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 $(1.667+\epsilon )$ to $(1.598+\epsilon )$, for any $\epsilon >0$. For TTP-4, we improve the approximation ratio from $(1.750+\epsilon )$ to $(1.700+\epsilon )$. By a refined analysis of the Hamiltonian cycle construction, we also improve the approximation ratio of TTP-k from $(\frac{5k-7}{2k}+\epsilon )$ to $(\frac{5k^2-4k+3}{2k(k+1)}+\epsilon )$ for any constant $k\ge 5$.