We continue the study of covering complexity of constraint satisfaction problems (CSPs) initiated by Guruswami, Hastad and Sudan [SIAM J. Computing, 31(6):1663--1686, 2002] and Dinur and Kol [In Proc. $28$th IEEE Conference on Computational Complexity, 2013]. The covering number of a CSP instance $\Phi$, denoted by $\nu(\Phi)$ is the smallest number of assignments to the variables of $\Phi$, such that each constraint of $\Phi$ is satisfied by at least one of the assignments. We show the following results regarding how well efficient algorithms can approximate the covering number of a given CSP instance. - Assuming a covering unique games conjecture, introduced by Dinur and Kol, we show that for every non-odd predicate $P$ over any constant sized alphabet and every integer $K$, it is NP-hard to distinguish between $P$-CSP instances (i.e., CSP instances where all the constraints are of type $P$) which are coverable by a constant number of assignments and those whose covering number is at least $K$. Previously, Dinur and Kol, using the same covering unique games conjecture, had shown a similar hardness result for every non-odd predicate over the Boolean alphabet that supports a pairwise independent distribution. Our generalization yields a complete characterization of CSPs over constant sized alphabet $\Sigma$ that are hard to cover since CSP's over odd predicates are trivially coverable with $|\Sigma|$ assignments. - For a large class of predicates that are contained in the $2k$-LIN predicate, we show that it is quasi-NP-hard to distinguish between instances which have covering number at most two and covering number at least $\Omega(\log\log n)$. This generalizes the $4$-LIN result of Dinur and Kol that states it is quasi-NP-hard to distinguish between $4$-LIN-CSP instances which have covering number at most two and covering number at least $\Omega(\log \log\log n)$.

We consider the problem of distributed corruption detection in networks. In this model, each vertex of a directed graph is either truthful or corrupt. Each vertex reports the type (truthful or corrupt) of each of its outneighbors. If it is truthful, it reports the truth, whereas if it is corrupt, it reports adversarially. This model, first considered by Preparata, Metze, and Chien in 1967, motivated by the desire to identify the faulty components of a digital system by having the other components checking them, became known as the PMC model. The main known results for this model characterize networks in which \emph{all} corrupt (that is, faulty) vertices can be identified, when there is a known upper bound on their number. We are interested in networks in which the identity of a \emph{large fraction} of the vertices can be identified. It is known that in the PMC model, in order to identify all corrupt vertices when their number is $t$, all indegrees have to be at least $t$. In contrast, we show that in $d$ regular-graphs with strong expansion properties, a $1-O(1/d)$ fraction of the corrupt vertices, and a $1-O(1/d)$ fraction of the truthful vertices can be identified, whenever there is a majority of truthful vertices. We also observe that if the graph is very far from being a good expander, namely, if the deletion of a small set of vertices splits the graph into small components, then no corruption detection is possible even if most of the vertices are truthful. Finally, we discuss the algorithmic aspects and the computational hardness of the problem.

The population recovery problem is a basic problem in noisy unsupervised learning that has attracted significant research attention in recent years [WY12,DRWY12, MS13, BIMP13, LZ15,DST16]. A number of different variants of this problem have been studied, often under assumptions on the unknown distribution (such as that it has restricted support size). In this work we study the sample complexity and algorithmic complexity of the most general version of the problem, under both bit-flip noise and erasure noise model. We give essentially matching upper and lower sample complexity bounds for both noise models, and efficient algorithms matching these sample complexity bounds up to polynomial factors.

For any unsatisfiable CNF formula $F$ that is hard to refute in the Resolution proof system, we show that a gadget-composed version of $F$ is hard to refute in any proof system whose lines are computed by efficient communication protocols—or, equivalently, that a monotone function associated with $F$ has large monotone circuit complexity. Our result extends to monotone real circuits, which yields new lower bounds for the Cutting Planes proof system. -------------- An extended abstract of this paper appeared in the Proceedings of the 50th Symposium on Theory of Computing (STOC'18).

In the correlated sampling problem, two players are given probability distributions $P$ and $Q$, respectively, over the same finite set, with access to shared randomness. Without any communication, the two players are each required to output an element sampled according to their respective distributions, while trying to minimize the probability that their outputs disagree. A well known strategy due to Kleinberg--Tardos and Holenstein, with a close variant (for a similar problem) due to Broder, solves this task with disagreement probability at most $2 \delta/(1+\delta)$, where $\delta$ is the total variation distance between $P$ and $Q$. This strategy has been used in several different contexts, including sketching algorithms, approximation algorithms based on rounding linear programming relaxations, the study of parallel repetition and cryptography. In this paper, we give a surprisingly simple proof that this strategy is essentially optimal. Specifically, for every $\delta \in (0,1)$, we show that any correlated sampling strategy incurs a disagreement probability of essentially $2\delta/(1+\delta)$ on some inputs $P$ and $Q$ with total variation distance at most $\delta$. This partially answers a recent question of Rivest. Our proof is based on studying a new problem that we call constrained agreement. Here, the two players are given subsets $A \subseteq [n]$ and $B \subseteq [n]$, respectively, and their goal is to output an element $i \in A$ and $j \in B$, respectively, while minimizing the probability that $i \neq j$. We prove tight bounds for this question, which in turn imply tight bounds for correlated sampling. Though we settle basic questions about the two problems, our formulation leads to more fine grained questions that remain open.

In this paper we study the (Bichromatic) Maximum Inner Product Problem (Max-IP), in which we are given sets A and B of vectors, and the goal is to find a ∈ A and b ∈ B maximizing inner product a · b. Max-IP is very basic and serves as the base problem in the recent breakthrough of [Abboud et al., FOCS 2017] on hardness of approximation for polynomial-time problems. It is also used (implicitly) in the argument for hardness of exact e2-Furthest Pair (and other important problems in computational geometry) in poly-log-log dimensions in [Williams, SODA 2018]. We have three main results regarding this problem.• Characterization of Multiplicative Approximation. First, we study the best multiplicative approximation ratio for Boolean Max-IP in sub-quadratic time. We show that, for Max-IP with two sets of n vectors from {0, 1}d, there is an n2−ω(1) time (d/logn)ω(1) - multiplicative-approximating algorithm, and we show this is conditionally optimal, as such a (d/logn)o(1)-approximating algorithm would refute SETH. Similar characterization is also achieved for additive approximation for Max-IP.• 2O(log* n)-dimensional Hardness for Exact Max-IP Over The Integers. Second, we revisit the hardness of solving Max-IP exactly for vectors with integer entries. We show that, under SETH, for Max-IP with sets of n vectors from Zd for some d = 2O(log* n), every exact algorithm requires n2−o(1) time. With the reduction from [Williams, SODA 2018], it follows that e2-Furthest Pair and Bichromatic e2-Closest Pair in 2O(log* n) dimensions require n2−o(1) time.• Connection with NP · UPP Communication Protocols. Last, We establish a connection between conditional lower bounds for exact Max-IP with integer entries and NP · UPP communication protocols for Set-Disjointness, parallel to the connection between conditional lower bounds for approximating Max-IP and MA communication protocols for Set-Disjointness.The lower bound in our first result is a direct corollary of the new MA protocol for Set-Disjointness introduced in [Rubinstein, STOC 2018], and our algorithms utilize the polynomial method and simple random sampling. Our second result follows from a new dimensionality self reduction from the Orthogonal Vectors problem for n vectors from {0, 1}d to n vectors from Ze where e = 2O(log* d), dramatically improving the previous reduction in [Williams, SODA 2018]. The key technical ingredient is a recursive application of Chinese Remainder Theorem.As a side product, we obtain an MA communication protocol for Set-Disjointness with complexity [EQUATION], slightly improving the [EQUATION] log n) bound [Aaronson and Wigderson, TOCT 2009], and approaching the [EQUATION] lower bound [Klauck, CCC 2003].Moreover, we show that (under SETH) one can apply the [EQUATION] BQP communication protocol for Set-Disjointness to prove near-optimal hardness for approximation to Max-IP with vectors in {− 1, 1}d. This answers a question from [Abboud et al., FOCS 2017] in the affirmative.

$ $ We prove that for every $n$ and $1 < t < n$ any $t$-out-of-$n$ threshold secret sharing scheme for one-bit secrets requires share size $\log(t + 1)$. Our bound is tight when $t = n - 1$ and $n$ is a prime power. In 1990 Kilian and Nisan proved the incomparable bound $\log(n - t + 2)$. Taken together, the two bounds imply that the share size of Shamir's secret sharing scheme (Comm. ACM 1979) is optimal up to an additive constant even for one-bit secrets for the whole range of parameters $1 < t < n$. More generally, we show that for all $1 < s < r < n$, any ramp secret sharing scheme with secrecy threshold $s$ and reconstruction threshold $r$ requires share size $\log((r + 1)/(r - s))$. As part of our analysis we formulate a simple game-theoretic relaxation of secret sharing for arbitrary access structures. We prove the optimality of our analysis for threshold secret sharing with respect to this method and point out a general limitation.