Complementary home-away patterns in CD-TTP(3)

Determine whether every constant-distance Traveling Tournament Problem with maximum trip length three, CD-TTP(3), has an optimal solution containing a complementary home-away pattern set in which each team is paired with another team having the exact opposite home-away sequence.

Background

For constant-distance instances, minimizing total travel is equivalent to minimizing the number of travel legs. The paper reports a conjecture that optimal solutions can always be restricted to timetables whose teams are organized into complementary home-away pattern pairs. Whether this structural property can always be imposed without sacrificing optimality remains unresolved.

References

Moreover, when minimizing travel legs, \citet{VanBulck2023} conjecture that it suffices to consider timetables where every team is paired with another that plays the exact opposite sequence of home and away games (referred to as a complementary Home/Away Pattern (HAP) set).

\begin{open} Determine whether there always exists an optimal solution to CD-TTP(3) that has a complementary HAP set. \end{open}

— The Traveling Tournament Problem: An Overview  (2609.03612 - Bulck et al., 3 Sep 2026) in Section 4, paragraph “Minimum number of legs”