随机策略proof的设施选址：两个及以上设施

中文摘要 AI 辅助

我们针对多设施选址问题设计并分析随机策略proof机制，采用功利主义社会成本目标，即智能体到最近设施的距离之和。对于两个设施，Pairwise-Distance机制以报告位置对的距离为概率权重采样设施位置，该机制在包括欧氏空间和希尔伯特空间在内的托勒密空间上是策略proof的，近似比为4。由此得到的Hybrid-Distance机制是固定概率混合机制：以概率λ*=(5+4√3)/23选择经典Proportional机制[Lu等人，EC'10]，以概率1-λ*选择Pairwise-Distance机制，该机制在托勒密空间上是策略proof的，且近似比为(74+4√3)/23≈3.5186，打破了[Lu等人，EC'10]提出的长期存在的因子4基准。我们通过研究更多设施补充两设施结果：第一，对于n个智能体和k=n-1个设施，提出Inverse-Square机制，以报告的最近邻距离的倒数平方为概率权重省略一个报告，将设施设置在剩余所有报告处，该机制在任意度量空间上是策略proof的，近似比为Θ(√n)，优于[Escoffier等人，ADT'11]提出的先前最佳比值n/2；第二，对于直线上的k个设施，提出Gap-Product机制，将设施设置在k个报告处，以连续选中报告之间的间隙乘积为权重，当k=3时，该机制是策略proof的，近似比为6，将先前依赖n的保证[Fotakis和Tzamos，EC'13]替换为常数，而当k≥4时，该机制不具备策略proof性。

英文摘要

We design and analyze randomized strategyproof mechanisms for multi-facility location under the utilitarian social-cost objective, the sum of the agents' distances to their nearest facilities. For two facilities, the Pairwise-Distance mechanism locates facilities at a pair of reported locations sampled with probability proportional to their distance. It is strategyproof on Ptolemaic spaces, including Euclidean and Hilbert spaces as special cases, and has an approximation ratio of \(4\). The resulting Hybrid-Distance mechanism is a fixed-probability mixture: it selects the classical Proportional mechanism [Lu et al., EC'10] with probability \(λ^*=\frac{5+4\sqrt3}{23}\) and Pairwise-Distance with probability $1-λ^*$. It is strategyproof on Ptolemaic spaces and has a tight approximation ratio of \(\frac{74+4\sqrt3}{23}\approx3.5186\), breaking the long-standing factor-\(4\) benchmark of [Lu et al., EC'10]. We complement the two-facility results by studying more facilities. First, for \(n\) agents and \(k=n-1\) facilities, we introduce the Inverse-Square mechanism, which omits one report with probability proportional to the inverse square of its nearest-neighbor distance and locates facilities at all remaining reports. It is strategyproof on any metric space and has an approximation ratio of \(Θ(\sqrt{n})\), improving the previous best-known ratio of \(\frac{n}{2}\) [Escoffier et al., ADT'11]. Second, for $k$ facilities on the line, we introduce the Gap-Product mechanism, which locates facilities at \(k\) reports and weights each set by the product of the gaps between consecutive selected reports. When \(k=3\), it is strategyproof and has a \(6\)-approximation, replacing the previous \(n\)-dependent guarantee [Fotakis and Tzamos, EC'13] by a constant, whereas it is not strategyproof for any \(k\ge4\).

发表机构

• University of Nebraska–Lincoln（内布拉斯加大学林肯分校）
• Rensselaer Polytechnic Institute（伦斯勒理工学院）
• Beijing Normal University–Zhuhai（北京师范大学珠海校区）
• Beijing Normal–Hong Kong Baptist University（北京师范大学-香港浸会大学联合国际学院）

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