Parameterized tractability under restricted transition structures

Determine whether tractability holds for sparse, reversible, rapidly mixing, or bounded-treewidth transition structures, or whether computational hardness persists under alternative parameterizations of chronological shared realization.

Background

The paper establishes computational hardness for deciding exact and approximate chronological shared realizability, but Proposition fixed-parameter-obstruction shows that the present reduction cannot keep all relevant width parameters fixed. The authors therefore leave unresolved the complexity of the problem under structural restrictions on the transition dynamics or under alternative parameterizations.

The question concerns whether properties such as sparsity, reversibility, rapid mixing, or bounded treewidth make chronological closure efficiently decidable, or whether the hardness phenomenon survives these restrictions.

References

It remains open whether tractability holds for sparse, reversible, rapidly mixing, or bounded-treewidth transition structures, or whether hardness persists under alternative parameterizations.

— Exact Local Optimality Does Not Compose: The Complexity of Chronological Realization  (2609.19707 - Zhao, 17 Sep 2026) in Section 6, Discussion and Open Problems, item “Fixed-parameter tractability”

It remains an intriguing question which dynamical families guarantee polynomially encodable and exactly verifiable discrete, rational, or algebraic certificates for exact realizability itself.

— Exact Local Optimality Does Not Compose: The Complexity of Chronological Realization  (2609.19707 - Zhao, 17 Sep 2026) in Section 6, Discussion and Open Problems, item “Continuous versus discrete certificates”

It is open whether stronger conditioning assumptions or alternative geometric sources yield sharper robustness exponents while retaining the chronological hardness gap.

— Exact Local Optimality Does Not Compose: The Complexity of Chronological Realization  (2609.19707 - Zhao, 17 Sep 2026) in Section 6, Discussion and Open Problems, item “Improved robust transfer”

The general problem is a recurrent analogue of zero-error source coding with decoder side information \citep{witsenhausen1976zeroerror,alon1996source}; we do not characterize its computational complexity.

— Minimal Recurrent Behavioral Memory for Imitation under Partial Observability  (2609.25757 - Li et al., 22 Sep 2026) in Appendix A.4, Section “The general case: closed compatible state assignments”