发表机构
University of Nebraska–Lincoln; Rensselaer Polytechnic Institute; Beijing Normal University–Zhuhai; Beijing Normal–Hong Kong Baptist University（内布拉斯加大学林肯分校; 伦斯勒理工学院; 北京师范大学珠海校区; 北京师范大学-香港浸会大学联合国际学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究在 $\mathbb{R}^d$ 中带 $\ell_p$ 距离的设施选址，提出带预测的坐标中位数机制，给出二维精确保证和高维渐近紧界，平衡一致性与鲁棒性。
AI 中文摘要
我们研究在 $\mathbb{R}^d$ 中定位单个设施以最小化代理到设施的 $\ell_p$ 距离之和的学习增强机制设计。我们分析了带预测的坐标中位数（CMP）机制，该机制在 $n$ 个报告位置处添加 $cn$ 个位于预测最优设施位置的虚拟代理，并返回它们的坐标中位数。参数 $c\in[0,1)$ 表示对预测的置信度；当两者独立于报告固定时，CMP 已知是策略proof的。我们在两种设置下确定了其在正确预测（一致性）和任意预测（鲁棒性）下的近似保证。首先，对于 $d=2$，我们为每个 $p\in[1,+\infty]$ 建立了精确保证。对于 $1<p<+\infty$，一致性和鲁棒性分别为 $\Bigl(1+\bigl(\frac{1-c}{1+c}\bigr)^{\frac{p}{p-1}}\Bigr)^{\frac{p-1}{p}}$ 和 $\Bigl(1+\bigl(\frac{1+c}{1-c}\bigr)^{\frac{p}{p-1}}\Bigr)^{\frac{p-1}{p}}$。我们使用每个坐标中的中位数条件来比较机制成本与最优成本，并利用代理仅位于三个不同位置的实例建立紧性。其次，对于 $1<p<+\infty$，我们获得了对每个 $d\ge1$ 都成立的维度无关的一致性和鲁棒性上界。对于每个固定的 $p$ 和 $c$，我们构造实例族，其比率在 $d\to\infty$ 时接近各自的上界，证明了渐近紧性。我们还为每个维度中的 $p=1$ 建立了精确保证，并为 $p=+\infty$ 建立了渐近紧界。我们的高维结果在 $c=0$ 时恢复了 Gravin 和 Jia (STOC 2025) 的无预测界，并在 $p=2$ 时恢复了任意维欧几里得空间中的学习增强界。
英文摘要
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 $p=2$.
