最大成本策略性设施选址：随机化的局限
Maximum-Cost Strategic Facility Location: The Limits of Randomization
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中文总结 AI 辅助
该研究针对欧氏空间策略性设施选址问题，证明了d≥2时期望策略防护机制的近似比下界，表明随机化无法在所有维度和智能体规模下实现对确定性机制近似比的普适常数级改进。
中文摘要 AI 辅助
我们研究欧氏空间$\n\mathbb R^d$\n中的策略性设施选址问题：机制基于$n$个智能体上报的位置选择单个设施，目标是最小化任意智能体到设施的最大距离。确定性策略防护机制的最优近似比为2，随机化能否在该因子上实现普适常数级改进一直是核心开放问题。我们证明，当$d \geq 2$时，所有期望意义下策略防护机制的近似比至少为\n\\[\n α(\mathcal M) \ge 2 - e^{-\Theta(\sqrt d)} - O\left(n^{-2/(d-1)}\right).\n\\]\n因此，对于任意普适常数$\varepsilon>0$，不存在期望意义下策略防护的机制能对所有$n$和$d$一致保证$(2-\varepsilon)$近似。
英文摘要
We consider strategic facility location in Euclidean space $\mathbb R^d$, where a mechanism selects a single facility based on the reported locations of $n$ agents and seeks to minimize the maximum distance from any agent to the facility. The optimal approximation ratio of deterministic strategyproof mechanisms is $2$. Whether randomization can yield a universal constant improvement over this factor has remained a major open question. We show that for $d \geq 2$, every strategyproof-in-expectation mechanism has approximation ratio at least \[ α(\mathcal M) \ge 2 - e^{-Θ(\sqrt d)} - O\left(n^{-2/(d-1)}\right). \] Therefore, no strategyproof-in-expectation mechanism can guarantee a $(2-\varepsilon)$-approximation uniformly over all $n$ and $d$, for any universal constant $\varepsilon>0$.
