Approximation guarantees for Hub Covering Problems
Hub Covering Problems are a subclass of Hub Location Problems. The objective is to select a set of hubs that enable paths between given origin-destination delivery tasks, while minimizing the total setup cost of the hubs. Two constraints must be satisfied: each path must include one or two hubs, and depending on the problem variant, the total path length or the length of the longest edge must not exceed prescribed limits. Such problems arise in a variety of practical applications, including urban planning, cargo delivery systems, airline networks, telecommunication networks, and e-mobility. Although Hub Covering Problems are especially important from a practical point of view, no systematic study of their approximability exists. In particular, it remains open whether some variants can be reduced to others in polynomial time, and no approximation bound is known. We close this gap by establishing a hierarchy among these problems, demonstrating that certain variants are indeed special cases of others. For four variants, we give a polynomial-time approximation algorithm, and for the other eight variants, we show that no polynomial-time algorithm can decide whether a solution exists unless P=NP.