Determine the complexity threshold for locally balanced cuts

Determine the distributed round complexity of computing an \alpha-locally balanced cut for guarantees between \frac14-\varepsilon and \frac12, in particular by establishing whether a (\frac12-\varepsilon)-locally balanced cut can be computed in \operatorname{poly}(\log n) rounds independently of \Delta or whether a dependence on \Delta becomes unavoidable above an intermediate threshold.

Background

The paper constructs a (\frac14-\varepsilon)-locally balanced cut in \widetilde O(\log2 n) deterministic rounds independently of the maximum degree \Delta. In contrast, computing an exactly locally maximum cut, corresponding to the guarantee \alpha=1/2, requires \Omega(\min{\Delta,\sqrt n}) rounds. The unresolved question concerns the transition between these regimes and whether the threshold for degree-independent polylogarithmic complexity lies at, or below, a guarantee arbitrarily close to one half.

References

What happens between these two regimes? Can one compute a $\left(\frac12-\varepsilon\right)$-locally balanced cut in $\poly(\log n)$ rounds independently of $\Delta$, or is there an intermediate threshold beyond which a dependence on $\Delta$ is unavoidable?

— Introvert Clustering for Distributed Graph Algorithms  (2609.10044 - Chang et al., 9 Sep 2026) in Section 6, Conclusions and Open Problems