Randomized Strategic Facility Location with Predictions

TL;DR

A deeper understanding of the strategic facility location problem is provided by exploring the power of randomization as well as the impact of different types of predictions on the performance of truthful learning-augmented mechanisms.

Abstract

In the strategic facility location problem, a set of agents report their locations in a metric space and the goal is to use these reports to open a new facility, minimizing an aggregate distance measure from the agents to the facility. However, agents are strategic and may misreport their locations to influence the facility's placement in their favor. The aim is to design truthful mechanisms, ensuring agents cannot gain by misreporting. This problem was recently revisited through the learning-augmented framework, aiming to move beyond worst-case analysis and design truthful mechanisms that are augmented with (machine-learned) predictions. The focus of this prior work was on mechanisms that are deterministic and augmented with a prediction regarding the optimal facility location. In this paper, we provide a deeper understanding of this problem by exploring the power of randomization as well as the impact of different types of predictions on the performance of truthful learning-augmented mechanisms. We study both the single-dimensional and the Euclidean case and provide upper and lower bounds regarding the achievable approximation of the optimal egalitarian social cost.

Figures (3)

• Figure 1: The original mechanism $f(.)$ and the new OnlyM mechanism $f'(.)$ in the proof of Lemma $1$, where $p_\ell = \mathbb{P}[f(x_L,x_R)\in (x_L,M)]$ and $p_r = \mathbb{P}[f(x_L,x_R)\in (M,x_R)]$, $\pi_\ell= \mathbb{E}[f(x_L,x_R) ~|~ f(x_L,x_R)\in (x_L,M)]$ and $\pi_r= \mathbb{E}[f(x_L,x_R) ~|~ f(x_L,x_R)\in (M,x_R)]$, and $q_\ell$ and $q_r$ are such that $\pi_\ell=q_\ell x_L+ (1-q_\ell)M$ and $\pi_r = q_r x_R+ (1-q_r)M$.
• Figure 2: Instances $\mathbf{x} = \langle x_{1} = (0, 0), x_2 = \cdots = x_{n-1} = (1/2, 0), x_{n} = (1, 0) \rangle$, and $\mathbf{x}' = \langle x'_1 = (0, 0), x'_2 = \cdots = x'_{n-1} = (1/2, 0), x'_n = (2, 0) \rangle$ in the proof of Theorem $2$. It is assumed that $d(x_n, f(\mathbf{x})) \geq 1/2$ and $x^{\star} \in \arg\min_{y: d(x_n, y) \geq 1/2} C(y, \mathbf{x})$
• Figure 3: $\mathcal{G} = (x_{e_1} + x_{e_2} + x_{e_3})/3$ is the centroid of $x_{e_1}, \ldots, x_{e_3}$, which is the intersection of the three medians. $H$ is the orthocenter, which is the intersection of the three altitudes, and $O$ is the center of the circumcircle, which is the intersection of the three perpendicular bisectors. In any triangle, the circumcenter ($O$), the centroid ($\mathcal{G}$), and the orthocenter ($H$) are collinear, forming the Euler line. Moreover, $d(O, \mathcal{G}) = \frac{1}{2} d(\mathcal{G}, H)$.

Theorems & Definitions (27)

• Theorem 1
• Definition 1: $\textsc{OnlyM}$ mechanisms
• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• proof : Proof of Theorem \ref{['thm: Ran-lowerbound-line-predX']}