Tightness of independent lower bounds for time-relaxed TTP

Determine whether there exists a value α, possibly dependent on the number of teams n and the maximum trip length k, such that the independent lower bound ILB(k) is tight for the time-relaxed Traveling Tournament Problem RTTP(k) with 2n−2+α time slots.

Background

The independent lower bound decomposes the Traveling Tournament Problem into independent routing problems for the individual teams. For the time-relaxed TTP, additional time slots or byes may permit the individual minimum routing costs to be combined into a globally feasible timetable. The paper attributes to Bao et al. an unresolved conjecture concerning the sufficiency of additional time slots and formulates the broader question in terms of an α-dependent number of slots.

References

An interesting conjecture by \citet{Bao2010} that, to the best of our knowledge, is still open is whether ILB(3) becomes tight when there are sufficiently more time slots than games per team. \begin{open} \label{open:rttp} Determine whether there exists an $\alpha$ such that the ILB($k$) is tight for RTTP($k$) with $2n-2 + \alpha$ time slots, where $\alpha$ may depend on $n$ and $k$. \end{open}

— The Traveling Tournament Problem: An Overview  (2609.03612 - Bulck et al., 3 Sep 2026) in Section 4, paragraph “Independent lower bounds”