Current Developments in Mathematics
On the proof of the 2-to-2 Games Conjecture
Pages: 43 – 94
This article gives an overview of the recent proof of the 2‑to‑2 Games Conjecture in [68, 39, 38, 69] (with additional contributions from [75, 18, 67]). The proof requires an understanding of expansion in the Grassmann graph.
The 2‑to‑2 Games Conjecture is a lesser variant of the more wellknown Unique Games Conjecture in Theoretical Computer Science. These conjectures have applications to “hardness of approximation”, namely, the phenomenon that for many NP-hard problems, even computing approximate solutions to them remains NP-hard. They have several additional connections to computational complexity, algorithm design, analysis, and geometry.
The proof of the 2‑to‑2 Games Conjecture proves the Unique Games Conjecture “half-way” (at least in a certain technical sense) and in author’s opinion, provides a strong evidence in favor of the latter.
Research supported by NSF grant CCF-1813438, Simons Collaboration on Algorithms and Geometry grant, and the Simons Investigator Award.
Published 13 October 2021