Exact comparison complexity on arbitrary cyclic topologies

Determine an exact characterization of the comparison complexity, and more generally the comparison–addition resource frontier, for exact labeled single-source shortest paths on arbitrary directed topologies containing cycles.

Background

The paper completely characterizes the addition coordinate for every directed topology and proves an exact rectangular comparison–addition region for directed acyclic graphs. For cyclic graphs, it establishes explicit nonrectangular behavior on the two-spoke graph H_2 and an entropy-tight asymptotic tradeoff for the shared-hub family H_k.

These family-specific results do not yield a closed formula for arbitrary cyclic topologies. In particular, the authors leave unresolved how to characterize exact comparison complexity, or the full two-resource feasible region, beyond the exhibited cyclic examples.

References

Even within the present model, an exact comparison characterization for arbitrary cyclic topologies is open.

— Beyond Distance Ordering: Resource Complexity and Universal Optimality of Exact Labeled Directed Shortest Paths  (2609.04825 - Cai, 4 Sep 2026) in Section 6, 'Interpretation of the resource theory'; Section 7, paragraph 'Scope of the resource characterizations'