We present an algorithm for strongly refuting smoothed instances of all Boolean CSPs. The smoothed model is a hybrid between worst and average-case input models, where the input is an arbitrary instance of the CSP with only the negation patterns of t
he literals re-randomized with some small probability. For an $n$-variable smoothed instance of a $k$-arity CSP, our algorithm runs in $n^{O(ell)}$ time, and succeeds with high probability in bounding the optimum fraction of satisfiable constraints away from $1$, provided that the number of constraints is at least $tilde{O}(n) (frac{n}{ell})^{frac{k}{2} - 1}$. This matches, up to polylogarithmic factors in $n$, the trade-off between running time and the number of constraints of the state-of-the-art algorithms for refuting fully random instances of CSPs [RRS17]. We also make a surprising new connection between our algorithm and even covers in hypergraphs, which we use to positively resolve Feiges 2008 conjecture, an extremal combinatorics conjecture on the existence of even covers in sufficiently dense hypergraphs that generalizes the well-known Moore bound for the girth of graphs. As a corollary, we show that polynomial-size refutation witnesses exist for arbitrary smoothed CSP instances with number of constraints a polynomial factor below the spectral threshold of $n^{k/2}$, extending the celebrated result for random 3-SAT of Feige, Kim and Ofek [FKO06].
Random subspaces $X$ of $mathbb{R}^n$ of dimension proportional to $n$ are, with high probability, well-spread with respect to the $ell_p$-norm (for $p in [1,2]$). Namely, every nonzero $x in X$ is robustly non-sparse in the following sense: $x$ is $
varepsilon |x|_p$-far in $ell_p$-distance from all $delta n$-sparse vectors, for positive constants $varepsilon, delta$ bounded away from $0$. This $ell_p$-spread property is the natural counterpart, for subspaces over the reals, of the minimum distance of linear codes over finite fields, and, for $p = 2$, corresponds to $X$ being a Euclidean section of the $ell_1$ unit ball. Explicit $ell_p$-spread subspaces of dimension $Omega(n)$, however, are not known except for $p=1$. The construction for $p=1$, as well as the best known constructions for $p in (1,2]$ (which achieve weaker spread properties), are analogs of low density parity check (LDPC) codes over the reals, i.e., they are kernels of sparse matrices. We study the spread properties of the kernels of sparse random matrices. Rather surprisingly, we prove that with high probability such subspaces contain vectors $x$ that are $o(1)cdot |x|_2$-close to $o(n)$-sparse with respect to the $ell_2$-norm, and in particular are not $ell_2$-spread. On the other hand, for $p < 2$ we prove that such subspaces are $ell_p$-spread with high probability. Moreover, we show that a random sparse matrix has the stronger restricted isometry property (RIP) with respect to the $ell_p$ norm, and this follows solely from the unique expansion of a random biregular graph, yielding a somewhat unexpected generalization of a similar result for the $ell_1$ norm [BGI+08]. Instantiating this with explicit expanders, we obtain the first explicit constructions of $ell_p$-spread subspaces and $ell_p$-RIP matrices for $1 leq p < p_0$, where $1 < p_0 < 2$ is an absolute constant.
We prove that with high probability over the choice of a random graph $G$ from the ErdH{o}s-Renyi distribution $G(n,1/2)$, a natural $n^{O(varepsilon^2 log n)}$-time, degree $O(varepsilon^2 log n)$ sum-of-squares semidefinite program cannot refute th
e existence of a valid $k$-coloring of $G$ for $k = n^{1/2 +varepsilon}$. Our result implies that the refutation guarantee of the basic semidefinite program (a close variant of the Lovasz theta function) cannot be appreciably improved by a natural $o(log n)$-degree sum-of-squares strengthening, and this is tight up to a $n^{o(1)}$ slack in $k$. To the best of our knowledge, this is the first lower bound for coloring $G(n,1/2)$ for even a single round strengthening of the basic SDP in any SDP hierarchy. Our proof relies on a new variant of instance-preserving non-pointwise complete reduction within SoS from coloring a graph to finding large independent sets in it. Our proof is (perhaps surprisingly) short, simple and does not require complicated spectral norm bounds on random matrices with dependent entries that have been otherwise necessary in the proofs of many similar results [BHK+16, HKP+17, KB19, GJJ+20, MRX20]. Our result formally holds for a constraint system where vertices are allowed to belong to multiple color classes; we leave the extension to the formally stronger formulation of coloring, where vertices must belong to unique colors classes, as an outstanding open problem.
Caches are a fundamental component of latency-sensitive computer systems. Recent work of [ASWB20] has initiated the study of delayed hits: a phenomenon in caches that occurs when the latency between the cache and backing store is much larger than the
time between new requests. We present two results for the delayed hits caching model. (1) Competitive ratio lower bound. We prove that the competitive ratio of the algorithm in [ASWB20], and more generally of any deterministic online algorithm for delayed hits, is at least Omega(kZ), where k is the cache size and Z is the delay parameter. (2) Antimonotonicity of the delayed hits latency. Antimonotonicity is a naturally desirable property of cache latency: having a cache hit instead of a cache miss should result in lower overall latency. We prove that the latency of the delayed hits model is not antimonotone by exhibiting a scenario where having a cache hit instead of a miss results in an increase in overall latency. We additionally present a modification of the delayed hits model that makes the latency antimonotone.
