Cores Characterize the One-Pass Streaming Complexity of CSPs

2026-10-05

AI Summary

This study addresses the challenge of characterizing the complexity of constraint satisfaction problems (CSPs) over arbitrary finite domains within the single-pass streaming model. For three computational objectives—SAT decision, minimization, and exact solving—the authors propose a unified theoretical framework based on the notion of a "core." By leveraging hard-pinning variable techniques to overcome limitations inherent to Boolean domains, and employing deterministic streaming algorithm design alongside reduction techniques for lower bound proofs, this work establishes tight complexity bounds for all objectives. Notably, even under randomized algorithms, these bounds differ by at most polylogarithmic factors. Consequently, this research achieves a precise characterization of the streaming complexity of CSPs over general domains.

Abstract

We study the one-pass streaming complexity of CSPs over a fixed finite relation $R$ under three natural objectives: $\textsf{SAT}(R)$, deciding whether all constraints can be satisfied; $\textsf{Min}$-$\textsf{CSP}(R)$, approximately minimizing the number of unsatisfied constraints; and $\textsf{Exact-CSP}(R)$, exactly computing the maximum number of satisfied constraints. Sharma and Velusamy (ESA 2026) characterized the streaming complexity of satisfiability for CSPs with literals using the non-redundancy parameter $\textsf{NRD}_n(R)$, which roughly measures the largest instance in which every constraint is independently necessary. For the most general setting without literals, they obtained the corresponding characterization for Boolean relations and showed obstacles to extending their techniques to larger domains. Kol, Paramonov, Saxena, and Yu (ITCS 2023) similarly characterized the streaming complexity of $\textsf{Exact-CSP}(R)$ for Boolean CSPs with literals in terms of the degree deg$(R)$ of the relation when written as a polynomial, while without literals, the corresponding result was known only for $\textsf{Max-Cut}$. For all three of the objectives we study, we give tight characterizations for unweighted CSP instances over arbitrary finite domains. The streaming algorithms we provide are deterministic, while the corresponding bounds are tight up to polylogarithmic factors, even against randomized algorithms. The central idea in all three results is a reduction of $R$ to its core, a canonical subrelation of $R$ which admits gadgets that allow for the hard-pinning (or fixing) of variables.

Problem

• Constraint Satisfaction Problems
• Streaming Complexity
• One-Pass Streaming
• Core Reduction
• Finite Domains

Innovation

• Streaming Complexity
• Constraint Satisfaction Problems
• Core Reduction
• Hard-Pinning Gadgets
• Arbitrary Finite Domains