Random local access for sampling k-SAT solutions (2409.03951v2)

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.

• The paper presents a sublinear-time algorithm that offers random local access to a nearly uniform distribution of k-SAT solutions.
• It leverages local computation techniques and α-marking to partition variable dependencies and ensure efficient sampling of small formula components.
• The approach extends local sampling models, implying potential improvements in distributed computation and optimization for complex combinatorial problems.

Random Local Access for Sampling $k$-\textsf{SAT} Solutions

The paper "Random local access for sampling $k$-\textsf{SAT} solutions" by Dingding Dong and Nitya Mani presents a sublinear time algorithm designed to provide random local access to the uniform distribution of satisfying assignments for an arbitrary $k$-CNF formula $\Phi$, especially at high clause densities. This essay will explore the main contributions, the methodologies employed, and the implications of these findings.

Introduction

Efficient sampling from exponentially large solution spaces is a notable challenge in computational theory. Particularly, the problem of sampling from the set of satisfying assignments of $k$-CNF formulas ($k$-SAT problems) is crucial. A $k$-CNF formula $\Phi$ consists of $n$ variables each appearing in up to $d$ clauses. This paper's primary focus is on approximating the uniform distribution over such satisfying assignments.

Background and Related Work

Previous advancements include:

1. Existence of Satisfying Assignments: The Lovász Local Lemma (LLL), established by Erdős and Lovász, ensures the existence of satisfying assignments for $k$-CNFs under specific conditions.
2. Efficient Computation of Assignments: Moser and Tardos provided a linear-time algorithm for constructing satisfying assignments, but efficient sampling remained less understood.
3. Sampling Algorithms: Moitra developed a deterministic algorithm to sample from the uniform distribution for $k$-CNFs with $d \lesssim 2^{ck}$ for $c \approx 1/60$. Follow-up work by Feng et al. improved this to $d \lesssim 2^{k/20}$ using Markov chains.

Main Contributions

This paper builds on the aforementioned work and explores the feasibility of random local access for $k$-SAT solutions by providing memory-less query access to variable assignments in sublinear time. Key contributions include:

1. Algorithm Design: A sublinear time algorithm that allows local access to a consistent global satisfying assignment closely approximating the uniform distribution.
2. Formal Definition: Extension of the Biswas, Rubinfeld, and Yodpinyanee model for local sampling to the context of $k$-SAT solutions.
3. Extension to Gibbs Distributions: While previous works focused on graph colorings and other distributions, this paper demonstrates applicability to $k$-CNFs, thus contributing to a long line of research on sublinear algorithms.

Methodology

The construction of the random local access sampling algorithm involves several sophisticated techniques:

1. Marking and Shattering: An $\alpha$-marking is used to conditionally partition the original formula into smaller components. Each component's size is probabilistically bounded to ensure efficient sampling.
2. Local Computation Algorithms: The algorithm leverages local computation algorithms (LCAs) to access variable assignments locally without needing to evaluate the entire formula.
3. Randomness: The algorithm introduces randomness in variable sampling to ensure that the partial assignments emulate a consistent satisfying assignment under the uniform distribution.

The Algorithm

The paper defines a detailed algorithm (\texttt{Algorithm 1}) to sample a variable assignment:

1. Marking Step: Determine if a variable belongs to the marking set.
2. Sampling: If the variable is marked, use marginal sampling (\textsf{MarginSample}) to determine its value, potentially invoking recursive calls to handle dependencies.
3. Component Calculation: For unmarked variables, calculate their connected component in the reduced dependency graph using \textsf{Conn}.
4. Uniform Sampling: Sample from the resulting small connected component using the \textsf{UniformSample} algorithm.

The algorithm guarantees that each query accesses only a sublinear number of variables, ensuring efficiency.

Results

Theoretical results in the paper include:

• Correctness: Proving that the \textsf{MarginSample} and \textsf{MarginOverflow} algorithms correctly sample from the uniform distribution, maintaining consistency across queries.
• Efficiency: Demonstrating that for any queried variable, the algorithm runs in sublinear time.

Implications and Future Directions

This work has significant implications both in theory and practice:

1. Efficient Sampling: Provides a new lens for efficient sampling from large solution spaces, which can notably improve algorithms in optimization and probabilistic reasoning.
2. Local Computation Models: Extends the local computation model to more complex distributions, potentially benefiting distributed computing and parallel processing.
3. Future Research: Opens avenues for improving clause density handling and extending the model to other classes of problems.

Conclusion

The paper makes substantial progress in the field of local access and sampling algorithms for $k$-SAT problems. The proposed algorithm achieves random local access in sublinear time while ensuring the output closely follows the uniform distribution of satisfying assignments. This work not only strengthens the theoretical underpinnings of local computation algorithms but also provides practical tools for efficient sampling in vast combinatorial spaces. Future research may further refine these techniques or explore their applications in broader contexts.