We consider the strategic facility location problem in $\ell_q(\mathbb R^d)$ spaces where the social cost is defined by an arbitrary $p$-norm of the individual costs. While the optimal approximation ratios for deterministic strategyproof mechanisms are well established in the $d = 1$ setting, the guarantees for multi-dimensional spaces under an arbitrary $p$-norm are less understood. In this work, we analyze the well-studied, strategyproof coordinate-wise median (CM) mechanism and provide approximation guarantees for these generalized social costs. For $d = 2$, we establish tight approximation ratios for all $p, q \geq 1$. In particular, we show that the CM mechanism is a $2^{1 - 1/ \max(p, q)}$-approximation, resolving a conjecture of Goel and Hann-Caruthers (Social Choice and Welfare, 2023). Furthermore, for $d\geq 3$, we give upper bounds on the approximation ratio of the CM mechanism for arbitrary $p$-norm social costs, generalizing the recent result of Gravin and Jia (STOC, 2025) for the utilitarian social cost. Remarkably, we show that this approximation ratio never exceeds 3, regardless of the dimension.

翻译：我们研究了$\ell_q(\mathbb R^d)$空间中的战略设施选址问题，其中社会成本由个体成本的任意$p$-范数定义。虽然在$d=1$的情形下，确定性策略证明机制的最优近似比已得到充分建立，但在任意$p$-范数下的多维空间中的保证却鲜为人知。在本文中，我们分析了被广泛研究的、策略证明的坐标-wise中位数（CM）机制，并为这些广义社会成本提供了近似保证。对于$d=2$，我们建立了对所有$p, q \geq 1$的紧近似比。特别地，我们证明了CM机制是一个$2^{1 - 1/ \max(p, q)}$-近似，解决了Goel和Hann-Caruthers（《社会选择与福利》，2023）的一个猜想。此外，对于$d\geq 3$，我们给出了任意$p$-范数社会成本下CM机制近似比的上界，推广了Gravin和Jia（STOC，2025）关于功利主义社会成本的最新结果。值得注意的是，我们证明了该近似比永远不会超过3，无论维度如何。