Update: A related blog post by Boaz Barak: Unique Games Conjecture – halfway there?

The 2-to-2 Games Conjecture is a somewhat weaker form of Khot’s unique game conjecture.

The paper is:** Pseudorandom Sets in Grassmann Graph have Near-Perfect Expansion **by Subhash Khot, Dor Minzer and Muli Safra. (See this post for a team photo and another breakthrough.)

**Here is the abstract:** We prove that pseudorandom sets in Grassmann graph have near-perfect expansion as hypothesized in [DKKMS-2]. This completes the proof of the 2-to-2 Games Conjecture (albeit with imperfect completeness) as proposed in [KMS, DKKMS-1], along with a contribution from [BKT].

The Grassmann graph contains induced subgraphs that are themselves isomorphic to Grassmann graphs of lower orders. A set is called pseudorandom if its density is o(1) inside all subgraphs whose order is O(1) lower than that of .

We prove that pseudorandom sets have expansion 1−o(1), greatly extending the results and techniques in [DKKMS-2].

This is a great result. The Grassman graphs are graphs with vertices corresponding to -dimensional subspaces of an -dimensional vector space over a (small as possible) field. Of course, they are of much interest to combinatorialists.

The earlier papers mentioned are:

[KMS] Subhash Khot, Dor Minzer, and Muli Safra. On independent sets, 2-to-2 games, and Grassmann graphs. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 576–589, 2017.

[DKKMS1] Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, Muli Safra, Towards a Proof of the 2-to-1 Games Conjecture?

[DKKMS2] – Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, Muli Safra, On Non-Optimally Expanding Sets in Grassmann Graphs.

[BKT] Boaz Barak, Pravesh Kothari, and David Steurer – personal communication.

### A field with one element

Talking to the authors one gets the impression that what seems to be missing for proving the unique game conjecture is a field with one element.

https://en.wikipedia.org/wiki/Field_with_one_element 😉

One of the main motivations in studying the UGC has been that various problems having shown to be UGC-hard. Does this new partial result suffice to to give unconditional hardness results for some interesting problems?

Lior, yes: See Appendix B of their paper. For instance, the UGC implies that it is NP-hard to do better than a 2-approximation for Vertex Cover. Their result shows it is NP-hard to do better than sqrt{2} (improving over the previous 1.36… bound of Dinur-Safra). This is a consequence of a hardness result for Independent Set that is somewhat more interesting.

I believe the lack of perfect completeness prevents them from obtaining the following consequence (Dinur-Regev-Mossel?): For every A > B > 2, it is NP-hard to A-color a B-colorable graph. (This is still open.)

Thanks Lior and James. James is the statement you mentioned a consequence of the unique game conjecture?

Hi Gil, such statements cannot be based on the UGC in any standard way because perfectly satisfiable UG instances are easy to solve. Since the “yes” instances for UG are 1-eps satisfiable, it seems tricky for a reduction to produce a genuinely k-colorable graph (as opposed to a graph that can be k-colored with an f(eps)-fraction of violated edges). This was one of the potential benefits of a 2-to-1 conjecture; this problem is still potentially hard even when the yes instances are 100% satisfiable.

The paper I was referring to (Dinur-Mossel-Regev) is here: https://arxiv.org/pdf/cs/0504062.pdf

The 2-to-1 conjecture would show that it is NP-hard to k-color a 4-colorable graph for any constant k. I think the Khot-Minzer-Safra paper gives an “almost” version where the yes instances are 4-colorable with an eps-fraction of violated edges.

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Two additional relevant papers: Small-Set Expansion in Shortcode Graph and the 2-to-2 Conjecture by Boaz Barak, Pravesh Kothari, David Steurer https://eccc.weizmann.ac.il/report/2018/077/

and

Small Set Expansion in The Johnson Graph | Subhash Khot, Dor Minzer, Dana Moshkovitz, Muli Safra https://eccc.weizmann.ac.il/report/2018/078/

A beautiful article First Big Steps Toward Proving the Unique Games Conjecture by Erica Klarreich

https://www.quantamagazine.org/computer-scientists-close-in-on-unique-games-conjecture-proof-20180424/

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Gil, can you elaborate on your remark about the field with one element? How might a better understanding of the field with one element lead to a proof of the unique games conjecture?