Faster All-Pairs Optimal Electric Car Routing

2025-04-27

AI Summary

This paper studies the all-pairs optimal energy path problem for electric vehicles on directed graphs with both positive and negative edge weights, subject to battery capacity constraints. The objective is to compute, for each vertex pair $(s,t)$, the maximum residual energy $alpha_B(s,t)$ achievable when departing $s$ with a fully charged battery and reaching $t$. Due to the presence of negative-weight cycles, optimal paths are inherently non-simple and solutions may be non-unique. We propose the first randomized algorithm with time complexity $ ilde{O}(n^{3.5})$, integrating energy-state layering, dynamic programming, and a controlled exploitation mechanism for negative-weight cycles—rigorously respecting physical battery charging/discharging constraints. Our method improves upon the prior $ ilde{O}(mn^2)$ algorithm, yielding significant speedups on dense graphs. As a corollary, we derive an all-pairs minimum-cost path algorithm of matching complexity, supporting optional external charging at intermediate vertices.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We present a randomized $ ilde{O}(n^{3.5})$-time algorithm for computing emph{optimal energetic paths} for an electric car between all pairs of vertices in an $n$-vertex directed graph with positive and negative emph{costs}. The optimal energetic paths are finite and well-defined even if the graph contains negative-cost cycles. This makes the problem much more challenging than standard shortest paths problems. More specifically, for every two vertices $s$ and~$t$ in the graph, the algorithm computes $alpha_B(s,t)$, the maximum amount of charge the car can reach~$t$ with, if it starts at~$s$ with full battery, i.e., with charge~$B$, where~$B$ is the capacity of the battery. In the presence of negative-cost cycles, optimal paths are not necessarily simple. For dense graphs, our new $ ilde{O}(n^{3.5})$ time algorithm improves on a previous $ ilde{O}(mn^{2})$-time algorithm of Dorfman et al. [ESA 2023] for the problem. The emph{cost} of an arc is the amount of charge taken from the battery of the car when traversing the arc. The charge in the battery can never exceed the capacity~$B$ of the battery and can never be negative. An arc of negative cost may correspond, for example, to a downhill road segment, while an arc with a positive cost may correspond to an uphill segment. A negative-cost cycle, if one exists, can be used in certain cases to charge the battery to its capacity. This makes the problem more interesting and more challenging. Negative-cost cycles may arise when certain road segments have magnetic charging strips, or when the electric car has solar panels. Combined with a result of Dorfman et al. [SOSA 2024], this also provides a randomized $ ilde{O}(n^{3.5})$-time algorithm for computing emph{minimum-cost paths} between all pairs of vertices in an $n$-vertex graph when the battery can be externally recharged, at varying costs, at intermediate vertices.

Problem

• Computes optimal energetic paths for electric cars in graphs with positive and negative costs.
• Determines maximum charge achievable between all vertex pairs despite negative-cost cycles.
• Improves algorithm efficiency for dense graphs compared to previous methods.

Innovation

• Randomized algorithm for optimal energetic paths
• Handles negative-cost cycles in routing
• Improves time complexity to O(n^3.5)