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

Abstract

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 −ε, and in the “no” case we show that this fraction is at most ε, 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−ε.
The most important implication of this work is (assuming the hypothesis) an NP-hardness gap of 1/2−ε vs. ε 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−ε 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 NP-hardness 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
Title of host publicationSTOC 2018 Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing
Pages376-389
Number of pages14
ISBN (Electronic)9781450355599
DOIs
StatePublished - 20 Jun 2018
Event50th Annual ACM Symposium on Theory of Computing - United States, CA, Los Angeles
Duration: 25 Jun 201829 Jun 2018
Conference number: 50th

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
Volume50
ISSN (Print)0737-8017

Conference

Conference50th Annual ACM Symposium on Theory of Computing
Abbreviated titleSTOC 2018
Period25/06/1829/06/18

All Science Journal Classification (ASJC) codes

  • Software

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