FLIGHT: Facility Location Integrating Generalized, Holistic Theory of Welfare

TL;DR

This work introduces FLIGHT, a unified framework for the Facility Location Problem that accommodates a broad class of welfare notions via an α-welfare function. By defining $W_\alpha(y,\mathbf{x})=\sum_i\alpha(y-x_i)$ and $P_\alpha(\mathbf{x})=\arg\max_y W_\alpha(y,\mathbf{x})$, the approach subsumes utilitarian, egalitarian, and Nash welfare as special cases and extends to $p$-mean welfare through appropriate choices of $\alpha$. The authors establish key structural properties—concavity in $y$, location invariance, and monotonic shift behavior—and derive approximation bounds that remain favorable as the agent count grows, with a constant bound in the large-$n$ limit. The probabilistic extension shows that the expected welfare $\mathbb{W}_\alpha^\mathcal{P}$ serves as the optimal estimator and that empirical welfare converges to this expectation, enabling robust, data-driven decision-making in large-scale settings. Overall, FLIGHT provides a flexible, tractable foundation for integrating learned or diverse welfare notions into facility location, with clear avenues for future work on sufficiency, strategyproofness, and tighter bounds.

Abstract

The Facility Location Problem (FLP) is a well-studied optimization problem with applications in many real-world scenarios. Past literature has explored the solutions from different perspectives to tackle FLPs. These include investigating FLPs under objective functions such as utilitarian, egalitarian, Nash welfare, etc. Also, there is no treatment for asymmetric welfare functions around the facility. We propose a unified framework, FLIGHT, to accommodate a broad class of welfare notions. The framework undergoes rigorous theoretical analysis, and we prove some structural properties of the solution to FLP. Additionally, we provide approximation bounds, which provide insight into an interesting fact: as the number of agents arbitrarily increases, the choice of welfare notion is irrelevant. Furthermore, the paper also includes results around concentration bounds under certain distributional assumptions over the preferred locations of agents.

Figures (1)

• Figure 1: Illustration of Theorem 8: Bounding the Deviation of the Welfare Peak from the Median. The longest arrow, representing the maximum deviation from the symmetric agent profile, provides the required upper bound.

Theorems & Definitions (25)

• Definition 1: Utilitarian Welfare
• Definition 2: Egalitarian Welfare
• Definition 3: Nash Welfare
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 1
• Lemma 2