Optimal Single-Pass Streaming Lower Bounds for Approximating CSPs

2026-04-09

AI Summary

This work addresses the absence of general tight space lower bounds for the gap version of Max-CSP in the single-pass streaming model. By introducing a single-pass streaming variant of the “distributed implicit partitioning” problem, the paper establishes a linear space lower bound applicable to any predicate family, provided the corresponding CSP admits a (γ,β) integrality gap instance for the basic linear programming relaxation. The key innovation lies in generalizing prior conditions—originally reliant on linear-algebraic structure—to a broader analytic class, significantly simplifying the proof and extending coverage to a much wider range of CSPs. Combining techniques from streaming complexity, LP duality theory, and reductions, the authors show that for any Max-CSP exhibiting an LP integrality gap, achieving a nontrivial approximation in the single-pass streaming model requires linear space, and this lower bound is essentially optimal within that model.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Data Stream Mining

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Applications of cryptography

Abstract

For an arbitrary family of predicates $\mathcal{F} \subseteq \{0,1\}^{[q]^k}$ and any $\epsilon>0$, we prove a single-pass, linear-space streaming lower bound against the gap promise problem of distinguishing instances of Max-CSP$({\mathcal{F}})$ with at most $\beta+\epsilon$ fraction of satisfiable constraints from instances of with at least $\gamma-\epsilon$ fraction of satisfiable constraints, whenever Max-CSP$({\mathcal{F}})$ admits a $(\gamma,\beta)$-integrality gap instance for the basic LP. This subsumes the linear-space lower bound of Chou, Golovnev, Sudan, Velingker, and Velusamy (STOC 2022), which applies only to a special subclass of CSPs with linear-algebraic structure. (Their result itself generalizes work of Kapralov and Krachun (STOC 2019) for Max-CUT.) Our approach identifies the right ``analytic''analogues of previously-used linear-algebraic conditions; this yields substantial simplifications while capturing a much larger class of problems. Our lower bound is essentially optimal for single-pass streaming, since: (1) All CSPs admit $(1-\epsilon)$-approximations in quasilinear space, and (2) sublinear-space streaming algorithms can simulate the LP (on bounded-degree instances), giving approximation algorithms when integrality gap instances do not exist. The starting point for our lower bound is a reduction from a"distributional implicit hidden partition''problem defined by Fei, Minzer, and Wang (STOC 2026) in the context of multi-pass streaming. Our result is an analogue of theirs in the single-pass setting, where we obtain a much stronger (and tight) space lower bound.

Problem

• streaming lower bounds
• constraint satisfaction problems
• approximation
• integrality gap
• single-pass

Innovation

• streaming lower bounds
• constraint satisfaction problems
• integrality gap
• linear programming
• single-pass algorithms