Towards a proof of the 2-to-1 games conjecture?

Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, Muli Safra

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


We present a polynomial time reduction from gap-3LIN to label cover with 2-to-1 constraints. In the “yes” case the fraction of satisfied constraints is at least 1 −, and in the “no” case we show that this fraction is at most, assuming a certain (new) combinatorial hypothesis on the Grassmann graph. In other words, we describe a combinatorial hypothesis that implies the 2-to-1 conjecture with imperfect completeness. The companion submitted paper [Dinur, Khot, Kindler, Minzer and Safra, STOC 2018] makes some progress towards proving this hypothesis. Our work builds on earlier work by a subset of the authors [Khot, Minzer and Safra, STOC 2017] where a slightly different hypothesis was used to obtain hardness of approximating vertex cover to within factor of 2 − . The most important implication of this work is (assuming the hypothesis) an NP-hardness gap of 1/2 − vs. for unique games. In addition, we derive optimal NP-hardness for approximating the max-cut-gain problem, NP-hardness of coloring an almost 4-colorable graph with any constant number of colors, and the same 2 − NP-hardness for approximate vertex cover that was already obtained based on a slightly different hypothesis. Recent progress towards proving our hypothesis [Barak, Kothari and Steurer, ECCC TR18-077], [Dinur, Khot, Kindler, Minzer and Safra, STOC 2018] directly implies some new unconditional NPhardness results. These include new points of NP-hardness for unique games and for 2-to-1 and 2-to-2 games. More recently, the full version of our hypothesis was proven [Khot, Minzer and Safra, ECCC TR18-006].

Original languageEnglish (US)
Title of host publicationSTOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing
EditorsMonika Henzinger, David Kempe, Ilias Diakonikolas
PublisherAssociation for Computing Machinery
Number of pages10
ISBN (Electronic)9781450355599
StatePublished - Jun 20 2018
Event50th Annual ACM Symposium on Theory of Computing, STOC 2018 - Los Angeles, United States
Duration: Jun 25 2018Jun 29 2018

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017


Other50th Annual ACM Symposium on Theory of Computing, STOC 2018
Country/TerritoryUnited States
CityLos Angeles


  • 2-to-2 games
  • Grassmann graph
  • PCP
  • Unique games

ASJC Scopus subject areas

  • Software


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