Related papers: The $k$-XORSAT threshold revisited

The runtime performance of modern SAT solvers on random $k$-CNF formulas is deeply connected with the 'phase-transition' phenomenon seen empirically in the satisfiability of random $k$-CNF formulas. Recent universal hashing-based approaches…

This paper provides a new conceptual perspective on survey propagation, which is an iterative algorithm recently introduced by the statistical physics community that is very effective in solving random k-SAT problems even with densities…

We determine the exact threshold of satisfiability for random instances of a particular NP-complete constraint satisfaction problem (CSP). This is the first random CSP model for which we have determined a precise linear satisfiability…

Matrix completion, i.e., the exact and provable recovery of a low-rank matrix from a small subset of its elements, is currently only known to be possible if the matrix satisfies a restrictive structural constraint---known as {\em…

We present efficient counting and sampling algorithms for random $k$-SAT when the clause density satisfies $\alpha \le \frac{2^k}{\mathrm{poly}(k)}.$ In particular, the exponential term $2^k$ matches the satisfiability threshold…

We study an Achlioptas-process version of the random k-SAT process: a bounded number of k-clauses are drawn uniformly at random at each step, and exactly one added to the growing formula according to a particular rule. We prove the…

The problem of identifying the satisfiability threshold of random $3$-SAT formulas has received a lot of attention during the last decades and has inspired the study of other threshold phenomena in random combinatorial structures. The…

Let F be a random k-SAT formula on n variables, formed by selecting uniformly and independently m = rn out of all possible k-clauses. It is well-known that if r>2^k ln 2, then the formula F is unsatisfiable with probability that tends to 1…

In this letter, we study the deterministic sampling patterns for the completion of low rank matrix, when corrupted with a sparse noise, also known as robust matrix completion. We extend the recent results on the deterministic sampling…

We consider the recovery of a nonnegative vector x from measurements y = Ax, where A is an m-by-n matrix whos entries are in {0, 1}. We establish that when A corresponds to the adjacency matrix of a bipartite graph with sufficient…

This paper studies the hierarchy of sparse matrix Moment-SOS relaxations for solving sparse polynomial optimization problems with matrix constraints. First, we prove a sufficient and necessary condition for the sparse hierarchy to be tight.…

We determine the rank of a random matrix A over a finite field with prescribed numbers of non-zero entries in each row and column. As an application we obtain a formula for the rate of low-density parity check codes. This formula verifies a…

The structural phase transitions and computational complexity of random 3-SAT instances are traditionally described using thermodynamic analogies from statistical physics, such as Replica Symmetry Breaking and energy landscapes. While…

We prove that for any $k\geq3$ for clause/variable ratios up to the Gibbs uniqueness threshold of the corresponding Galton-Watson tree, the number of satisfying assignments of random $k$-SAT formulas is given by the `replica symmetric…

The Random K-Satisfiability Problem, consisting in verifying the existence of an assignment of N Boolean variables that satisfy a set of M=alpha N random logical clauses containing K variables each, is studied using the replica symmetric…

Using methods and ideas from statistical mechanics, we propose a simple method for obtaining rigorous upper bounds for satisfiability transition in random boolean expressions composed of N variables and M clauses with K variables per…

Given a matrix A \in R^{m x n}, we present a randomized algorithm that sparsifies A by retaining some of its elements by sampling them according to a distribution that depends on both the square and the absolute value of the entries. We…

The low-rank matrix approximation problem with respect to the entry-wise $\ell_{\infty}$-norm is the following: given a matrix $M$ and a factorization rank $r$, find a matrix $X$ whose rank is at most $r$ and that minimizes $\max_{i,j}…

For large clause-to-variable ratio, typical K-SAT instances drawn from the uniform distribution have no solution. We argue, based on statistical mechanics calculations using the replica and cavity methods, that rare satisfiable instances…

We propose an information-theoretic framework for matrix completion. The theory goes beyond the low-rank structure and applies to general matrices of "low description complexity". Specifically, we consider $m\times n$ random matrices…