The APX-hardness of the Traveling Tournament Problem

TL;DR

The paper proves that the Traveling Tournament Problem with at-most-$k$ constraint (TTP-$k$) is APX-hard for every fixed $k\ge 3$, by constructing a gap-preserving L-reduction from a boosted restricted $k$-tour cover problem to TTP-$k$. It introduces Boosted Restricted $k$-TC, shows its APX-hardness, and carefully builds intermediate instances $I'$ and $J$ to relate optimal values and approximation ratios. The reduction employs a sophisticated scheduling construction with super-teams and a tailored TTP-2 subroutine to enforce the at-most-$k$ constraint while bounding travel distances, yielding a constant-factor gap transfer. This work closes the complexity status gap for TTP-$k$ across fixed $k\ge 3$ and lays groundwork for further exploration of PTAS possibilities in non-metric or non-Euclidean settings.

Abstract

The Traveling Tournament Problem (TTP-$k$) is a well-known benchmark problem in sports scheduling, which asks us to design a double round-robin schedule such that each pair of teams plays one game in each other's home venue, no pair of teams plays each other on two consecutive days, each team plays at most $k$ consecutive home games or away games, and the total traveling distance of all the $n$ teams is minimized. TTP-$k$ allows a polynomial-time approximation scheme when $k=2$ and becomes APX-hard when $k\geq n-1$. In this paper, we reduce the gap by showing that TTP-$k$ is APX-hard for any fixed $k\geq3$.

Figures (5)

• Figure 1: An illustration of the relations for instances $I$, $I'$, and $J$, where cycle $C$ denotes the depot position, cycle $A$ denotes $m-(n-1)$ vertices on the depot, and cycle $B$ denotes $m(m^2-1)-1$ vertices on the depot: the ellipse can be seen as instance $I$ (containing 1 depot vertex, denoted by the black node, and $n-1$ non-depot vertices, denoted by the white nodes), the ellipse with vertices in $A$ (resp., $A$ and $B$) can be seen as instance $I'$ (resp., $J$).
• Figure 2: The super-game schedule in the first block.
• Figure 3: The super-game schedule in the second block.
• Figure 4: The super-game schedule in the third block.
• Figure 5: An illustration of the special TTP-2 algorithm, where we only show the first three days for $m$ teams.

Theorems & Definitions (32)

• Definition 1: williamson2011design
• Theorem 2
• Definition 3: $k$-TC
• Lemma 4: asano1996coveringasano1996covering+
• Definition 5: Restricted $k$-TC
• Lemma 6
• proof
• Definition 7: Boosted Restricted $k$-TC
• Lemma 8
• proof