Log-space solvability for general instances of Recon(D)

Determine whether the reconfiguration problem Recon(D) for a reflexive digraph cycle D can be solved in logarithmic space for general instance digraphs, rather than only for cyclic instances.

Background

The paper proves that Recon(D) is solvable in logarithmic space when the instance digraph is itself a cycle. For more general instances, prior results reduce the problem to checking the relevant condition on individual cycle instances belonging to a basis of the instance's cycle space. The authors identify a space-complexity obstacle: deterministically discovering such a cycle basis appears to require storing a spanning tree, which exceeds logarithmic space. Consequently, extending the logarithmic-space result from cyclic instances to arbitrary instances remains unresolved.

References

A natural next question is whether Recon(D) can be solved in log-space for more general instances. From [2] it is enough to check the conditions for a single cycle instance on every cycle in a basis of the cycle space.

— Reflexive Digraph Reconfiguration by Orientation Strings  (2501.01599 - Pullas et al., 3 Jan 2025) in Section 5, Concluding Remarks