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

TL;DR

This work introduces a portfolio-based approach to multiobjective optimization where the objective class $\mathbf{C}$ may be infinite. It defines $\alpha$-approximate portfolios, showing how to construct compact menus that approximate any objective in $\mathbf{C}$, for both conic combinations $\mathbf{C}_1$ and monotonically interpolating norms $\mathbf{C}_2$ (including $L_p$ norms). A key application is Fair Subsidized Facility Location (FSFL), for which the authors develop a bicriteria approximation and prove that portfolios of size $O\left(\log \left(\tfrac{t}{\min(1, \delta)}\right)\right)$ exist, each solution being $(2\delta + \theta)$-subsidized and offering provable approximation guarantees across all $L_p$ norms. The framework is complemented by experiments on Mississippi data showing substantial reductions in medical deserts with modest subsidies and small facility additions, illustrating the practical impact for policymakers. Overall, the paper provides a general, principled method to balance fairness and efficiency across diverse objectives with a manageable set of representative solutions.

Abstract

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.

Figures (6)

• Figure 1: A screenshot from our online tool depicting medical deserts in Mississippi, USA. Note that the majority Black blockgroups (35.42% of all blockgroups) make up 61.03% of all medical deserts. The tool can be accessed at https://usa-medical-deserts.streamlit.app/.
• Figure 2: An illustrative example for $k$-Clustering with three client groups $C_1, C_2, C_3$ (in blue, green, and purple, respectively) that partition client set $C$. We seek to open one facility $f$ anywhere in $C$. The optimal solution for the classical objective $\sum_{j \in C} \texttt{dist}_{j, f}$ opens facility $f$ near the center of the blue group. If we minimize the $L_p$ norm of vector $\left(\frac{1}{|C_s|} \sum_{j \in C_s} \texttt{dist}_{j, f}\right)_{s = 1, 2, 3}$ of average group distances, then $f$ moves closer to the center of all groups as $p$ increases from $1$ to $\infty$. The adjacent table shows average group distances for optimal solutions to different objectives.
• Figure 3: An illustration for the mesh for $\mathcal{H}_i$ used in the proof of Theorem \ref{['thm: portfolios-convex-combinations']}.
• Figure 4: An example to illustrate Algorithm \ref{['alg: core-clients']}. (left) The graph $G = (C, E)$ with $\Delta$ values for vertices. Initially, the algorithm chooses core client $a = {\arg\min}_{j \in C} \Delta_j$ and forms an arboresence rooted at $a$ on clients $\{a, b, c, d, f\}$. Then, the algorithm chooses core client $g$ and forms the arborescence on clients $\{g, h\}$, and finally, the algorithm chooses the core client $i$. (right) The core clients $C^*$ (shaded) and $\texttt{paths}$, represented through arborescences rooted at the core clients.
• Figure 5: Portfolios of suggested locations for $k = 10$ new pharmacies using the FSFL model in the state of Mississippi, USA, in addition to existing CVS, Walmart, and Walgreens pharmacies. Each column shows the portfolio for a given subsidy parameter $\delta \in \{0.005, 0.01, 0.02, 0.05\}$ for approximation factor $\alpha = 1.15$. While different solutions recommend opening facilities in different locations, all solutions significantly reduce the number of medical deserts.

Theorems & Definitions (25)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Corollary 1
• Theorem 5
• Theorem 5
• Claim 1
• Theorem 5
• Theorem 5