Locating Depots for Capacitated Vehicle Routing

Abstract

We study a location-routing problem in the context of capacitated vehicle routing. The input to k-LocVRP is a set of demand locations in a metric space and a fleet of k vehicles each of capacity Q. The objective is to locate k depots, one for each vehicle, and compute routes for the vehicles so that all demands are satisfied and the total cost is minimized. Our main result is a constant-factor approximation algorithm for k-LocVRP. To achieve this result, we reduce k-LocVRP to the following generalization of k median, which might be of independent interest. Given a metric (V,d), bound k and parameter ρ ∈ ℝ₊, the goal in the k median forest problem is to find S ⊆ V with |S| = k minimizing:

$$\sum_{u\in V} d(u,S) \quad + \quad \rho\cdot d\big(\,\mbox{MST}(V/S)\,\big),$$

where d(u,S) =  min _w ∈ S d(u,w) and $\mbox{MST}(V/S)$ is a minimum spanning tree in the graph obtained by contracting S to a single vertex. We give a (3 + ε)-approximation algorithm for k median forest, which leads to a (12 + ε)-approximation algorithm for k-LocVRP, for any constant ε > 0. The algorithm for k median forest is t-swap local search, and we prove that it has locality gap $3+\frac2t$; this generalizes the corresponding result for k median [3].

Finally we consider the k median forest problem when there is a different (unrelated) cost function c for the MST part, i.e. the objective is $\sum_{u\in V} d(u,S) \,+ \,c (\,\mbox{MST}(V/S)\,)$. We show that the locality gap for this problem is unbounded even under multi-swaps, which contrasts with the c = d case. Nevertheless, we obtain a constant-factor approximation algorithm, using an LP based approach along the lines of [12].