In this paper, the author puts forward a variation of Feige's Hypothesis, which claims that it is hard on average refuting Unbalanced Max 3-XOR under biased assignments on a natural distribution. Under this hypothesis, the author strengthens the previous known hardness for approximating Minimum Unique Game, $5/4-\epsilon$, by proving that ... more >>>

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 $Gr_{global}$ contains induced subgraphs $Gr_{local}$ that are themselves ... more >>>

Unique Games Conjecture (UGC), proposed by [Khot02], lies in the center of many inapproximability results. At the heart of UGC lies approximability of MAX-CUT, which is a special instance of Unique Game.[KhotKMO04, MosselOO05] showed that assuming Unique Games Conjecture, it is NP-hard to distinguish between MAX-CUT instance that has a ... more >>>

Dinur, Khot, Kindler, Minzer and Safra (2016) recently showed that the (imperfect completeness variant of) Khot's 2 to 2 games conjecture follows from a combinatorial hypothesis about the soundness of a certain ``Grassmanian agreement tester''.
In this work, we show that the hypothesis of Dinur et al follows from a ... more >>>

The $2$-to-$2$ Games Theorem of [KMS-1, DKKMS-1, DKKMS-2, KMS-2] implies that it is NP-hard to distinguish between Unique Games instances with assignment satisfying at least $(\frac{1}{2}-\varepsilon)$ fraction of the constraints $vs.$ no assignment satisfying more than $\varepsilon$ fraction of the constraints, for every constant $\varepsilon>0$. We show that the reduction ... more >>>

We propose a variant of the $2$-to-$1$ Games Conjecture that we call the Rich $2$-to-$1$ Games Conjecture and show that it is equivalent to the Unique Games Conjecture. We are motivated by two considerations. Firstly, in light of the recent proof of the $2$-to-$1$ Games Conjecture, we hope to understand ... more >>>

We reduce the problem of proving a "Boolean Unique Games Conjecture" (with gap $1-\delta$ vs. $1-C\delta$, for any $C> 1$, and sufficiently small $\delta>0$) to the problem of proving a PCP Theorem for a certain non-unique game.
In a previous work, Khot and Moshkovitz suggested an inefficient candidate reduction (i.e., ... more >>>