# Capacitated Vehicle Routing with Non-uniform Speeds

The *capacitated vehicle routing problem* (CVRP) [21] involves distributing (identical) items from a depot to a set of demand locations in the shortest possible time, using a single capacitated vehicle. We study a generalization of this problem to the setting of multiple vehicles having non-uniform speeds (that we call *Heterogenous CVRP*), and present a constant-factor approximation algorithm.

The technical heart of our result lies in achieving a constant approximation to the following TSP variant (called *Heterogenous TSP*). Given a metric denoting distances between vertices, a depot *r* containing *k* vehicles having speeds {*λ* _{ i }}_{ i = 1} ^{ k }, the goal is to find a tour for each vehicle (starting and ending at *r*), so that every vertex is covered in some tour and the maximum completion time is minimized. This problem is precisely Heterogenous CVRP when vehicles are uncapacitated.

The presence of non-uniform speeds introduces difficulties for employing standard tour-splitting techniques. In order to get a better understanding of this technique in our context, we appeal to ideas from the 2-approximation for minimum makespan scheduling in unrelated parallel machines of Lenstra et al. [19]. This motivates the introduction of a new approximate MST construction called *Level-Prim*, which is related to *Light Approximate Shortest-path Trees* [18]. The last component of our algorithm involves partitioning the Level-Prim tree and matching the resulting parts to vehicles. This decomposition is more subtle than usual since now we need to enforce correlation between the lengths of the parts and their distances to the depot