Dynamic approximate min-cut or shortest-path maintenance on expander graphs

Establish whether a fully dynamic undirected graph that always remains an expander can support subpolynomial-time maintenance of a (1+ε)-approximate s-t minimum cut or shortest-path value.

Background

The paper identifies a broader gap in dynamic graph algorithms for graphs constrained to remain expanders. Although the paper develops efficient dynamic data structures for effective resistance and commute-time queries on undirected expanders, it does not resolve whether similarly efficient data structures exist for approximate s-t minimum-cut or shortest-path queries.

The unresolved question concerns fully dynamic updates while preserving the expander condition at every time step, and asks for subpolynomial update or maintenance time with a (1+ε)-approximation guarantee.

References

In particular, it is open whether one can maintain, in subpolynomial time, $(1 + )$-approximate $s$-$t$ min-cut or shortest path in fully dynamic undirected graphs that always remain expanders.

— Fully Dynamic Random-Walk Quantities on Undirected Expanders and $κ$-Uniform Directed Graphs  (2610.17712 - Peng et al., 13 Oct 2026) in Section 1, Introduction