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, 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 PTAS 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 $k\geq3$.

翻译：巡回赛问题（TTP-$k$）是体育赛程安排中一个著名的基准问题，该问题要求设计一个双循环赛程，使得每对队伍在对方主场各进行一场比赛，每支队伍最多连续进行$k$场主场或客场赛程，并且所有$n$支队伍的总旅行距离最小化。TTP-$k$在$k=2$时允许多项式时间近似方案（PTAS），而在$k\geq n-1$时是APX-难的。本文通过证明对于任意固定的$k\geq3$，TTP-$k$均为APX-难，从而缩小了这一差距。