Multiple TSP ($\mathrm{mTSP}$) is a important variant of $\mathrm{TSP}$ where a set of $k$ salesperson together visit a set of $n$ cities. The $\mathrm{mTSP}$ problem has applications to many real life applications such as vehicle routing. Rothkopf introduced another variant of $\mathrm{TSP}$ called many-visits TSP ($\mathrm{MV\mbox{-}TSP}$) where a request $r(v)\in \mathbb{Z}_+$ is given for each city $v$ and a single salesperson needs to visit each city $r(v)$ times and return back to his starting point. A combination of $\mathrm{mTSP}$ and $\mathrm{MV\mbox{-}TSP}$ called many-visits multiple TSP $(\mathrm{MV\mbox{-}mTSP})$ was studied by B\'erczi, Mnich, and Vincze where the authors give approximation algorithms for various variants of $\mathrm{MV\mbox{-}mTSP}$. In this work, we show a simple linear programming (LP) based reduction that converts a $\mathrm{mTSP}$ LP-based algorithm to a LP-based algorithm for $\mathrm{MV\mbox{-}mTSP}$ with the same approximation factor. We apply this reduction to improve or match the current best approximation factors of several variants of the $\mathrm{MV\mbox{-}mTSP}$. Our reduction shows that the addition of visit requests $r(v)$ to $\mathrm{mTSP}$ does $\textit{not}$ make the problem harder to approximate even when $r(v)$ is exponential in number of vertices. To apply our reduction, we either use existing LP-based algorithms for $\mathrm{mTSP}$ variants or show that several existing combinatorial algorithms for $\mathrm{mTSP}$ variants can be interpreted as LP-based algorithms. This allows us to apply our reduction to these combinatorial algorithms as well achieving the improved guarantees.

翻译：多旅行商问题（$\mathrm{mTSP}$）是经典旅行商问题的一个重要变体，其中$k$个销售员共同访问$n$个城市。$\mathrm{mTSP}$问题具有广泛的实际应用，例如车辆路径规划。Rothkopf引入了另一个TSP变体——多访问旅行商问题（$\mathrm{MV\mbox{-}TSP}$），其中每个城市$v$给出一个请求次数$r(v)\in \mathbb{Z}_+$，单个销售员需要访问每个城市$r(v)$次并返回起点。多访问多旅行商问题（$\mathrm{MV\mbox{-}mTSP}$）是$\mathrm{mTSP}$与$\mathrm{MV\mbox{-}TSP}$的结合，由Bérczi、Mnich和Vincze研究，他们给出了$\mathrm{MV\mbox{-}mTSP}$多种变体的近似算法。本文提出了一种基于线性规划（LP）的简单简化方法，将基于LP的$\mathrm{mTSP}$算法转化为具有相同近似因子的$\mathrm{MV\mbox{-}mTSP}$算法。我们应用此简化改进或匹配当前$\mathrm{MV\mbox{-}mTSP}$若干变体的最佳近似因子。该简化表明，即使请求次数$r(v)$随顶点数呈指数增长，将请求次数$r(v)$加入$\mathrm{mTSP}$后，问题的近似难度并未增加。为应用此简化，我们既可直接利用现有的基于LP的$\mathrm{mTSP}$变体算法，也可证明现有多种$\mathrm{mTSP}$变体的组合算法可被解释为基于LP的算法。这使得我们也能将这些组合算法纳入简化框架，从而实现改进的近似保证。