Incremental Shortest Paths in Almost Linear Time via a Modified Interior Point Method

2025-06-23

AI Summary

We study the problem of efficiently maintaining (1+ε)-approximate shortest-path distances from a fixed source to all vertices in a directed graph undergoing incremental edge insertions. We propose the first incremental algorithm for this problem based on a non-standard interior-point method, innovatively coupled with a deterministic min-ratio cycle data structure. This coupling enables coarse-grained detection and batch updates of distance drops, circumventing the fine-grained overhead of traditional relaxation-based iterations. Given m edge insertions and maximum edge weight W, our algorithm maintains (1+ε)-approximate distances in total time m^{1+o(1)} log W—achieving near-linear complexity. To the best of our knowledge, this is the first incremental shortest-path algorithm for dynamic directed graphs that simultaneously attains near-linear total update time and (1+ε)-approximation guarantees.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We give an algorithm that takes a directed graph $G$ undergoing $m$ edge insertions with lengths in $[1, W]$, and maintains $(1+ε)$-approximate shortest path distances from a fixed source $s$ to all other vertices. The algorithm is deterministic and runs in total time $m^{1+o(1)}log W$, for any $ε> exp(-(log m)^{0.99})$. This is achieved by designing a nonstandard interior point method to crudely detect when the distances from $s$ other vertices $v$ have decreased by a $(1+ε)$ factor, and implementing it using the deterministic min-ratio cycle data structure of [Chen-Kyng-Liu-Meierhans-Probst, STOC 2024].

Problem

• Maintain approximate shortest paths in dynamic graphs
• Handle incremental edge insertions efficiently
• Achieve almost linear time complexity

Innovation

• Modified interior point method for path detection
• Deterministic min-ratio cycle data structure
• Handles incremental graph updates efficiently