe that the Shortcode graph provides a useful view of both the hypothesis and the reduction, and might be suitable for obtaining new hardness reductions.

Original languageEnglish (US)
Title of host publication10th Innovations in Theoretical Computer Science, ITCS 2019
EditorsAvrim Blum
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770958
DOIs
StatePublished - Jan 1 2019
Event10th Innovations in Theoretical Computer Science, ITCS 2019 - San Diego, United States
Duration: Jan 10 2019Jan 12 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume124
ISSN (Print)1868-8969

Conference

Conference10th Innovations in Theoretical Computer Science, ITCS 2019
CountryUnited States
CitySan Diego
Period1/10/191/12/19

All Science Journal Classification (ASJC) codes

  • Software

Keywords

  • Grassmann Graph
  • Shortcode
  • Small-Set Expansion
  • Unique Games Conjecture

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    Barak, B., Kothari, P. K., & Steurer, D. (2019). Small-set expansion in shortcode graph and the 2-to-2 conjecture. In A. Blum (Ed.), 10th Innovations in Theoretical Computer Science, ITCS 2019 [9] (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 124). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ITCS.2019.9