The Traveling Tournament Problem: Improved Algorithms Based on Cycle Packing

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 $(6/5+\varepsilon)$.