Provably Small Portfolios for Multiobjective Optimization with Application to Subsidized Facility Location

Many multiobjective real-world problems, such as facility location and bus routing, become more complex when optimizing the priorities of multiple stakeholders. These are often modeled using infinite classes of objectives, e.g., $L_p$ norms over group distances induced by feasible solutions in a fixed domain. Traditionally, the literature has considered explicitly balancing `equity' (or min-max) and `efficiency' (or min-sum) objectives to capture this trade-off. However, the structure of solutions obtained by such modeling choices can be very different. Taking a solution-centric approach, we introduce the concept of provably small set of solutions $P$, called a {\it portfolio}, such that for every objective function $h(\cdot)$ in the given class $\mathbf{C}$, there exists some solution in $P$ which is an $α$-approximation for $h(\cdot)$. Constructing such portfolios can help decision-makers understand the impact of balancing across multiple objectives. Given a finite set of base objectives $h_1, \ldots, h_N$, we give provable algorithms for constructing portfolios for (1) the class of conic combinations $\mathbf{C} = \{\sum_{j \in [N]}λ_j h_j: λ\ge 0\}$ and for (2) any class $\mathbf{C}$ of functions that interpolates monotonically between the min-sum efficiency objective (i.e., $h_1 + \ldots + h_N$) and the min-max equity objective (i.e., $\max_{j \in [N]} h_j$). Examples of the latter are $L_p$ norms and top-$\ell$ norms. As an application, we study the Fair Subsidized Facility Location (FSFL) problem, motivated by the crisis of medical deserts caused due to pharmacy closures. FSFL allows subsidizing facilities in underserved areas using revenue from profitable locations. We develop a novel bicriteria approximation algorithm and show a significant reduction of medical deserts across states in the U.S.