Linear-Programming based Approximation Algorithms for Multi-Vehicle Minimum Latency Problems

We consider various {\em multi-vehicle versions of the minimum latency problem}. There is a fleet of $k$ vehicles located at one or more depot nodes, and we seek a collection of routes for these vehicles that visit all nodes so as to minimize the total latency incurred, which is the sum of the client waiting times. We obtain an $8.497$-approximation for the version where vehicles may be located at multiple depots and a $7.183$-approximation for the version where all vehicles are located at the same depot, both of which are the first improvements on this problem in a decade. Perhaps more significantly, our algorithms exploit various LP-relaxations for minimum-latency problems. We show how to effectively leverage two classes of LPs---{\em configuration LPs} and {\em bidirected LP-relaxations}---that are often believed to be quite powerful but have only sporadically been effectively leveraged for network-design and vehicle-routing problems. This gives the first concrete evidence of the effectiveness of LP-relaxations for this class of problems. The $8.497$-approximation the multiple-depot version is obtained by rounding a near-optimal solution to an underlying configuration LP for the problem. The $7.183$-approximation can be obtained both via rounding a bidirected LP for the single-depot problem or via more combinatorial means. The latter approach uses a bidirected LP to obtain the following key result that is of independent interest: for any $k$, we can efficiently compute a rooted tree that is at least as good, with respect to the prize-collecting objective (i.e., edge cost + number of uncovered nodes) as the best collection of $k$ rooted paths. Our algorithms are versatile and extend easily to handle various extensions involving: (i) weighted sum of latencies, (ii) constraints specifying which depots may serve which nodes, (iii) node service times.