P-versus-NP-hard dichotomy for RCSPs with infinite restrictions

Establish whether, for every finite structure H and every possibly infinite structure R, the restricted CSP RCSP(H,R) is always either in P or NP-hard.

Background

The finite-domain, finite-restriction case is resolved in the paper by reducing restricted CSPs to finite-domain CSPs. The unrestricted infinite-restriction setting is not covered by that theorem.

This problem asks whether an intermediate complexity classification can occur for restricted CSPs whose restriction structure is infinite.

References

Is it true that for every finite structure $$ and every (possibly infinite) $$ the problem $\RCSP(,)$ is either in $\cP$ or $\NP$-hard? (Compare to Theorem~\ref{thm:finite-RCSP-dichotomy}).

— Restricted CSPs and F-free Digraph Algorithmics  (2502.17596 - Guzmán-Pro et al., 24 Feb 2025) in Conclusion and outlook, Section 8