Playing Unique Games on Certified Small-Set Expanders

We give an algorithm for solving unique games (UG) instances whose constraints correspond to edges of graphs with a sum-of-squares (SoS) small-set-expansion certificate. As corollaries, we obtain the first polynomial-time algorithms for solving UG on the noisy hypercube and the short code graphs. 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 graph. All of our results achieve an approximation of $1-\epsilon$ vs $\delta$ for UG instances, where $\delta > 0$ depends on the expansion parameters of the graph but is independent of the alphabet size. Specifically, say that a regular graph $G=(V,E)$ is a $(\mu,\eta)$ small-set expander (SSE) if for every subset $S \subseteq V$ with $|S| \leq \mu |V|$, the edge-expansion of $S$ is at least $\eta$. We say that $G$ is a $d$-certified $(\mu,\eta)$-SSE if there is a degree-d SoS certificate for this fact (based on 2 to 4 hypercontractivity). We prove that there is a $|V|^{f(d,\mu,\eta)}$ time algorithm $A$ (based on the SoS hierarchy) such that for every $\eta>0$ and $d$-certified $(\mu, \eta)$-SSE $G$, if $I$ is a $1-\eta^2/100$ satisfiable affine UG instance over $G$ then $A(I)$ is an assignment satisfying at least some positive fraction $\delta = \delta(\mu,\eta)$ of $I$'s constraints. As a corollary, we get a polynomial-time algorithm $A$ such that if $I$ is a $1-\epsilon$ satisfiable instance over the $\alpha$-noisy hypercube or short code graph, then $A(I)$ outputs an assignment satisfying an $\exp(-O(\sqrt{\epsilon}/\alpha))$ fraction of the constraints. Our techniques can be extended even to graphs that are not SSE, and in particular we obtain a new efficient algorithm for solving UG instances over the Johnson graph.Comment: To appear in STOC 202

High Dimensional Expanders: Eigenstripping, Pseudorandomness, and Unique Games

Higher order random walks (HD-walks) on high dimensional expanders (HDX) have seen an incredible amount of study and application since their introduction by Kaufman and Mass [KM16], yet their broader combinatorial and spectral properties remain poorly understood. We develop a combinatorial characterization of the spectral structure of HD-walks on two-sided local-spectral expanders [DK17], which offer a broad generalization of the well-studied Johnson and Grassmann graphs. Our characterization, which shows that the spectra of HD-walks lie tightly concentrated in a few combinatorially structured strips, leads to novel structural theorems such as a tight $\ell_2$-characterization of edge-expansion, as well as to a new understanding of local-to-global algorithms on HDX. Towards the latter, we introduce a spectral complexity measure called Stripped Threshold Rank, and show how it can replace the (much larger) threshold rank in controlling the performance of algorithms on structured objects. Combined with a sum-of-squares proof of the former $\ell_2$-characterization, we give a concrete application of this framework to algorithms for unique games on HD-walks, in many cases improving the state of the art [RBS11, ABS15] from nearly-exponential to polynomial time (e.g. for sparsifications of Johnson graphs or of slices of the $q$-ary hypercube). Our characterization of expansion also holds an interesting connection to hardness of approximation, where an $\ell_\infty$-variant for the Grassmann graphs was recently used to resolve the 2-2 Games Conjecture [KMS18]. We give a reduction from a related $\ell_\infty$-variant to our $\ell_2$-characterization, but it loses factors in the regime of interest for hardness where the gap between $\ell_2$ and $\ell_\infty$ structure is large. Nevertheless, we open the door for further work on the use of HDX in hardness of approximation and unique games.Comment: An old version of this paper appeared under the title "High Dimensional Expanders: Random Walks, Pseudorandomness, and Unique Games." New version contains UG Algorithm for HD-walks over two-sided local-spectral expanders, tighter structural results, and simplified proof