Small-Set Expansion in Shortcode Graph and the 2-to-2 Conjecture

TL;DR

This paper links the 2-to-2 conjecture to the structure of non-expanding sets in shortcode and Grassman graphs, completing the proof of the conjecture and suggesting new avenues for research.

Contribution

It establishes the equivalence between the inverse shortcode hypothesis and the non-expanding sets characterization, completing the proof of the 2-to-2 conjecture.

Findings

Proves the inverse shortcode hypothesis implies the 2-to-2 conjecture.

Shows the inverse shortcode hypothesis is equivalent to a Grassman graph characterization.

Completes the proof of the 2-to-2 conjecture with imperfect completeness.

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 the hypothesis of Dinur et. al. follows from a conjecture we call the "Inverse Shortcode Hypothesis" characterizing the non-expanding sets of the degree-two shortcode 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 "Inverse Shortcode 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. Moreover, we believe that the shortcode graph provides a useful view of both the hypothesis and the reduction, and might be useful in extending it further.