Faster computation of minimum-radius balanced disk separators

Develop an algorithm that finds, for a given graph and vertex subset, a minimum-radius balanced disk separator in o(nm) time.

Background

The paper shows that a balanced disk separator of radius at most the layering cluster-diameter can be found in linear time, while a minimum-radius balanced disk separator can be found in O(nm) time by considering all possible centers. The authors explicitly identify the lack of a sub-O(nm) method for finding a center whose disk contains a selected layering cluster as an unresolved algorithmic issue.

References

It is not clear how to find in o(nm) time a vertex v such that C\subseteq D_r(v) (we may need the entire distance matrix of G to do that).

— Graph parameters that are coarsely equivalent to tree-length  (2502.00951 - Dragan, 2 Feb 2025) in Section 3.2, subsection “McCarty-width of a graph”