√n-space lower bound for single-pass streaming Max-Monarchy

Prove that for every k ≥ 5 and ε > 0, any single-pass streaming algorithm that outputs a (1/2 + ε)-approximation to Max-CSP with predicate family Π = {Monarchy ∘ Not_b : b ∈ {0,1}^k} must use Ω(√n) bits of memory.

Background

CGS+22-monarchy showed that for k ≥ 5, any single-pass sketch achieving (1/2 + ε)-approximation for the Monarchy predicate requires Ω(√n) space. This conjecture extends the sketching lower bound to general streaming algorithms, aligning with efforts to determine whether the CGSV24 dichotomy characterizes streaming algorithms beyond sketching.

References

We naturally conjecture an analogue of Conjecture for this problem: Conjecture For every k ≥ 5 and ε > 0, every single-pass streaming algorithm which (1/2+ε)-approximates Max uses Ω(√ n) space.

— Nine lower bound conjectures on streaming approximation algorithms for CSPs (2510.10714 - Singer, 12 Oct 2025) in Conjecture, Section 6 (More o(√n)-space streaming lower bounds)