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 a characterization of continuous, strategyproof, and anonymous mechanisms, and compute the total and maximum cost approximation ratios achievable by these mechanisms.

翻译：本文基于经典设施选址问题，研究了一种变体：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选址决定的规模因子。除了一般类型的连续规模函数外，我们还针对分段线性规模函数给出了相关结果，这类函数能有效近似或建模许多现实场景中的规模效应。我们聚焦于总成本和最大成本这两个目标，描述了最优解的计算方法。随后转向近似机制设计场景，观察到智能体的偏好可能不再具有单峰性。因此，我们刻画了确保智能体具有单峰偏好的规模函数所需满足的条件。在这些条件下，我们得到了连续、防策略且匿名机制的特征刻画，并计算了这些机制在总成本和最大成本上可达到的近似比。