AI summaryⓘ
The authors prove that for a broad class of constraint satisfaction problems (CSPs), any streaming algorithm that tries to tell apart nearly satisfiable instances from poorly satisfiable ones must use a lot of memory if it only sees the data once. This generalizes previous lower bounds that only applied to special CSPs with linear-algebraic properties. Their technique simplifies and expands prior approaches by using more general analytical conditions. The lower bound matches the best possible for one-pass streaming since better approximations require more memory or multiple passes. Their proof builds on a complex problem from recent multi-pass streaming work, adapting it to the single-pass case with stronger memory demands.
Max-CSPstreaming algorithmslinear-space lower boundintegrality gapbasic LP relaxationsingle-pass streamingapproximation algorithmsconstraint satisfaction problemsdistributional implicit hidden partitionspace complexity
Abstract
For an arbitrary family of predicates $\mathcal{F} \subseteq \{0,1\}^{[q]^k}$ and any $ε> 0$, 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 $β+ε$ fraction of satisfiable constraints from instances of with at least $γ-ε$ fraction of satisfiable constraints, whenever Max-CSP$({\mathcal{F}})$ admits a $(γ,β)$-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 $(1-ε)$-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.