Multi-pass Max-Cut requires √n total space–passes

Establish that for every ε > 0, any k-pass, s-space streaming algorithm that outputs a (1/2 + ε)-approximation to the Max-Cut value on an undirected graph given as an edge stream must satisfy ks = Ω(√n).

Background

FMW25 proved a ks = Ω(n{1/3}) lower bound for multi-pass Max-Cut approximation. There is folklore evidence that certain hard instances become distinguishable in O(√n) space with a few passes (via random walks), suggesting √n is a critical threshold. This conjecture seeks to push the multi-pass lower bound to this threshold, clarifying the precise trade-off between passes and space.

References

Conjecture For every ε > 0, every k-pass, s-space streaming algorithm which (1/2 + ε)-approximates Max-Cut has ks = Ω(√n).

— Nine lower bound conjectures on streaming approximation algorithms for CSPs (2510.10714 - Singer, 12 Oct 2025) in Conjecture (label: conj:multi-pass max-cut:sqrt(n)-space-pass), Section 5 (Multi-pass streaming lower bounds)