Primal-Dual Schema and Lagrangian Relaxation for the k-Location-Routing Problem

The location-routing problem arises in the context of providing integrated support for logistics in a number of transportation settings, where given a set of requests and potential depot locations, one must simultaneously decide where to locate depots as well as how to route tours so that all requests are connected to an open depot. This problem can be formulated either with specific costs incurred for choosing to open each depot, or with an upper bound

k

on the number of open depots, which we call the

k-location-routing problem

.

We develop a primal-dual schema and use Lagrangian relaxation to provide a 2-approximation algorithm for the

k

-location-routing problem; no constant performance guarantee was known previously for this problem. This strengthens previous work of Goemans & Williamson who gave a 2-approximation algorithm for the variant in which there are opening costs, but no limit on the number of depots. We give a new primal-dual algorithm and a strengthened analysis that proves a so-called Lagrangian-preserving performance guarantee. In contrast to the results of Jain & Vazirani for the uncapacitated facility location and

k

-median problems, our results have the surprising property that our performance guarantee for the

k

-location-routing problem matches the guarantee for the version in which there are depot opening costs; furthermore, this relies on a simple structural property of the algorithm that allows us to identify the critical Lagrangian value for the opening cost with a single execution of the primal-dual algorithm, rather than invoking a bisection search.