We give an approximately uniform sampler for satisfying assignments of constraint satisfaction problems that satisfy $4\mathrm e p(\Delta+1)^2\le1$, where $p$ is the largest constraint-violation probability under the uniform product distribution, and $\Delta$ is the maximum degree of the dependency graph. The algorithm invokes the recent efficient approximate counting algorithm of Liu, Wang, Yin, Zhang, and Zhou~\cite{CountingLLL} as a subroutine and returns a satisfying assignment sampled within total-variation distance $\varepsilon$ of the uniform distribution in $(n+m/\varepsilon)^{O(k\Delta\log D)}$ time, where $n$ and $m$ are the numbers of variables and constraints, $D$ is the common domain size, and $k$ bounds the constraint arity. We also give an asymmetric product-form condition under which both counting and sampling are efficient. To supply the counts needed in the additional regime, we extend the marginal-expansion proof of Liu et al.
We study approximately uniform sampling of satisfying assignments from random $k$-SAT formulas. For every sufficiently large $k$ and density $0<\alpha \le 2^k/k^{16}$, we prove that, with high probability over the formula, there is a sampler whose output distribution is within total variation distance $\varepsilon$ of...
Consider a constraint satisfaction problem $\mathbf{C}$ on finitely many independent random variables with dependency graph $G$. Let $p_a$ be the violation probability of a constraint $a\in \mathbf{C}$ and $N_G^2 (a)$ the set of constraints at distance one or two from $a$ in $G$. Suppose that, there exists $x\in (0,1)^...
Vishesh Jain, Clayton Mizgerd, H. Pham· 0 citations
We show that there exists a class of boolean functions C such that $(i)$ there is a distribution-independent statistical query algorithm for learning C that makes a polynomial number of queries of inverse polynomial tolerance and $(ii)$ for any set of functions $\Phi_1, \dots, \Phi_r$ such that for all $f \in$ C we can...
We give a randomized algorithm that, for an $m\times n$ matrix whose columns have Euclidean norm at most one, finds a signing with discrepancy below $99$ in expected $\widetilde O(L_{\mathrm{in}}+n^\omega)$ bit operations, where $L_{\mathrm{in}}$ is the input encoding length and $\omega$ is the exponent of matrix multi...
It is proved that, unless SETH fails, for any $\varepsilon, \delta>0$, $k \in \mathbb{N}$, and function $f\colon \mathbb{N}\to \mathbb{N}$, there is no data structure which maintains a $(2-\varepsilon)-approximation of the degeneracy of $G$ with initialization time and amortized update time.
We prove that, for every constant $\varepsilon>0$ and every function $f:\Sigma^n\to\{0, 1\}$, there is a coalition of $O(n/\sqrt{\log n})$ coordinates and a target output $b\in\{0, 1\}$ such that, after the remaining coordinates are sampled uniformly and independently, the coalition can choose its values to make the ou...
