In the Tricolored Euclidean Traveling Salesperson problem, we are given~$k=3$ sets of points in the plane and are looking for disjoint tours, each covering one of the sets. Arora (1998) famously gave a PTAS based on ``patching'' for the case $k=1$ and, recently, Dross et al.~(2023) generalized this result to~$k=2$. Our contribution is a $(5/3+\epsilon)$-approximation algorithm for~$k=3$ that further generalizes Arora's approach. It is believed that patching is generally no longer possible for more than two tours. We circumvent this issue by either applying a conditional patching scheme for three tours or using an alternative approach based on a weighted solution for $k=2$.

翻译：在三色欧几里得旅行商问题中，给定平面上的$k=3$个点集，目标是寻找不相交的回路，每条回路覆盖其中一个点集。Arora (1998) 针对$k=1$的情形，基于“拼接”技术提出了一个著名的多项式时间近似方案，而Dross等人 (2023) 近期将该结果推广至$k=2$。我们的贡献是针对$k=3$提出一个$(5/3+\epsilon)$-近似算法，进一步推广了Arora的方法。一般认为对于超过两条回路的拼接通常不再可行。我们通过两种方式规避这一问题：要么对三条回路采用条件性拼接方案，要么使用基于$k=2$加权解的替代方法。