Min-max cover of a graph with a small number of parts

We consider a variety of vehicle routing problems. The input consists of an undirected graph and edge lengths. Customers located at the nodes have to be visited by a set of vehicles. Two important parameters are k the number of vehicles, and the longest distance traveled by a vehicle denoted by λ. Here, we consider k to be a given bound on the maximum number of vehicles, and thus the decision maker cannot increase its value. Therefore, the goal will be to minimize λ. We study different variations of this problem, where for instance instead of servicing the customers using paths, we can serve them using spanning trees or cycles. For all these variations, we present new approximation algorithms with FPT time (where k is the parameter) which improve the known approximation guarantees for these problems.