Talk by Dana Moshkovitz: The Unique Games Conjecture – Proved (!!?)

In my “Mathocalypse” post, I mentioned that my wife Dana—who’s been working on Subhash Khot’s Unique Games Conjecture since ~2009, for the entire time I’ve known her—has for the past few weeks (while on sabbatical at the Simons Institute in Berkeley) been spending essentially all of her time laboring to understand OpenAI’s recent proof of the conjecture. Having now studied the proof closely in collaboration with Subhash Khot himself, Johan Håstad, Dor Minzer, and other colleagues, and (naturally) GPT-6 Astra, Dana probably now understands this crown jewel of OpenAI’s release as well as anyone does. This week, she gave two well-received whiteboard talks on the proof at Simons Institute. So, I and others encouraged her to give the same talks next week at UT Austin when she’s back here, and Dana agreed! Because of the large timely interest, we’re going to do a live Zoom option, and will also record the talk for later sharing on YouTube. Please note: While all are welcome, this is a 2-hour technical talk. The first hour will review key ideas from hardness of approximation, while the second hour will describe the new construction. —SA

Title: The Unique Games Conjecture – Proved (!!?)

Speaker: Dana Moshkovitz (UT Austin)

When: Wednesday Nov. 4, 3:30-5:30pm US Central (with a brief break in the middle)

Where: UT Austin, Gates Dell Complex 6.302. Live Zoom will be available here, and a recording will be available later.

Abstract: The Unique Games Conjecture of Khot says that it is NP-hard, given a system of linear equations of the form xi-xj=bij (mod p) over variables x1…xn, where 99% fraction of the equations can be satisfied, to find an assignment satisfying 1% of the equations (here p is a sufficiently large constant).

This conjecture implies optimal inapproximability results for many optimization problems.

A few weeks ago OpenAI released a writeup with a proof of the Unique Games Conjecture, as well as a Lean formalization of the proof. The writeup, like others released that day, is not written in a way that’s especially readable to human mathematicians.

My goal in this talk is to offer my mental map of the proof, in a way that’s helpful to theoretical computer scientists who are familiar with the area. I think of the innovation as a new code construction, “Hadamard with a noise test.” I’ll sketch how it fits into the NP-hardness reduction. Then I’ll describe the code construction fully and sketch its soundness analysis.

I’ll also point out a couple of ingredients of the proof that are of independent interest, like a simple quadratic that behaves “like a linear,” and a probability bound using a trick with harmonic numbers.