Minimizing Latency of Capacitated k -Tours

We study variants of the capacitated vehicle routing problem. In the multiple depot capacitated k - travelling repairmen problem (MD-C k TRP), we have a collection of clients to be served by one vehicle in a fleet of k identical vehicles based at given depots. Each client has a given demand that must be satisfied, and each vehicle can carry a total of at most Q demand before it must resupply at its original depot. We wish to route the vehicles in a way that obeys the constraints while minimizing the average time (latency) required to serve a client. This generalizes the Multi-depot k -Travelling Repairman Problem (MD- k TRP) (Chaudhuri et al. in 44th IEEE-FOCS, pp 36–45, 2003 ; Post and Swamy in 26th ACM-SIAM SODA, pp 512–531, 2015 ) to the capacitated vehicle setting, and while it has been previously studied (Lysgaard and Wholk in Eur J Oper Res 236(3):800–810, 2014 ), no approximation algorithm with a proven ratio is known. We give a 42.49-approximation to this general problem, and refine this constant to 25.49 when clients have unit demands. As far as we are aware, these are the first constant-factor approximations for capacitated vehicle routing problems with a latency objective. We achieve these results by developing a framework allowing us to solve a wider range of latency problems, and crafting various orienteering-style oracles for use in this framework. We also show a simple LP rounding algorithm has a better approximation ratio for the maximum coverage problem with groups (MCG), first studied by Chekuri and Kumar (Approximation, randomization, and combinatorial optimization, algorithms and techniques, pp 72–83, 2004 ), and use it as a subroutine in our framework. Our approximation ratio for MD-C k TRP when restricted to uncapacitated setting matches the best known bound for it (Post and Swamy in 26th ACM-SIAM SODA, pp 512–531, 2015 ). With our framework, any improvements to our oracles or our MCG approximation will result in improved approximations to the corresponding k -TRP problem.