In the Many-visits Path TSP, we are given a set of $n$ cities along with their pairwise distances (or cost) $c(uv)$, and moreover each city $v$ comes with an associated positive integer request $r(v)$.
The goal is to find a minimum-cost path, starting at city $s$ and ending at city $t$, that visits each city $v$ exactly $r(v)$ times.
We present a $\frac32$-approximation algorithm for the metric Many-visits Path TSP, that runs in time polynomial in $n$ and poly-logarithmic in the requests $r(v)$.
Our algorithm can be seen as a far-reaching generalization of the $\frac32$-approximation algorithm for Path TSP by Zenklusen (SODA 2019), which answered a long-standing open problem by providing an efficient algorithm which matches the approximation guarantee of Christofides' algorithm from 1976 for metric TSP.
One of the key components of our approach is a polynomial-time algorithm to compute a connected, degree bounded multigraph of minimum cost.
We tackle this problem by generalizing a fundamental result of Király, Lau and Singh (Combinatorica, 2012) on the Minimum Bounded Degree Matroid Basis problem, and devise such an algorithm for general polymatroids, even allowing element multiplicities.
Our result directly yields a $\frac32$-approximation to the metric Many-visits TSP, as well as a $\frac32$-approximation for the problem of scheduling classes of jobs with sequence-dependent setup times on a single machine so as to minimize the makespan.
本文提出了度量路径旅行商问题（Path TSP）的一个$1.5$ -近似.最近对路径TSP的所有改进都关键地利用了An，Kleinberg和Shmoys [Journal of the ACM，2015]所示的结构属性，即相对于Held-Karp解决方案的窄切割形成链。我们显着偏离这些方法，显示处理较大的$s$ - $t$切割的好处，即使它们的结构少得多。更准确地说，我们证明了Traub和Vygen最近引入的动态规划思想的变体[SODA，2018]通过利用Karger关于近最小切割数量的开创性结果，足以处理更大尺寸的切割。这避免了Traub和Vygen使用的动态规划的递归应用，并导致一个相当简单的算法，避免了近似保证中的额外误差项。我们匹配仍然不败的$1.5$ -近似保证Christofides的TSP算法。因此，路径TSP的逼近性的任何进一步的进展也将导致TSP的改进。
