Learning-Augmented Strategyproof Facility Location in $\mathbb{R}^d$ with $\ell_p$ Distances

2026-09-11

AI Summary

研究了在d维空间中，通过添加预测位置虚拟代理的坐标中位数机制来最小化设施到各点的lp距离总和的方法，并分析了其在不同设置下的近似保证。

Abstract

We study learning-augmented mechanism design for locating a single facility in $\mathbb{R}^d$ to minimize the sum of the agents' $\ell_p$ distances to the facility. We analyze the coordinate-wise median with predictions (CMP) mechanism, which adds $cn$ virtual agents at a predicted optimal facility location to the $n$ reported locations and returns their coordinate-wise median. The parameter $c\in[0,1)$ represents confidence in the prediction; CMP is known to be strategyproof when both are fixed independently of the reports. We determine its approximation guarantees under correct predictions (consistency) and arbitrary predictions (robustness) in two settings. First, for $d=2$, we establish exact guarantees for every $p\in[1,+\infty]$. For $1<p<+\infty$, the consistency and robustness are respectively $\Bigl(1+\bigl(\frac{1-c}{1+c}\bigr)^{\frac{p}{p-1}}\Bigr)^{\frac{p-1}{p}}$ and $\Bigl(1+\bigl(\frac{1+c}{1-c}\bigr)^{\frac{p}{p-1}}\Bigr)^{\frac{p-1}{p}}$. We use the median condition in each coordinate to compare the mechanism's cost with the optimal cost, and establish tightness using instances with agents at only three distinct locations. Second, for $1<p<+\infty$, we obtain dimension-independent consistency and robustness upper bounds valid for every $d\ge1$. For each fixed $p$ and $c$, we construct families of instances whose ratios approach the respective upper bounds as $d\to\infty$, proving asymptotic tightness. We also establish exact guarantees for $p=1$ in every dimension and asymptotically tight bounds for $p=+\infty$. Our high-dimensional results recover the prediction-free bounds of Gravin and Jia (STOC 2025) when $c=0$ and their learning-augmented bounds in arbitrary-dimensional Euclidean spaces when $p=2$.

Problem

• facility location
• learning-augmented
• strategyproof
• ell_p distances
• coordinate-wise median

Innovation

• learning-augmented mechanism design
• facility location
• coordinate-wise median with predictions (CMP)
• approximation guarantees
• dimension-independent bounds