We take the classic facility location problem and consider a variation, in which each agent's individual cost function is equal to their distance from the facility multiplied by a scaling factor which is determined by the facility placement. In addition to the general class of continuous scaling functions, we also provide results for piecewise linear scaling functions which can effectively approximate or model the scaling of many real world scenarios. We focus on the objectives of total and maximum cost, describing the computation of the optimal solution. We then move to the approximate mechanism design setting, observing that the agents' preferences may no longer be single-peaked. Consequently, we characterize the conditions on scaling functions which ensure that agents have single-peaked preferences. Under these conditions, we find results on the total and maximum cost approximation ratios achievable by strategyproof and anonymous mechanisms.

翻译：我们考虑经典设施选址问题的一个变体，其中每个智能体的个体成本函数等于其到设施的距离乘以由设施位置决定的缩放因子。除了一般类别的连续缩放函数外，我们还针对分段线性缩放函数给出了结果，这类函数能有效近似或模拟许多现实场景中的缩放效应。我们聚焦于总成本和最大成本目标，描述了最优解的计算方法。随后转向近似机制设计环境，观察到智能体的偏好可能不再具有单峰特性。因此，我们刻画了保证智能体偏好具有单峰特性的缩放函数条件。在这些条件下，我们得到了关于策略证明且匿名机制所能实现的总成本和最大成本近似比的结果。