Vehicle Routing with Subtours

When delivering items to a set of destinations, one can save time and cost by passing a subset to a sub-contractor at any point en route. We consider a model where a set of items are initially loaded in one vehicle and should be distributed before a given deadline ∆. In addition to travel time and time for deliveries, we assume that there is a fixed delay for handing over an item from one vehicle to another. We will show that it is easy to decide whether an instance is feasible, i.e., whether it is possible to deliver all items before the deadline ∆. We then consider computing a feasible tour of minimum cost, where we incur a cost per unit distance traveled by the vehicles, and a setup cost for every used vehicle. Our problem arises in practical applications and generalizes classical problems such as shallow-light trees and the bounded-latency problem. Our main result is a polynomial-time algorithm that, for any given > 0 and any feasible instance, computes a solution that delivers all items before time (1 + )∆ and has cost O(1 + 1 )OPT, where OPT is the minimum cost of any feasible solution. Known algorithms for special cases begin with a cheap solution and decompose it where the deadline is violated. This alone is insufficient for our problem. Instead, we also need a fast solution to start with, and a key feature of our algorithm is a careful combination of cheap and fast solutions. We show that our result is best possible in the sense that any improvement would lead to progress on 25-year-old questions on shallow-light trees.