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We give improved approximations for two metric \textsc{Traveling Salesman Problem} (TSP) variants. In \textsc{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 \textsc{Precedence-Constrained TSP} ($PTSP$) in which the precedence constraints are given by a single chain on a subset of nodes. In \textsc{$k$-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 $\alpha+1$ approximation where $\alpha$ 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 \textsc{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. |
