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In some professional sports leagues, inter-league games are scheduled among multiple divisions or conferences. This inspired us to study the $p$-partite Traveling Tournament Problem ($p$-partite TTP), where teams are partitioned into $p$ leagues, and each team plays games against teams from different leagues. Previously, only the case of $p=2$, known as the Bipartite TTP or BTTP, has been introduced and studied (AAAI 2011 and IJCAI 2024).
In this paper, we show that the $p$-partite TTP is NP-hard for any fixed $p \geq 3$, and we propose an efficient algorithm based on a solution to the TSP. Furthermore, we prove that the algorithm achieves a novel approximation ratio of $\frac{8}{3} + O(\frac{1}{n})$ when $p=3$. We also conduct experiments demonstrating that the algorithm produces practical schedules with significantly reduced total travel distances, highlighting its effectiveness in generating high-quality multipartite tournament schedules.