Near-optimal Max-Cut approximation requires many passes or near-linear space

Show that for every constant C > 0, there exists an ε > 0 such that any streaming algorithm that outputs a (1 − ε)-approximation to the Max-Cut value on an undirected graph given as an edge stream either uses Ω(n^C) passes or Ω(n) space.

Background

Beyond O(√n) space, the landscape of multi-pass streaming algorithms for Max-Cut is poorly understood. This conjecture posits that achieving approximation ratios close to 1 requires either polynomially many passes or linear space, reinforcing the fundamental difficulty of compressing Max-Cut instances sufficiently to preserve near-optimal cut values.

References

Conjecture For every C > 0, there exists some ε > 0 such that every streaming algorithm which (1−ε)-approximates Max-Cut uses Ω(nC) passes or Ω(n) space.

— Nine lower bound conjectures on streaming approximation algorithms for CSPs (2510.10714 - Singer, 12 Oct 2025) in Conjecture (label: conj:multi-pass max-cut:beating 1/2), Section 5 (Multi-pass streaming lower bounds)