Single-pass adversarial-order Max-Cut requires n log n space

Prove that for every ε > 0, any single-pass adversarial-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 log n) bits of memory.

Background

The best known single-pass adversarial-order lower bound for Max-Cut is Ω(n) space (KK19), while a generic sparsification-based upper bound achieves (1−ε)-approximation for all CSPs using O(n log n/ε2) space. Strengthening the lower bound to Ω(n log n) would match the sparsification upper bound scale and further clarify the optimal streaming space for nontrivial Max-Cut approximation.

References

Another conjecture about lower bounds for Max-Cut with single-pass algorithms is the following: Conjecture For every ε > 0, every single-pass adversarial-ordering streaming algorithm which (1/2+ε)-approximates Max-Cut uses Ω(n log 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 log n-space), Section 4 (Single-pass, linear(ish)-space streaming lower bounds)