We introduce a new class of network allocation games called graphical distance preservation games. Here, we are given a graph, called a topology, and a set of agents that need to be allocated to its vertices. Moreover, every agent has an ideal (and possibly different) distance in which to be from some of the other agents. Given an allocation of agents, each one of them suffers a cost that is the sum of the differences from the ideal distance for each agent in their subset. The goal is to decide whether there is a stable allocation of the agents, i.e., no agent would like to deviate from their location. Specifically, we consider three different stability notions: envy-freeness, swap stability, and jump stability. We perform a comprehensive study of the (parameterized) complexity of the problem in three different dimensions: the topology of the graph, the number of agents, and the structure of preferences of the agents.

翻译：本文引入一类新的网络分配博弈，称为图距离保持博弈。在此类博弈中，给定一个称为拓扑结构的图以及一组需要分配到其顶点上的智能体。此外，每个智能体都有一个理想（且可能互异）的距离，用于描述其与特定其他智能体之间应保持的间隔。给定智能体的一种分配方案，每个智能体会承担一项成本，该成本等于其与关联子集中每个其他智能体的实际距离和理想距离之差的绝对值总和。本研究的目标是判定是否存在稳定的智能体分配方案，即没有智能体有意愿偏离其当前位置。具体而言，我们考察三种不同的稳定性概念：无嫉妒稳定性、交换稳定性与跳跃稳定性。我们从三个维度对该问题的（参数化）复杂性进行了系统性研究：图的拓扑结构、智能体数量以及智能体偏好结构。