Connectivity of the round-robin timetable search space

Determine whether there exists a polynomial-time neighborhood that fully connects the search space of round-robin timetables while preserving the double round-robin structure after every consecutive move.

Background

Neighborhood-based heuristics modify feasible round-robin timetables through local moves such as PartialRoundSwap, PartialTeamSwap, and related operators. Although several neighborhoods have been developed, the paper notes that even after introducing GPTS it is unresolved whether the search space is connected. The open problem asks for a polynomial-time move system that guarantees connectivity while maintaining the 2RR structure throughout the search.

References

Even with the inclusion of GPTS to the neighborhood structure, it is unknown whether the search space of round-robin tournament scheduling is connected (see \citet{Ribeiro2025}). This leads to the following open question.

\begin{open} Does there exist a polynomial-time neighborhood that fully connects the round-robin timetable search space, while preserving the 2RR structure among consecutive moves? \end{open}

— The Traveling Tournament Problem: An Overview  (2609.03612 - Bulck et al., 3 Sep 2026) in Section 6, subsection “Heuristic Methods”