AI 中文总结
该研究在Ptolemaic度量空间中提出随机策略proof双设施选址机制，将最优近似比从4降至约3.667，同时将最优下界从1.045提升至约1.207，突破了此前的4近似比障碍。
AI 中文摘要
我们研究度量空间中无转移支付的策略proof机制设计下的双设施选址问题：机制基于智能体报告的位置选择两个设施点，每个智能体承担其到较近设施的距离，目标是最小化期望社会成本；若没有智能体能通过谎报位置获益，则该机制是策略proof的。此前，随机策略proof机制达到的最优近似比为4，由Lu、Sun、Wang和Zhu（EC 2010）提出的Proportional机制实现，最优下界为1.045，由Lu、Wang和Zhou（WINE 2009）给出，自那时起，即使在直线ℝ上，这两个界也未被突破。我们改进了这两个界：主要结果是在所有Ptolemaic度量空间（包含所有欧氏空间的丰富类）上，提出一种随机策略proof机制，其近似比为11/3≈3.667；该机制在Proportional机制与新机制Global Pair之间随机选择，Global Pair以智能体间距离成正比的概率抽取无序智能体对，并在其报告位置开设设施；尽管Global Pair和Proportional各自的近似比均为4，但二者在互补实例上达到最坏情况近似比，通过在二者间随机选择平衡这些互补弱点，从而突破4近似比障碍；下界方面，我们构造了新的双配置实例，得出下界为(1+√2)/2≈1.207，优于此前的1.045。
英文摘要
We study strategyproof mechanism design without transfers for the two-facility location problem in metric spaces. A mechanism selects two facility locations based on agents' reported locations; each agent incurs her distance to the nearer facility, and the objective is to minimize the expected social cost. A mechanism is strategyproof if no agent ever benefits from misreporting her location. The best approximation ratio achieved by a randomized strategyproof mechanism has been $4$, attained by the Proportional mechanism of Lu, Sun, Wang, and Zhu (EC 2010), and the best lower bound has been $1.045$, due to Lu, Wang, and Zhou (WINE 2009). Neither bound has moved since then, even on the line $\mathbb{R}$. We improve both bounds. Our main result is a randomized strategyproof mechanism with approximation ratio $11/3 \approx 3.667$ on every Ptolemaic metric space, a rich class containing all Euclidean spaces. The mechanism randomizes between the Proportional mechanism and a new mechanism that we call Global Pair. Global Pair draws an unordered pair of agents with probability proportional to their distance and opens facilities at their reported locations. Although Global Pair and Proportional each have approximation ratio $4$, the two mechanisms attain their worst-case approximation ratios on complementary instances. Randomizing between them balances these complementary weaknesses and breaks the $4$-approximation barrier. On the lower-bound side, we construct a new two-profile instance that yields a lower bound of $(1+\sqrt{2})/2 \approx 1.207$, improving upon the previous lower bound of $1.045$.
Comments23 pages, 1 figure
