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arXiv 2405.12876cs.DS
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利用桥引理近似旅行商问题

Approximating Traveling Salesman Problems Using a Bridge Lemma

University of Wrocław（弗罗茨瓦夫大学）；University of Alberta（阿尔伯塔大学）；University of Augsburg（奥格斯堡大学）；University of Sheffield（谢菲尔德大学）

Martin Böhm, Zachary Friggstad, Tobias Mömke, Joachim Spoerhase

备注：
v3: No textual changes since v2, only a license change
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AI总结：

本文改进了顺序TSP和k人TSP路径的近似算法，通过桥引理的改进方法，分别获得1.878和2.214的近似比，突破了传统方法的限制。

摘要：

我们为两个度量旅行商问题变种提供了改进的近似算法。在有序TSP（OTSP）中，给定节点子集的线性顺序$o_1, \ldots, o_k$，TSP解必须满足$o_{i+1}$在$o_i$之后被访问。这是具有单链 precedence 约束的 precedence-constrained TSP（PTSP）的特殊情况。在k-人TSP路径（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。我们为OTSP获得$3/2 + e^{-1} < 1.878$的近似比，首次超越了传统$α+1$近似比（其中$α$是当前最佳TSP近似比）。我们还为k-TSPP获得$1 + 2 \cdot e^{-1/2} < 2.214$的近似比，首次超越了传统3近似比。这些算法均使用了对桥引理的改进版本，该版本最初用于改进Steiner树近似算法[Byrka et al., 2013]。我们的变体表明，如果随机采样一组非终端节点使其成为终端节点，且每个非终端节点有常数概率被采样，则以给定终端节点为根的最便宜森林的成本将显著减少。我们相信这种桥引理的观点将在本文之外进一步用于改进车辆路径近似算法。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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