## 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

#### 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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