Adapt parallel Glauber dynamics or correlation decay for random local access to k-CNF solutions

Develop an adaptation of parallel Glauber dynamics or algorithmic correlation decay techniques, combined with a query-oblivious local marking scheme that ensures each clause has at least αk marked and at least αk unmarked variables (for some fixed α in (0,1/2)), to obtain random local access algorithms for the uniform distribution over satisfying assignments of bounded-degree k-CNF formulas at clause densities comparable to the regime d ≤ 2^k/400. Establish that the resulting algorithms provide memory-less, consistent local access whose outputs are distributed close to uniform over satisfying assignments.

Background

The paper establishes a sublinear-time, memory-less random local access algorithm for sampling variable assignments consistent with a uniformly random satisfying assignment of a bounded-degree k-CNF formula, in an exponential clause density regime (e.g., d ≤ 2^k/400). The approach hinges on constructing a query-oblivious local marking that shatters the formula into small components and then performing localized sampling.

In the concluding remarks, the authors propose a different route inspired by prior work on local access via parallel dynamics in graph colorings. They conjecture that parallel Glauber dynamics or algorithmic correlation decay methods, when integrated with their local marking scheme, could yield alternative sampling algorithms that also achieve random local access for k-CNF satisfying assignments at comparable densities.