Random local access for sampling k-SAT solutions

Abstract

We present a sublinear time algorithm that gives random local access to the uniform distribution over satisfying assignments to an arbitrary k-CNF formula $\Phi$, at exponential clause density. Our algorithm provides memory-less query access to variable assignments, such that the output variable assignments consistently emulate a single global satisfying assignment whose law is close to the uniform distribution over satisfying assignments to $\Phi$. Such models were formally defined (for the more general task of locally sampling from exponentially sized sample spaces) in 2017 by Biswas, Rubinfeld, and Yodpinyanee, who studied the analogous problem for the uniform distribution over proper q-colorings. This model extends a long line of work over multiple decades that studies sublinear time algorithms for problems in theoretical computer science. Random local access and related models have been studied for a wide variety of natural Gibbs distributions and random graphical processes. Here, we establish feasiblity of random local access models for one of the most canonical such sample spaces, the set of satisfying assignments to a k-CNF formula.