We give an algorithm for solving unique games (UG) instances whenever low-degree sum-of-squares proofs certify good bounds on the small-set-expansion of the underlying constraint graph via a hypercontractive inequality. Our algorithm is in fact more
versatile, and succeeds even when the constraint graph is not a small-set expander as long as the structure of non-expanding small sets is (informally speaking) characterized by a low-degree sum-of-squares proof. Our results are obtained by rounding emph{low-entropy} solutions -- measured via a new global potential function -- to sum-of-squares (SoS) semidefinite programs. This technique adds to the (currently short) list of general tools for analyzing SoS relaxations for emph{worst-case} optimization problems. As corollaries, we obtain the first polynomial-time algorithms for solving any UG instance where the constraint graph is either the emph{noisy hypercube}, the emph{short code} or the emph{Johnson} graph. The prior best algorithm for such instances was the eigenvalue enumeration algorithm of Arora, Barak, and Steurer (2010) which requires quasi-polynomial time for the noisy hypercube and nearly-exponential time for the short code and Johnson graphs. All of our results achieve an approximation of $1-epsilon$ vs $delta$ for UG instances, where $epsilon>0$ and $delta > 0$ depend on the expansion parameters of the graph but are independent of the alphabet size.
Many previous Sum-of-Squares (SOS) lower bounds for CSPs had two deficiencies related to global constraints. First, they were not able to support a cardinality constraint, as in, say, the Min-Bisection problem. Second, while the pseudoexpectation of
the objective function was shown to have some value $beta$, it did not necessarily actually satisfy the constraint objective = $beta$. In this paper we show how to remedy both deficiencies in the case of random CSPs, by translating emph{global} constraints into emph{local} constraints. Using these ideas, we also show that degree-$Omega(sqrt{n})$ SOS does not provide a $(frac{4}{3} - epsilon)$-approximation for Min-Bisection, and degree-$Omega(n)$ SOS does not provide a $(frac{11}{12} + epsilon)$-approximation for Max-Bisection or a $(frac{5}{4} - epsilon)$-approximation for Min-Bisection. No prior SOS lower bounds for these problems were known.
For every $epsilon>0$, we give an $exp(tilde{O}(sqrt{n}/epsilon^2))$-time algorithm for the $1$ vs $1-epsilon$ emph{Best Separable State (BSS)} problem of distinguishing, given an $n^2times n^2$ matrix $mathcal{M}$ corresponding to a quantum measurem
ent, between the case that there is a separable (i.e., non-entangled) state $rho$ that $mathcal{M}$ accepts with probability $1$, and the case that every separable state is accepted with probability at most $1-epsilon$. Equivalently, our algorithm takes the description of a subspace $mathcal{W} subseteq mathbb{F}^{n^2}$ (where $mathbb{F}$ can be either the real or complex field) and distinguishes between the case that $mathcal{W}$ contains a rank one matrix, and the case that every rank one matrix is at least $epsilon$ far (in $ell_2$ distance) from $mathcal{W}$. To the best of our knowledge, this is the first improvement over the brute-force $exp(n)$-time algorithm for this problem. Our algorithm is based on the emph{sum-of-squares} hierarchy and its analysis is inspired by Lovetts proof (STOC 14, JACM 16) that the communication complexity of every rank-$n$ Boolean matrix is bounded by $tilde{O}(sqrt{n})$.
We prove that for every $epsilon>0$ and predicate $P:{0,1}^krightarrow {0,1}$ that supports a pairwise independent distribution, there exists an instance $mathcal{I}$ of the $mathsf{Max}P$ constraint satisfaction problem on $n$ variables such that no
assignment can satisfy more than a $tfrac{|P^{-1}(1)|}{2^k}+epsilon$ fraction of $mathcal{I}$s constraints but the degree $Omega(n)$ Sum of Squares semidefinite programming hierarchy cannot certify that $mathcal{I}$ is unsatisfiable. Similar results were previously only known for weaker hierarchies.
Halfspaces or linear threshold functions are widely studied in complexity theory, learning theory and algorithm design. In this work we study the natural problem of constructing pseudorandom generators (PRGs) for halfspaces over the sphere, aka spher
ical caps, which besides being interesting and basic geometric objects, also arise frequently in the analysis of various randomized algorithms (e.g., randomized rounding). We give an explicit PRG which fools spherical caps within error $epsilon$ and has an almost optimal seed-length of $O(log n + log(1/epsilon) cdot loglog(1/epsilon))$. For an inverse-polynomially growing error $epsilon$, our generator has a seed-length optimal up to a factor of $O( log log {(n)})$. The most efficient PRG previously known (due to Kane, 2012) requires a seed-length of $Omega(log^{3/2}{(n)})$ in this setting. We also obtain similar constructions to fool halfspaces with respect to the Gaussian distribution. Our construction and analysis are significantly different from previous works on PRGs for halfspaces and build on the iterative dimension reduction ideas of Kane et. al. (2011) and Celis et. al. (2013), the emph{classical moment problem} from probability theory and explicit constructions of emph{orthogonal designs} based on the seminal work of Bourgain and Gamburd (2011) on expansion in Lie groups.
