Fixed-parameter tractability for three ranks parameterized by temporal agony

Determine whether temporal agony with three ranks and one change point per vertex is fixed-parameter tractable when parameterized by the target temporal agony α.

Background

The paper establishes that temporal agony with three ranks and one change point per vertex is NP-hard in general, while it is solvable in polynomial time for every fixed value of the target agony α, placing the problem in XP with respect to α. The authors compare this situation with the known fixed-parameter tractability of Max 2-SAT parameterized by the number of unsatisfied clauses and identify fixed-parameter tractability for temporal agony as the main unresolved complexity question.

References

Since Max 2-SAT is known to be in FPT for this parameter, we leave it as the main open question whether with three ranks is also in FPT for~$\alpha$?

— Parameterized Complexity of Temporal Agony  (2608.20077 - Breitkopf et al., 20 Aug 2026) in Conclusion

It is unclear whether the case~$k=3$ is also NP-hard if the underlying digraph is a DAG.

— Parameterized Complexity of Temporal Agony  (2608.20077 - Breitkopf et al., 20 Aug 2026) in Section 4, final paragraph before Section 5 (Conclusion)