Which $L_p$ norm is the fairest? Approximations for fair facility location across all "$p$"

Swati Gupta, Jai Moondra, Mohit Singh

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 (i.e., total distance traveled by clients in such a group). This is important when planning the location of socially relevant facilities such as emergency rooms. In this work, we consider a \emph{fair} version of the problem by minimizing the Minkowski $p$-norm of the total distance traveled by clients across different socioeconomic groups and the cost of opening facilities, to penalize high access costs to open facilities across $r$ groups of clients. This generalizes classic facility location ($p =1$) and the minimization of the maximum total distance traveled by clients in any group ($p = \infty$). However, it is often unclear how to select a specific "$p$" to model the cost of unfairness. To get around this, we show the existence of a small portfolio of solutions where for any Minkowski $p$-norm, at least one of the solutions is a constant-factor approximation with respect to any $p$-norm. Moreover, we give efficient algorithms to find such a portfolio of solutions. We also introduce the notion of refinement across the solutions in the portfolio. This property ensures that once a facility is closed in one of the solutions, all clients assigned to it are reassigned to a single facility and not split across open facilities. We give $\text{poly}(r^{1/\sqrt{\log r}})$-approximation for refinement in general metrics and $O(\log r)$-approximation for the line metric, where $r$ is the number of (disjoint) client groups. The techniques introduced in the work are quite general, and we show an additional application to a hierarchical facility location problem.

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