Two works regarding the Unique Games (UG) Conjecture

  1. A new point of NP-hardness for Unique Games by Ryan O'Donnell, John Wright
  2. Hypercontractivity, Sum-of-Squares Proofs, and their Applications by Boaz Barak, Aram W. Harrow, Jonathan Kelner, David Steurer, Yuan Zhou
See STOC'12 proceedings, May 20, 2012

Oded's comments

1) A new point of NP-hardness for Unique Games

The focus is on a (popular) special case of UG that consists of a system of linear equations in two variables modulo $q$. Let $2LIN(c,s)$ be the problem of finding an assignment that satisfies an $s$ fraction of the equations, when given a system for which a $c$ fraction of the equations can be satisfied. The standard UGC refers to points of the form $(1-\eps,\eps)$, where $\eps>0$ is arbitrary small (and $q$ grows as $poly(1/\eps)$). Hardness results were known before for the point $(3/4,11/16+\eps)$ (its extrapolation with the points $(0,0)$ and $(1,1)$) and a region close to $(0,0)$. The new hardness result is for $(1/2,3/8+\eps)$, which is obtained by using a result for $(1/2+1/2q,3/8+5/8q+\eps)$.

The new construction is based on constructing a special purpose PCP (rather than a test) for the standard "matching dictator problem". Specifically, the proof oracle is an auxiliary function $h$, and rather than testing the two functions against each other, one test one of the two functions versus $h$. Indeed, a lesson to take home is that, when constructing PCPs, one may always replace testers by special purpose PCPs (of proximity).

2) Hypercontractivity, Sum-of-Squares Proofs, and their Applications

Here $UG(\eps)$ is a special case of $2LIN(1-\eps,\eps)$ as in (1). The presentation focused on the first result of the paper, but I find the second one more interesting. Its starting point is that the subexponential-time algorithm of Arora, Barak, and Steurer [2010] is viewed as evidence against UGC, since it is said that NP-hard problems should be exponentially hard (wrt their "natural" parameter). Personally, I am not convinced by this evidence (*), and the foregoing result adds concreteness to my principled skepticism: It shows that there exists a natural generalization of UG that (i) has a subexponential time algorithm, and (ii) has no polynomial-time algorithm provided that SAT requires exponential time (i.e., the "ETH").

*) Firstly, a problem may be hard and not NP-hard; secondly, I don't see why all NP-hard problems should be exponentially hard (wrt some "natural" parameter), although some may be so; and thirdly I don't see what is fundamental in exponential-time (as opposed to the inevitability of exhaustive search). Lastly, I think that there is something fundamentally wrong in trying to infer more basic issues (i.e., intractability) based on more advanced ones (i.e., inevitability of exhaustive search)

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