Fine-Grained Optimality of Partially Dynamic Shortest Paths and More

TL;DR

The paper establishes a tight conditional lower bound showing that any partially dynamic exact stSP algorithm on $n$-node graphs requires $n^{4-o(1)}$ time under the Minimum-Weight 4-Clique hypothesis, thereby separating the complexity of exact from approximate dynamic shortest paths. It provides a comprehensive study of dynamic variants beyond edge-weighted SSSP, including node-weighted shortest paths, bottleneck paths, and earliest arrivals, delivering new subquartic (and near-linear) algorithms in the partially dynamic setting and demonstrating conditional lower bounds based on OMv3 and 4-clique hypotheses. The work also delivers a fully dynamic APBP algorithm with $ ilde{O}(n^2)$ amortized update time, and develops intricate gadget-based reductions to propagate hardness from classical problems to dynamic path problems. Overall, it advances understanding of what can be efficiently maintained under graph updates and clarifies why fast approximations exist while exact dynamic solutions remain intractable under widely believed hypotheses, with implications for related path problems in practice. The results leverage a blend of algebraic techniques, batch-processing schemes, and carefully engineered reductions to illuminate the boundaries of dynamic shortest-path computation.

Abstract

Single Source Shortest Paths ($\textrm{SSSP}$) is among the most well-studied problems in computer science. In the incremental (resp. decremental) setting, the goal is to maintain distances from a fixed source in a graph undergoing edge insertions (resp. deletions). A long line of research culminated in a near-optimal deterministic $(1 + \varepsilon)$-approximate data structure with $m^{1 + o(1)}$ total update time over all $m$ updates by Bernstein, Probst Gutenberg and Saranurak [FOCS 2021]. However, there has been remarkably little progress on the exact $\textrm{SSSP}$ problem beyond Even and Shiloach's algorithm [J. ACM 1981] for unweighted graphs. For weighted graphs, there are no exact algorithms beyond recomputing $\textrm{SSSP}$ from scratch in $\widetilde{O}(m^2)$ total update time, even for the simpler Single-Source Single-Target Shortest Path problem ($\textrm{stSP}$). Despite this lack of progress, known (conditional) lower bounds only rule out algorithms with amortized update time better than $m^{1/2 - o(1)}$ in dense graphs. In this paper, we give a tight (conditional) lower bound: any partially dynamic exact $\textrm{stSP}$ algorithm requires $m^{2 - o(1)}$ total update time for any sparsity $m$. We thus resolve the complexity of partially dynamic shortest paths, and separate the hardness of exact and approximate shortest paths, giving evidence as to why no non-trivial exact algorithms have been obtained while fast approximation algorithms are known. Moreover, we give tight bounds on the complexity of combinatorial algorithms for several path problems that have been studied in the static setting since early sixties: Node-weighted shortest paths (studied alongside edge-weighted shortest paths), bottleneck paths (early work dates back to 1960), and earliest arrivals (early work dates back to 1958).

Figures (4)

• Figure 1: Overall structure of reduction of dynamic path problems. For each $d \in D$, the $A, B, C$ and $\hat{A}$ gadgets encode neighbors of $d$ in $A, B$, and $C$. Edges between gadgets encode edges between $A, B$, and $C$ in the Minimum-Weight $4$-Clique instance.
• Figure 2: An example of encoding a Minimum-Weight $4$-Clique instance into an incremental stSP instance with parallel edges. Solid black edges encode edges $A \times B$ with weight $\mathsf{wt}(a, b)$. Edges within gadgets are inserted in decreasing order of weight. Let $n = 4$. Solid red edges join neighbors of $d_2$ with weight $2 W + \mathsf{wt}(b_j, d_2)$ between $(b_j^{(1)}, b_j^{(2)})$ in $B$ and weight $2 n W + k W + \mathsf{wt}(a_k, d_2)$ between $(a_k^{(1)}, a_k^{(2)})$ in $A$. Dashed purple edges join neighbors of $d_1$ with weight $W + \mathsf{wt}(b_j, d_1)$ between $(b_j^{(1)}, b_j^{(2)})$ in $B$ and weight $n W + k W + \mathsf{wt}(a_k, d_1)$ between $(a_k^{(1)}, a_k^{(2)})$ in $A$. Dotted blue edges join neighbors of $d_0$ with weight $\mathsf{wt}(b_j, d_0)$ between $(b_j^{(1)}, b_j^{(2)})$ in $B$ and weight $k W + \mathsf{wt}(a_k, d_0)$ between $(a_k^{(1)}, a_k^{(2)})$ in $A$. The $(s, t)$-distance is maintained after each insertion.
• Figure 3: Removing parallel edges from the multi-edge gadget. On the left, solid red edges join neighbors of $d_2$ with weight $2 W + \mathsf{wt}(b_j, d_2)$. Dashed purple edges join neighbors of $d_1$ with weight $W + \mathsf{wt}(b_j, d_1)$. Dotted blue edges join neighbors of $d_0$ with weight $\mathsf{wt}(b_j, d_0)$. On the right, we insert an additional middle layer of $n = 4$ nodes. If $b_j$ is a neighbor of $d_i$, two edges of weight $i W + \frac{\mathsf{wt}(b_j, d_i)}{2}$ join $b_{j}^{(1)}, b_{j}^{(3)}$ via $b_{j + i \mod n}^{(2)}$. Note that each $b_{j}^{(2)}$ participates in at most one path of weight $< (2 i + 1) W$, where $d_i$ is the current smallest vertex of $d$. In particular, all paths with weight in this range are node disjoint, and must enter and leave the gadget from the copies of the same vertex $b \in B$. Furthermore, the simple graph only has $n$ additional nodes.
• Figure 4: Comparison of node-weighted gadget with an edge-weighted gadget. The gadget on the left is the edge-weighted gadget from \ref{['thm:s-t-sp-lb']}. We now create a $4$-layer node-weighted graph where the weight of $b_{j}^{(i)}$ vertices in $B_1, B_2, B_4$ are $j \cdot W$ for $i \in \{1\}$ and $j \cdot W^{3}$ for $i \in \{2, 4\}$ and the weight of $b_{j}^{(3)}$ vertices are $(4 n - 2 j) \cdot W^{3}$ for some large constant $W$. While the edges are no longer weighted, we format the edges as before to illustrate corresponding paths. Note that each $b_{j}^{(3)}$ participates in at most one path of weight $< (4n - 2i + 1) \cdot W^{3}$ where $d_i$ is the current smallest vertex of $d$. As in \ref{['fig:multi-edge-to-simple']}, all paths with weight in this range are vertex disjoint, and must enter and leave the gadget from copies of the same vertex $b \in B$. Furthermore, the simple graph only has $4n$ additional vertices.

Theorems & Definitions (83)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Proposition 2.1
• proof : Proof of \ref{['prop:multi-s-t-sp-lb-min-weight']}
• proof
• Lemma 2.3
• proof