Streaming Dichotomy Conjecture for single-pass Max-CSP relative to the BasicLP integrality gap

Establish the single-pass streaming dichotomy for Max-CSP(F) with respect to the integrality gap of the BasicLP: for every predicate family F and every ε>0, (i) prove there exists a single-pass streaming algorithm using o(n) space that achieves a (gap_BasicLP(F) − ε)-approximation, and (ii) prove that any single-pass streaming algorithm achieving a (gap_BasicLP(F) + ε)-approximation requires Ω(n) space.

Background

The integrality gap of the BasicLP for a CSP family F (denoted in the paper by 30_F) is closely tied to approximability in various models. The conjectured single-pass streaming dichotomy posits that this gap precisely characterizes the threshold between sublinear-space approximability and linear-space hardness.

Prior to this work, the upper bound side was known under constant-degree assumptions or in multipass models. This paper resolves the upper bound in the single-pass setting for all F, leaving the lower bound side (linear-space hardness for (gap+ε)-approximation) as the remaining part of the conjecture to be proved.

References

In particular, a major conjecture of the area is that in the single-pass streaming setting, for any fixed ε > 0, (i) an (30−ε)-approximation can be achieved with o(n) space, and that (ii) any (30+ε)-approximation requires Ω(n) space. Despite significant progress, both sides of the conjecture remain open.

— Single-Pass Streaming CSPs via Two-Tier Sampling  (2604.01575 - Azarmehr et al., 2 Apr 2026) in Abstract; also stated as Conjecture (Streaming Dichotomy) in Section 1 (Introduction)