Amiri and Wargalla (2020) proved the following local-to-global theorem in directed acyclic graphs (DAGs): if $G$ is a weighted DAG such that for each subset $S$ of 3 nodes there is a shortest path containing every node in $S$, then there exists a pair $(s,t)$ of nodes such that there is a shortest $st$-path containing every node in $G$. We extend this theorem to general graphs. For undirected graphs, we prove that the same theorem holds (up to a difference in the constant 3). For directed graphs, we provide a counterexample to the theorem (for any constant), and prove a roundtrip analogue of the theorem which shows there exists a pair $(s,t)$ of nodes such that every node in $G$ is contained in the union of a shortest $st$-path and a shortest $ts$-path. The original theorem for DAGs has an application to the $k$-Shortest Paths with Congestion $c$ (($k,c$)-SPC) problem. In this problem, we are given a weighted graph $G$, together with $k$ node pairs $(s_1,t_1),\dots,(s_k,t_k)$, and a positive integer $c\leq k$. We are tasked with finding paths $P_1,\dots, P_k$ such that each $P_i$ is a shortest path from $s_i$ to $t_i$, and every node in the graph is on at most $c$ paths $P_i$, or reporting that no such collection of paths exists. When $c=k$ the problem is easily solved by finding shortest paths for each pair $(s_i,t_i)$ independently. When $c=1$, the $(k,c)$-SPC problem recovers the $k$-Disjoint Shortest Paths ($k$-DSP) problem, where the collection of shortest paths must be node-disjoint. For fixed $k$, $k$-DSP can be solved in polynomial time on DAGs and undirected graphs. Previous work shows that the local-to-global theorem for DAGs implies that $(k,c)$-SPC on DAGs whenever $k-c$ is constant. In the same way, our work implies that $(k,c)$-SPC can be solved in polynomial time on undirected graphs whenever $k-c$ is constant.

翻译：Amiri与Wargalla（2020）在有向无环图（DAG）中证明了以下局部到全局定理：若加权有向无环图$G$满足对任意3个节点构成的子集$S$，均存在一条包含$S$中所有节点的最短路径，则存在节点对$(s,t)$使得存在一条包含$G$中所有节点的最短$st$-路径。我们将该定理推广至一般图。对于无向图，我们证明相同定理成立（常数3存在差异）。对于有向图，我们给出该定理的反例（针对任意常数），并证明其往返版本定理：存在节点对$(s,t)$使得$G$中每个节点均包含于某条最短$st$-路径与某条最短$ts$-路径的并集中。原DAG版本的定理在拥塞$c$的$k$最短路径（($k,c$)-SPC）问题中具有应用。该问题中，给定加权图$G$、$k$个节点对$(s_1,t_1),\dots,(s_k,t_k)$及正整数$c\leq k$，需寻找路径$P_1,\dots,P_k$使得每条$P_i$均为从$s_i$到$t_i$的最短路径，且图中每个节点至多出现在$c$条路径$P_i$中，否则报告不存在这样的路径集合。当$c=k$时，问题可通过独立寻找每对$(s_i,t_i)$的最短路径轻松求解。当$c=1$时，($k,c$)-SPC问题退化为$k$条不相交最短路径（$k$-DSP）问题，其中最短路径集合必须节点不相交。对于固定的$k$，$k$-DSP问题可在有向无环图和无向图上多项式时间内求解。先前研究表明，DAG的局部到全局定理蕴含当$k-c$为常数时，有向无环图上的($k,c$)-SPC问题可解。类似地，我们的工作表明当$k-c$为常数时，无向图上的($k,c$)-SPC问题可在多项式时间内求解。