  The Traveling Tournament Problem (TTP) is a well-known benchmark problem in
the field of tournament timetabling, 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, minimizing the total distance traveled by all $n$ teams
($n$ is even). TTP-$k$ is the problem with one more constraint that each team
can have at most $k$-consecutive home games or away games. In this paper, we
investigate schedules for TTP-$k$ and analyze the approximation ratio of the
solutions. Most previous schedules were constructed based on a Hamiltonian
cycle of the graph. We will propose a novel construction based on a $k$-cycle
packing. Then, combining our $k$-cycle packing schedule with the Hamiltonian
cycle schedule, we obtain improved approximation ratios for TTP-$k$ with deep
analysis. The case where $k=3$, TTP-3, is one of the most investigated cases.
We improve the approximation ratio of TTP-3 from $(1.667+\varepsilon)$ to
$(1.598+\varepsilon)$, for any $\varepsilon>0$. For TTP-$4$, we improve the
approximation ratio from $(1.750+\varepsilon)$ to $(1.700+\varepsilon)$. By a
refined analysis of the Hamiltonian cycle construction, we also improve the
approximation ratio of TTP-$k$ from $(\frac{5k-7}{2k}+\varepsilon)$ to
$(\frac{5k^2-4k+3}{2k(k+1)}+\varepsilon)$ for any constant $k\geq 5$. Our
methods can be extended to solve a variant called LDTTP-$k$ (TTP-$k$ where all
teams are allocated on a straight line). We show that the $k$-cycle packing
construction can achieve an approximation ratio of
$(\frac{3k-3}{2k-1}+\varepsilon)$, which improves the approximation ratio of
LDTTP-3 from $4/3$ to $(6/5+\varepsilon)$.
