Detecting Disjoint Shortest Paths in Linear Time and More

TL;DR

The paper investigates the complexity of the $k$-Disjoint Shortest Paths problem ($k$-DSP) and delivers new algorithmic and complexity results. It presents randomized, algebraic, linear-time algorithms for the decision version of $2$-DSP in weighted undirected graphs and DAGs, using polynomial encodings evaluated over a field of characteristic two with Schwartz–Zippel type reasoning. It also extends the framework to edge-disjoint variants and provides improved conditional lower bounds by reducing $k$-Clique to $k$-DSP with tighter graph-size blowups, including a reduction achieving $p = k+ ig floor k^2/4ig floor$, and constructs a $k$-EDSP algorithm in DAGs with runtime $O(mn^{k-1})$. Together, these contributions advance understanding of the exact time complexity landscape for disjoint shortest paths, and connect algorithmic breakthroughs to fine-grained complexity hypotheses. The results imply both near-optimal linear-time detection in key graph classes and stronger barriers against dramatic improvements in the general directed setting, guiding future research on search variants and broader graph-structure restrictions.

Abstract

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 $p\le 4031$.

Figures (8)

• Figure 1: To enumerate the family of disjoint pairs of paths on the left (the dashed borders around the paths indicate that the paths do not intersect), it suffices to enumerate all pairs of paths and subtract out those pairs in the family which intersect at some point.
• Figure 2: To enumerate the family of intersecting pairs of paths on the left, we can perform casework on the earliest intersection point $v$ for the paths (the dashed border on the subpaths on the right indicates that the paths do not intersect before $v$).
• Figure 3: If we work modulo two, then we can enumerate pairs of paths which have common first intersection at node $v$ by enumerating pairs of paths which intersect at $v$ and have the property that the vertices appearing immediately before $v$ on each path are distinct.
• Figure 4: Given paths $P_1$ and $P_2$ which intersect at nodes $a = \alpha(P_1, P_2)$ and $b=\beta(P_1, P_2)$, such that $a$ appears before $b$ on both paths, if we swap the $a$ to $b$ subpaths of of $P_1$ and $P_2$ to produce new paths $Q_1$ and $Q_2$ respectively, then these pairs $f(P_1, P_2) = f(Q_1, Q_2)$ have the same monomials. Moreover, swapping the $a$ to $b$ subpaths of $Q_1$ and $Q_2$ recovers $P_1$ and $P_2$.
• Figure 5: For $k=2$, the paths $\langle s_1, v, t_1\rangle$ and $\langle s_2, v, t_2\rangle$ in $G$ map to a single path in $G'$, pictured on the right, with respect to the topological order $s_1 \prec s_2 \prec v \prec t_1 \prec t_2$. Since both coordinates of the node $(v,v)$ are equal, we can step from $(v,v)$ to $(t_1, t_2)$ with a single edge in $G'$.

Theorems & Definitions (97)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4
• Theorem 5
• Theorem 7
• Corollary 8
• Theorem 9
• Corollary 10
• Proposition 11: Shortest Path DAGs