We give improved approximations for two metric Traveling Salesman Problem (TSP) variants. In Ordered TSP (OTSP) we are given a linear ordering on a subset of nodes $o_1, \ldots, o_k$. The TSP solution must have that $o_{i+1}$ is visited at some point after $o_i$ for each $1 \leq i < k$. This is the special case of Precedence-Constrained TSP ($PTSP$) in which the precedence constraints are given by a single chain on a subset of nodes. In $k$-Person TSP Path (k-TSPP), we are given pairs of nodes $(s_1,t_1), \ldots, (s_k,t_k)$. The goal is to find an $s_i$-$t_i$ path with minimum total cost such that every node is visited by at least one path. We obtain a $3/2 + e^{-1} < 1.878$ approximation for OTSP, the first improvement over a trivial $α+1$ approximation where $α$ is the current best TSP approximation. We also obtain a $1 + 2 \cdot e^{-1/2} < 2.214$ approximation for k-TSPP, the first improvement over a trivial $3$-approximation. These algorithms both use an adaptation of the Bridge Lemma that was initially used to obtain improved Steiner Tree approximations [Byrka et al., 2013]. Roughly speaking, our variant states that the cost of a cheapest forest rooted at a given set of terminal nodes will decrease by a substantial amount if we randomly sample a set of non-terminal nodes to also become terminals such provided each non-terminal has a constant probability of being sampled. We believe this view of the Bridge Lemma will find further use for improved vehicle routing approximations beyond this paper.

翻译：我们针对两个度量旅行商问题（TSP）变体给出了改进的近似算法。在有序旅行商问题（OTSP）中，给定节点子集 $o_1, \ldots, o_k$ 上的线性顺序。TSP 解必须满足对于每个 $1 \leq i < k$，节点 $o_{i+1}$ 在 $o_i$ 之后某点被访问。这是优先约束旅行商问题（PTSP）的特例，其中优先约束由节点子集上的单链给出。在 $k$ 人旅行商路径问题（k-TSPP）中，给定节点对 $(s_1,t_1), \ldots, (s_k,t_k)$。目标是找到总成本最小的 $s_i$-$t_i$ 路径，使得每个节点至少被一条路径访问。我们获得了 OTSP 的 $3/2 + e^{-1} < 1.878$ 近似比，这是对平凡 $α+1$ 近似（其中 $α$ 是当前最佳 TSP 近似比）的首次改进。我们还获得了 k-TSPP 的 $1 + 2 \cdot e^{-1/2} < 2.214$ 近似比，这是对平凡 $3$ 近似比的首次改进。这两种算法都采用了桥接引理的改编版本，该引理最初用于改进斯坦纳树近似 [Byrka 等，2013]。粗略地说，我们的变体表明：如果随机采样一组非终端节点作为新的终端节点，使得每个非终端节点具有恒定的采样概率，则扎根于给定终端节点集的最便宜森林的成本将显著下降。我们相信，这种桥接引理的视角将在本文之外为改进车辆路径问题的近似提供进一步的应用。