A $5$-approximation Algorithm for the Traveling Tournament Problem

TL;DR

This paper presents a new algorithm that guarantees a 5-approximate solution for the Traveling Tournament Problem, improving previous bounds and providing better guarantees for specific cases.

Contribution

The authors develop a 5-approximation algorithm for TTP-$k$, correcting prior flawed approaches and offering an improved ratio of 4 when $k \\geq n/2$.

Findings

Established a 5-approximation algorithm for all $k$ and $n$.

Improved the approximation ratio to 4 when $k \\geq n/2$.

Corrected previous flawed constructions and analyses.

Abstract

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 $4$.