  The Traveling Tournament Problem (TTP-$k$) is a well-known benchmark problem
in tournament timetabling, which asks us to design a double round-robin
schedule such that the total traveling distance of all $n$ teams is minimized
under the constraints that each pair of teams plays one game in each other's
home venue, and each team plays at most $k$-consecutive home games or away
games. Westphal and Noparlik (Ann. Oper. Res. 218(1):347-360, 2014) claimed a
$5.875$-approximation algorithm for all $k\geq 4$ and $n\geq 6$. However, there
were both flaws in the construction of the schedule and in the analysis. In
this paper, we show that there is a $5$-approximation algorithm for all $k$ and
$n$. Furthermore, if $k \geq n/2$, the approximation ratio can be improved to
$4$.
