One of the most studied extensions of the famous Traveling Salesperson Problem (TSP) is the {\sc Multiple TSP}: a set of $m\geq 1$ salespersons collectively traverses a set of $n$ cities by $m$ non-trivial tours, to minimize the total length of their tours. This problem can also be considered to be a variant of {\sc Uncapacitated Vehicle Routing} where the objective function is the sum of all tour lengths. When all $m$ tours start from a single common \emph{depot} $v_0$, then the metric {\sc Multiple TSP} can be approximated equally well as the standard metric TSP, as shown by Frieze (1983). The {\sc Multiple TSP} becomes significantly harder to approximate when there is a \emph{set} $D$ of $d \geq 1$ depots that form the starting and end points of the $m$ tours. For this case only a $(2-1/d)$-approximation in polynomial time is known, as well as a $3/2$-approximation for \emph{constant} $d$ which requires a prohibitive run time of $n^{\Theta(d)}$ (Xu and Rodrigues, \emph{INFORMS J. Comput.}, 2015). A recent work of Traub, Vygen and Zenklusen (STOC 2020) gives another approximation algorithm for {\sc Multiple TSP} running in time $n^{\Theta(d)}$ and reducing the problem to approximating TSP. In this paper we overcome the $n^{\Theta(d)}$ time barrier: we give the first efficient approximation algorithm for {\sc Multiple TSP} with a \emph{variable} number $d$ of depots that yields a better-than-2 approximation. Our algorithm runs in time $(1/\varepsilon)^{\mathcal O(d\log d)}\cdot n^{\mathcal O(1)}$, and produces a $(3/2+\varepsilon)$-approximation with constant probability. For the graphic case, we obtain a deterministic $3/2$-approximation in time $2^d\cdot n^{\mathcal O(1)}$.ithm for metric {\sc Multiple TSP} with run time $n^{\Theta(d)}$, which reduces the problem to approximating metric TSP.

翻译：著名旅行商问题（TSP）最受关注的扩展之一是多旅行商问题（\sc{Multiple TSP}）：$m \geq 1$ 名销售员通过 $m$ 条非平凡环游路线共同遍历 $n$ 个城市，以最小化所有路线总长度。该问题也可视为目标函数为所有路线长度之和的无容量车辆路径问题（\sc{Uncapacitated Vehicle Routing}）的变体。当所有 $m$ 条路线均始于同一公共仓库 $v_0$ 时，正如 Frieze (1983) 所示，度量空间下的多旅行商问题可被近似到与标准度量 TSP 相同的精度。然而，当存在一个由 $d \geq 1$ 个仓库构成的集合 $D$ 作为 $m$ 条路线的起点和终点时，多旅行商问题的近似难度显著增加。针对该情形，目前已知的多项式时间算法仅能达到 $(2-1/d)$-近似比，而对于常数 $d$ 存在一个 $3/2$-近似算法，但其运行时间 $n^{\Theta(d)}$ 过高而不可行（Xu 和 Rodrigues, \emph{INFORMS J. Comput.}, 2015）。Traub、Vygen 和 Zenklusen 近期的工作（STOC 2020）提出了另一个运行时间为 $n^{\Theta(d)}$ 的多旅行商问题近似算法，该算法将问题简化为 TSP 近似。本文突破了 $n^{\Theta(d)}$ 的时间障碍：我们首次针对具有变数仓库数量 $d$ 的多旅行商问题提出了一个高效的近似算法，其近似比优于 2。我们的算法运行时间为 $(1/\varepsilon)^{\mathcal O(d\log d)}\cdot n^{\mathcal O(1)}$，并以恒定概率产生 $(3/2+\varepsilon)$-近似解。对于图论情形，我们在时间 $2^d\cdot n^{\mathcal O(1)}$ 内获得确定性 $3/2$-近似算法。