Socially Fair and Hierarchical Facility Location Problems

The classic facility location problem seeks to open a set of facilities to minimize the cost of opening the chosen facilities and the total cost of connecting all the clients to their nearby open facilities. Such an objective may induce an unequal cost over certain socioeconomic groups of clients, e.g., the average distance traveled by clients who do not have health insurance. To reduce the disproportionate impact of opening new facilities such as emergency rooms, we consider minimizing the Minkowski $p$-norm of the total distance traveled by each client group and the cost of opening facilities. We show that there is a small portfolio of solutions where for any norm, at least one of the solutions is a constant-factor approximation with respect to any $p$-norm, thereby alleviating the need for deciding on a particular value of $p$ to define what might be "fair". We also give a lower bound on the cardinality of such portfolios. We further introduce the notion of weak and strong refinements for the facility location problem, where the former requires that the set of facilities open for a lower $p$-norm is a superset of those open for higher $p$-norms, and the latter further imposes a partition refinement over the assignment of clients to open facilities in different norms. We give an $O(1)$-approximation for weak refinements, $\text{poly}(r^{1/\sqrt{\log r}})$-approximation for strong refinement in general metrics and $O(\log r)$-approximation for the tree metric, where $r$ is the number of (disjoint) client groups. We show that our techniques generalize to hierarchical versions of the facility location problem, which may be of independent interest.