The traveling tournament problem (TTP) is to minimize the total traveling distance of all teams in a double round-robin tournament. In this paper, we focus on TTP-2, in which each team plays at most two consecutive home games and at most two consecutive away games. For the case where the number of teams n ≡ 2 (mod 4), Zhao and Xiao (2022) presented a (1+5/n)-approximation algorithm. This is a randomized algorithm running in O(n³) time, and its derandomized version runs in O(n⁴) time. In this paper, we present a faster deterministic algorithm running in O(n³) time, with approximation ratio 1+9/n. This ratio improves the previous approximation ratios of the deterministic algorithms with the same time complexity.

This paper presents a continuous approximation model for analyzing the effect of road pricing on the total travel distance. An analytical expression for the total travel distance in road pricing is obtained for a circular city with a radial-arc network. The analytical expression demonstrates how the size of the toll area, the toll level, and the travel cost affect the total travel distance. The result shows that both a high toll and a large toll area are required to reduce the total travel distance. The model explicitly considers through, inward, outward, and city traffic. The effect of road pricing on each traffic can then be examined separately. Road pricing has little impact on the total travel distance of through traffic because through traffic can make a detour around the toll area.