Faster All-Pairs Optimal Electric Car Routing

We present a randomized $\tilde{O}(n^{3.5})$-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 $\tilde{O}(n^{3.5})$ time algorithm improves on a previous $\tilde{O}(mn^{2})$-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 $\tilde{O}(n^{3.5})$-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.