AI 中文总结
研究离散异构双设施选址的确定性防策略机制，针对线性图设计最优\(4/3\)近似机制填补比率差距，在线性图外为连通图设计\(2\)近似机制并证明\(K_{1,3}\)下界为\(3/2\)。
AI 中文摘要
我们研究离散异构双设施选址的确定性防策略机制。在模型中，\(n\)个代理占据连通图的不同节点，私下报告对两个设施的非空认可偏好，设施须置于不同节点。代理的成本是其到认可设施的总距离，目标是最小化社会成本。对于线性图，先前确定性防策略机制的最佳近似比率在\(4/3\)和\(17/4\)之间，我们通过设计最优\(4/3\)近似机制填补差距，该机制结合固定奇偶中位数规则（\(n\geq7\)时适用）及剩余较小情况的防策略局部覆盖。在线性图之外，我们为每个连通图设计确定性防策略\(2\)近似机制，并证明在图\(K_{1,3}\)上的下界为\(3/2\)。
英文摘要
We study deterministic strategyproof mechanisms for discrete heterogeneous two-facility location. In our model, $n$ agents occupy distinct nodes of a connected graph and privately report non-empty approval preferences over two facilities, which must be placed at distinct nodes. The cost of an agent is her total distance from the facilities she approves, and the objective is to minimize the social cost (the total cost of the agents). For the line graph, the best possible approximation ratio of deterministic strategyproof mechanisms was previously shown to lie between $4/3$ and $17/4$. We close this gap by designing an optimal $4/3$-approximate mechanism that combines a fixed-parity median rule, which suffices for instances with $n\ge7$, and strategyproof local overrides for the remaining smaller cases. Beyond the line, we design a deterministic strategyproof $2$-approximate mechanism for every connected graph and prove a lower bound of $3/2$ on the graph $K_{1,3}$.
