We initiate the study of a fundamental combinatorial problem: Given a capacitated graph $$G=(V,E)$$ G = ( V , E ) , find a shortest walk (“route”) from a source $${s\in V}$$ s ∈ V to a destination $$t\in V$$ t ∈ V that includes all vertices specified by a set $$WP \subseteq V$$ W P ⊆ V : the waypoints . This W aypoint R outing P roblem finds immediate applications in the context of modern networked systems. Our main contribution is an exact polynomial-time algorithm for graphs of bounded treewidth. We also show that if the number of waypoints is logarithmically bounded, exact polynomial-time algorithms exist even for general graphs. Our two algorithms provide an almost complete characterization of what can be solved exactly in polynomial time: we show that more general problems (e.g., on grid graphs of maximum degree 3, with slightly more waypoints) are computationally intractable.

中文翻译：

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穿越航点

我们开始研究一个基本的组合问题：给定一个容量图 $$G=(V,E)$$ G = ( V , E ) ，从源 $${s\ 找到最短的步行（“路线”） in V}$$ s ∈ V 到目的地 $$t\in V$$ t ∈ V 包括由集合 $$WP \subseteq V$$ WP ⊆ V 指定的所有顶点：航点。这个Waypoint Routing 问题在现代网络系统的上下文中找到了直接的应用。我们的主要贡献是针对有界树宽图的精确多项式时间算法。我们还表明，如果航点的数量是对数有界的，那么即使对于一般图形也存在精确的多项式时间算法。我们的两种算法几乎完整地描述了可以在多项式时间内精确解决的问题：我们展示了更一般的问题（例如，在最大次数为 3 的网格图上，