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

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].