Linear Programming based Reductions for Multiple Visit TSP and Vehicle Routing Problems

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 Bérczi, 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.