A Graph Matching Based Approach for the Multi-Depot Capacitated Vehicle Routing Problem
TL;DR
It is proved that for tours of up to two targets the matching formulation solves the MDCVRP exactly in polynomial time for any number of depots, and that both algorithms are constant-factor approximations, with a tight factor of two, in the structured regimes.
Abstract
The Multi-Depot Capacitated Vehicle Routing Problem (MDCVRP) asks for minimum-cost delivery tours from several capacitated depots to a set of customers. Like most vehicle-routing variants it is NP-hard, so practical solvers must trade solution quality against speed. We revisit this trade-off through the lens of graph matching. Adapting a matching-based construction first developed for the Traveling Tournament Problem, we present two algorithms, Cluster-First and Match-First, that reduce routing to a sequence of minimum-weight matchings. This is more than a heuristic. We prove that for tours of up to two targets the matching formulation solves the MDCVRP exactly in polynomial time for any number of depots, and that both algorithms are constant-factor approximations, with a tight factor of two, in the structured regimes. This matching optimum coincides with the exact combinatorial-auction optimum, so the auction serves as a strong quality baseline. On instances of 1000 customers and 20 depots our methods match or slightly beat that baseline in tour length while running two to three orders of magnitude faster, in tens of milliseconds against tens of seconds, a scale at which exact and auction-based solvers become impractical. Because Cluster-First routes each depot independently, the approach also re-routes cheaply when new customers arrive.