Limits of the FMU'22 semi-streaming approach for (1+ε)-approximate matching

Determine whether the semi-streaming framework and techniques of Fischer, Mitrović, and Uitto (STOC 2022) for computing a (1+ε)-approximate maximum matching in general graphs can be optimized, without fundamental changes to that approach, to achieve a pass complexity strictly better than O(ε^{-12}).

Background

The paper improves the pass complexity for semi-streaming (1+ε)-approximate maximum matching from O(ε^{-19}) (FMU'22) to O(ε^{-6}) by representing search structures as alternating trees with blossoms, leading to simpler analysis and stronger bounds. In comparing with FMU'22, the authors argue that certain analytical refinements—careful parameter tuning, adding scales, and replacing a maximal set of augmenting paths with a maximum set—could plausibly reduce the exponent in FMU'22’s pass complexity down to 12.

However, they explicitly note uncertainty about whether the FMU'22 approach, as it stands, can surpass this O(ε^{-12}) barrier without fundamentally changing its methodology. This raises a concrete unresolved question about the inherent limitations of that prior framework and whether further optimizations within the same approach are possible.