We investigate semi-streaming algorithms for the Traveling Salesman Problem (TSP). Specifically, we focus on a variant known as the $(1,2)$-TSP, where the distances between any two vertices are either one or two. Our primary emphasis is on the closely related Maximum Path Cover Problem, which aims to find a collection of vertex-disjoint paths that cover the maximum number of edges in a graph. We propose an algorithm that, for any $\epsilon > 0$, achieves a $(\frac{2}{3}-\epsilon)$-approximation of the maximum path cover size for an $n$-vertex graph, using $\text{poly}(\frac{1}{\epsilon})$ passes. This result improves upon the previous $\frac{1}{2}$-approximation by Behnezhad et al. [ICALP 2024] in the semi-streaming model. Building on this result, we design a semi-streaming algorithm that constructs a tour for an instance of $(1,2)$-TSP with an approximation factor of $(\frac{4}{3} + \epsilon)$, improving upon the previous $\frac{3}{2}$-approximation actor algorithm by Behnezhad et al. [ICALP 2024] (Although it is not explicitly stated in the paper that their algorithm works in the semi-streaming model, it is easy to verify). Furthermore, we extend our approach to develop an approximation algorithm for the Maximum TSP (Max-TSP), where the goal is to find a Hamiltonian cycle with the maximum possible weight in a given weighted graph $G$. Our algorithm provides a $(\frac{7}{12} - \epsilon)$-approximation for Max-TSP in $\text{poly}(\frac{1}{\epsilon})$ passes, improving on the previously known $(\frac{1}{2}-\epsilon)$-approximation obtained via maximum weight matching in the semi-streaming model.

翻译：本文研究旅行商问题（TSP）的半流算法。具体而言，我们关注一种称为$(1,2)$-TSP的变体，其中任意两顶点间的距离非1即2。我们的核心重点放在与之紧密相关的最大路径覆盖问题上，该问题旨在寻找覆盖图中最多边的顶点不相交路径集合。我们提出一种算法，对于任意$\epsilon > 0$，可在$\text{poly}(\frac{1}{\epsilon})$轮遍历内，对$n$顶点图实现最大路径覆盖规模的$(\frac{2}{3}-\epsilon)$近似。该结果改进了Behnezhad等人[ICALP 2024]在半流模型中提出的$\frac{1}{2}$近似算法。基于此结果，我们设计了一种半流算法，可为$(1,2)$-TSP实例构建近似比为$(\frac{4}{3} + \epsilon)$的环游，优于Behnezhad等人[ICALP 2024]先前提出的$\frac{3}{2}$近似算法（尽管原文未明确说明其算法适用于半流模型，但易验证其适用性）。进一步地，我们将该方法拓展至最大旅行商问题（Max-TSP），该问题旨在给定加权图$G$中寻找具有最大可能权重的哈密顿环。我们的算法在$\text{poly}(\frac{1}{\epsilon})$轮遍历内为Max-TSP提供$(\frac{7}{12} - \epsilon)$近似，改进了先前通过半流模型中最大权匹配获得的$(\frac{1}{2}-\epsilon)$近似结果。