In the $k$-Disjoint Shortest Paths ($k$-DSP) problem, we are given a weighted graph $G$ on $n$ nodes and $m$ edges with specified source vertices $s_1, \dots, s_k$, and target vertices $t_1, \dots, t_k$, and are tasked with determining if $G$ contains vertex-disjoint $(s_i,t_i)$-shortest paths. For any constant $k$, it is known that $k$-DSP can be solved in polynomial time over undirected graphs and directed acyclic graphs (DAGs). However, the exact time complexity of $k$-DSP remains mysterious, with large gaps between the fastest known algorithms and best conditional lower bounds. In this paper, we obtain faster algorithms for important cases of $k$-DSP, and present better conditional lower bounds for $k$-DSP and its variants. Previous work solved 2-DSP over weighted undirected graphs in $O(n^7)$ time, and weighted DAGs in $O(mn)$ time. For the main result of this paper, we present linear time algorithms for solving 2-DSP on weighted undirected graphs and DAGs. Our algorithms are algebraic however, and so only solve the detection rather than search version of 2-DSP. For lower bounds, prior work implied that $k$-Clique can be reduced to $2k$-DSP in DAGs and undirected graphs with $O((kn)^2)$ nodes. We improve this reduction, by showing how to reduce from $k$-Clique to $k$-DSP in DAGs and undirected graphs with $O((kn)^2)$ nodes. A variant of $k$-DSP is the $k$-Disjoint Paths ($k$-DP) problem, where the solution paths no longer need to be shortest paths. Previous work reduced from $k$-Clique to $p$-DP in DAGs with $O(kn)$ nodes, for $p= k + k(k-1)/2$. We improve this by showing a reduction from $k$-Clique to $p$-DP, for $p=k + \lfloor k^2/4\rfloor$. Under the $k$-Clique Hypothesis from fine-grained complexity, our results establish better conditional lower bounds for $k$-DSP for all $k\ge 4$, and better conditional lower bounds for $p$-DP for all $p\le 4031$.

翻译：在$k$条不相交最短路径问题（$k$-DSP）中，给定一个包含$n$个节点和$m$条边的加权图$G$，并指定源顶点$s_1, \dots, s_k$和目标顶点$t_1, \dots, t_k$，需要判断$G$中是否存在顶点不相交的$(s_i,t_i)$-最短路径。对于任意常数$k$，已知$k$-DSP可在无向图和有向无环图（DAG）上通过多项式时间求解。然而，$k$-DSP的精确时间复杂度仍属未知，现有最快算法与最佳条件性下界之间存在巨大差距。本文针对$k$-DSP的重要情形提出了更快的算法，并为$k$-DSP及其变体给出了更优的条件性下界。先前的工作在加权无向图上以$O(n^7)$时间求解2-DSP，在加权DAG上以$O(mn)$时间求解。作为本文的主要结果，我们提出了在加权无向图和DAG上求解2-DSP的线性时间算法。然而，这些算法基于代数方法，因此仅能解决2-DSP的检测问题而非搜索问题。在下界方面，先前研究表明可将$k$-Clique归约至DAG和无向图中具有$O((kn)^2)$个节点的$2k$-DSP。我们改进了这一归约，展示了如何将$k$-Clique归约至DAG和无向图中具有$O((kn)^2)$个节点的$k$-DSP。$k$-DSP的一个变体是$k$条不相交路径问题（$k$-DP），其中解路径无需为最短路径。先前工作将$k$-Clique归约至DAG中具有$O(kn)$个节点的$p$-DP，其中$p= k + k(k-1)/2$。我们将其改进为将$k$-Clique归约至$p$-DP，其中$p=k + \lfloor k^2/4\rfloor$。基于细粒度复杂度中的$k$-Clique假设，我们的结果为所有$k\ge 4$的$k$-DSP建立了更优的条件性下界，并为所有$p\le 4031$的$p$-DP建立了更优的条件性下界。