In this paper we look at $k$-stroll, point-to-point orienteering, as well as the deadline TSP problem on graphs with bounded doubling dimension and bounded treewidth and present approximation schemes for them. Given a weighted graph $G=(V,E)$, start node $s\in V$, distances $d:E\rightarrow \mathbb{Q}^+$ and integer $k$. In the $k$-stroll problem the goal is to find a path starting at $s$ of minimum length that visits at least $k$ vertices. The dual problem to $k$-stroll is the rooted orienteering in which instead of $k$ we are given a budget $B$ and the goal is to find a walk of length at most $B$ starting at $s$ that visits as many vertices as possible. In the P2P orienteering we are given start and end nodes $s,t$ for the path. In the deadline TSP we are given a deadline $D(v)$ for each $v\in V$ and the goal is to find a walk starting at $s$ that visits as many vertices as possible before their deadline. The best approximation for rooted or P2P orienteering is $(2+\epsilon)$-approximation [12] and $O(\log n)$-approximation for deadline TSP [3]. There is no known approximation scheme for deadline TSP for any metric (not even trees). Our main result is the first approximation scheme for deadline TSP on metrics with bounded doubling dimension. To do so we first show if $G$ is a metric with doubling dimension $\kappa$ and aspect ratio $\Delta$, there is a $(1+\epsilon)$-approximation that runs in time $n^{O\left(\left(\log\Delta/\epsilon\right)^{2\kappa+1}\right)}$. We then extend these to obtain an approximation scheme for deadline TSP when the distances and deadlines are integer which runs in time $n^{O\left(\left(\log \Delta/\epsilon\right)^{2\kappa+2}\right)}$. For graphs with treewidth $\omega$ we show how to solve $k$-stroll and P2P orienteering exactly in polynomial time and a $(1+\epsilon)$-approximation for deadline TSP in time $n^{O((\omega\log\Delta/\epsilon)^2)}$.

翻译：本文研究具有有界加倍维数和有界树宽的图上的k-漫步、点对点巡游以及截止日期TSP问题，并给出它们的近似方案。给定加权图$G=(V,E)$、起点$s\in V$、距离函数$d:E\rightarrow \mathbb{Q}^+$和整数$k$。在k-漫步问题中，目标是找到一条从$s$出发、长度最短且至少访问$k$个顶点的路径。k-漫步的对偶问题是有根巡游问题：此时给定预算$B$而非$k$，目标是找到一条从$s$出发、长度不超过$B$且访问尽可能多顶点的行走。在点对点巡游中，路径的起点和终点$s,t$均已给定。在截止日期TSP问题中，每个顶点$v\in V$有对应截止日期$D(v)$，目标是找到一条从$s$出发、在截止日期前访问尽可能多顶点的行走。有根巡游或点对点巡游的最佳近似比为$(2+\epsilon)$-近似[12]，而截止日期TSP的近似比为$O(\log n)$[3]。目前任何度量空间（甚至树）上的截止日期TSP均无已知的近似方案。本文的主要结果是为具有有界加倍维数的度量空间上的截止日期TSP给出首个近似方案。为此，我们首先证明：若$G$是加倍维数为$\kappa$、纵横比为$\Delta$的度量空间，则存在时间复杂度为$n^{O\left(\left(\log\Delta/\epsilon\right)^{2\kappa+1}\right)}$的$(1+\epsilon)$-近似算法。进一步，我们将此结果推广至距离和截止日期均为整数的情况，得到时间复杂度为$n^{O\left(\left(\log \Delta/\epsilon\right)^{2\kappa+2}\right)}$的截止日期TSP近似方案。对于树宽为$\omega$的图，我们给出多项式时间内精确求解k-漫步和点对点巡游的方法，以及对截止日期TSP的$(1+\epsilon)$-近似算法，其时间复杂度为$n^{O((\omega\log\Delta/\epsilon)^2)}$。