优化投票和设施选址的多个同时目标
Optimizing Multiple Simultaneous Objectives for Voting and Facility Location
摘要：
我们研究了经典的设施位置设置，在这里我们给出了$n$客户端和$m$可能的设施位置在一些任意度量空间，并希望选择一个位置来建立一个设施。在空间社会选择中也出现了完全相同的设置，其中选民是客户端，目标是选择候选人或结果，从选民到结果的距离表示选民的这一结果的成本（例如，基于他们的意识形态差异）。与大多数以前的工作不同，我们不关注单个目标来优化（例如，从客户端到设施的总距离或最大距离等），而是试图同时优化几个不同的目标。更具体地说，我们考虑$l$ -centrum目标族，其中包括总距离，最大距离和许多其他目标。我们对任何一对这样的目标（例如，max和sum）与它们的最优结果相比可以同时近似。特别是，我们表明，对于任何这样的一对目标，它总是可以选择一个结果，同时接近两个目标的一个因素内的$1+\sqrt{2}$，并给出了一个精确的表征，这个因素如何提高优化的两个目标变得更加相似。对于$q>2$不同的椎体目标，我们表明，它总是可以近似所有的$q$这些目标在一个小的常数，这个常数接近3作为$q\rightarrow \infty$。我们的研究结果表明，当优化只有几个同时的目标，它总是可以形成一个结果，这是一个显着优于3近似所有这些目标。
英文摘要：
We study the classic facility location setting, where we are given $n$ clients and $m$ possible facility locations in some arbitrary metric space, and want to choose a location to build a facility. The exact same setting also arises in spatial social choice, where voters are the clients and the goal is to choose a candidate or outcome, with the distance from a voter to an outcome representing the cost of this outcome for the voter (e.g., based on their ideological differences). Unlike most previous work, we do not focus on a single objective to optimize (e.g., the total distance from clients to the facility, or the maximum distance, etc.), but instead attempt to optimize several different objectives simultaneously. More specifically, we consider the $l$-centrum family of objectives, which includes the total distance, max distance, and many others. We present tight bounds on how well any pair of such objectives (e.g., max and sum) can be simultaneously approximated compared to their optimum outcomes. In particular, we show that for any such pair of objectives, it is always possible to choose an outcome which simultaneously approximates both objectives within a factor of $1+\sqrt{2}$, and give a precise characterization of how this factor improves as the two objectives being optimized become more similar. For $q>2$ different centrum objectives, we show that it is always possible to approximate all $q$ of these objectives within a small constant, and that this constant approaches 3 as $q\rightarrow \infty$. Our results show that when optimizing only a few simultaneous objectives, it is always possible to form an outcome which is a significantly better than 3 approximation for all of these objectives.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
