Single-pass random-order Max-Cut requires linear space

Prove that for every ε > 0, any single-pass random-order streaming algorithm that outputs a (1/2 + ε)-approximation to the Max-Cut value on an undirected graph given as an edge stream must use Ω(n) bits of memory.

Background

Existing single-pass lower bounds for Max-Cut either assume adversarial ordering to obtain Ω(n) space (KK19) or assume random ordering to obtain Ω(√n) space (KKS15). This conjecture aims to unify these results by establishing an Ω(n) lower bound even under random ordering, thereby closing a gap in our understanding of single-pass streaming hardness for Max-Cut.

References

Conjecture For every ε > 0, every single-pass random-order streaming algorithm which (1/2+ε)-approximates Max-Cut uses Ω(n) space.

— Nine lower bound conjectures on streaming approximation algorithms for CSPs (2510.10714 - Singer, 12 Oct 2025) in Conjecture (label: conj:single-pass max-cut:n-space random-order), Section 4 (Single-pass, linear(ish)-space streaming lower bounds)