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。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的（可能互异的）距离，用于设定其与部分其他智能体之间的间隔。在给定智能体分配方案时，每个智能体承担的成本为其与子集中其他智能体实际距离和理想距离差异的总和。研究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，我们考虑三种不同的稳定性概念：无嫉妒稳定性、交换稳定性和跳跃稳定性。我们从三个不同维度对该问题的（参数化）复杂性进行了系统研究：图的拓扑结构、智能体数量以及智能体偏好结构。