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

TL;DR

This work tackles fairness in facility location by balancing access costs across multiple groups via $L_p$ norms and introduces the concept of portfolios—compact sets of solutions that approximate every objective in a norm class. It develops both existence and algorithmic results showing logarithmic-sized portfolios and polynomial-time constructions, and extends the framework to rolling-budget scenarios through refinements implemented by DiscountedLookahead and related methods. The paper provides theoretical upper and lower bounds on portfolio sizes, and demonstrates practical efficacy with two US county experiments and a planning tool for healthcare access. The combination of portfolios and refinements offers policymakers a pragmatic, theoretically grounded way to navigate competing fairness notions while accommodating progressive investments. The methods generalize beyond static fairness to dynamic, budget-friendly planning, with strong implications for equitable facility placement and healthcare access.

Abstract

Fair facility location problems try to balance access costs to open facilities borne by different groups of people by minimizing the $L_p$ norm of these group distances. However, there is no clear choice of "$p$" in the current literature. We present a novel approach to address the challenge of choosing the right notion of fairness. We introduce the concept of portfolios, a set of solutions that contains an approximately optimal solution for each objective in a given class of objectives, such as $L_p$ norms. This concept opens up new possibilities for getting around the "right" notion of fairness for many problems. For $r$ client groups, we demonstrate portfolios of size $Θ(\log r)$ for the facility location and $k$-clustering problems, with an $O(1)$-approximate solution for each $L_p$ norm. Further, motivated by the Justice40 Initiative that provides rolling budget investments, we impose a refinement-like structure on the portfolio. We develop novel approximation algorithms for these structured portfolios and show experimental evidence of their performance in two US counties. We also present a planning tool that provides potential ways to expand access to US healthcare facilities, which might be of independent interest to policymakers.

Figures (16)

• Figure 1: A screenshot from our online https://usa-medical-deserts.streamlit.app/ indicating potential medical deserts in Mississippi, USA. The tool can be found at https://usa-medical-deserts.streamlit.app/
• Figure 2: An illustrative example for $k$-clustering with three client groups $X_1, X_2, X_3$ (in blue, green, and purple respectively) that partition client set $X$. We seek to open one facility anywhere in $X$. The optimal solution for classical objective $\sum_{j \in X} d(j, f)$ opens facility $f$ near the center of the blue group. If we minimize the $L_p$ norm of vector $\left(\frac{1}{|X_s|} \sum_{j \in X_s} d(j, f)\right)_{s = 1, 2, 3}$ of average group distances, then $f$ moves closer to the center of all groups as $p$ increases from $1$ to $\infty$. The table below shows average group distances for optimal solutions to different objectives:
• Figure 3: An illustration for refinements for $k$-clustering. Consider budgets $k_1 = 3, k_2 = 7$. Each circle is a client and each cross is an open facility. (left) A solution with $k_1 = 3$ facilities with colors representing assignment to facilities. (center) A solution with $k_2 = 7$ open facilities, obtained by opening $4$ more facilities. However, this does not form a refinement for $k$-clustering since the set of clients colored orange, pink, and yellow intersect the dotted boundaries. (right) A different assignment for the same open facilities that forms a refinement.
• Figure 4: An example to show that the greedy algorithm is $\Omega(2^l)$-approximate for the reassignment problem with $l$ facility sets. Client locations are drawn as circles and facility locations are drawn as a cross. A copy of the line is drawn for each $s \in [l]$ for clarity. The greedy algorithm assigns $\Pi_l(0) = 1$ since the closest facility in $F_l$ to the client at $x = 0$ is at $x = 1$. The closest facility to $x = 1$ in $F_{l - 1}$ is at $x = 3$, so the algorithm assigns $\Pi_{l - 1}(0) = 3$, and so on until $\Pi_1(0) = 2^{l} - 1$. However, the closest facility to $x = 0$ in $F_1$ is at $-(1 + \epsilon)$, so that $\frac{d(0, \Pi_1(0))}{\min_{f \in F_1} d(0, f)} \simeq 2^l - 1$.
• Figure 5: An example to illustrate intervals $N_\alpha(f, k)$, with $l = 2$ levels and an additional auxiliary level $F_0 = \{f_0\}$; $\alpha = 2l = 4$. Facilities $f_0, f_5, f_{15}, f_{25} \in F_2$ are at $x = 0, 5, 15, 25$ respectively with $N_\alpha(f_0, 0) = (-\infty, 1]$, $N_\alpha(f_5, 0) = [4, 7]$, $N_\alpha(f_{15}, 0) = [13, 17]$, and $N_\alpha(f_{25}, 0) = [23, \infty)$. For facilities $F_1$, $N_\alpha(f_0, 1) = (-\infty, 5]$ and $N_\alpha(f_{25}, 1) = [20, \infty)$.

Theorems & Definitions (49)

• Definition 1: Fair Facility Location (FFL) and Fair $k$-Clustering (FC)
• Definition 2: Portfolios for Fair Facility Location and Fair $k$-Clustering
• Definition 3: Monotonically Interpolating Norm Class
• Definition 4: Refinements
• Definition 5: $\alpha$-Reassignments
• Theorem 1
• Theorem 2
• proof
• Theorem 3
• proof : Proof for fair facility location.