Finding an envy-free allocation of indivisible resources to agents is a central task in many multiagent systems. Often, non-trivial envy-free allocations do not exist, and, when they do, finding them can be computationally hard. Classical envy-freeness requires that every agent likes the resources allocated to it at least as much as the resources allocated to any other agent. In many situations this assumption can be relaxed since agents often do not even know each other. We enrich the envy-freeness concept by taking into account (directed) social networks of the agents. Thus, we require that every agent likes its own allocation at least as much as those of all its (out)neighbors. This leads to a "more local" concept of envy-freeness. We also consider a "strong" variant where every agent must like its own allocation more than those of all its (out)neighbors. We analyze the classical and the parameterized complexity of finding allocations that are complete and, at the same time, envy-free with respect to one of the variants of our new concept. To this end, we study different restrictions of the agents' preferences and of the social network structure. We identify cases that become easier (from $\Sigma^\textrm{p}_2$-hard or NP-hard to polynomial-time solvability) and cases that become harder (from polynomial-time solvability to NP-hard) when comparing classical envy-freeness with our graph envy-freeness. Furthermore, we spot cases where graph envy-freeness is easier to decide than strong graph envy-freeness, and vice versa. On the route to one of our fixed-parameter tractability results, we also establish a connection to a directed and colored variant of the classical SUBGRAPH ISOMORPHISM problem, thereby extending a known fixed-parameter tractability result for the latter.

中文翻译：

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尊重社交网络的免费分配

在许多多代理系统中，找到不可羡慕的不可分割资源分配给代理是一项中心任务。通常，不平凡的无羡慕分配不存在，并且当存在时，要找到它们很难进行计算。经典的无羡慕度要求每个代理都喜欢分配给它的资源，至少喜欢分配给任何其他代理的资源。在许多情况下，此假设可以放宽，因为代理通常甚至彼此都不认识。通过考虑代理商的（定向）社交网络，我们丰富了嫉妒-自由的概念。因此，我们要求每个代理都至少喜欢与其所有（外）邻居的分配相同的分配。这导致了“更加本地化”的嫉妒概念。我们也认为“强” 每个代理都必须比其所有（外）邻居更喜欢自己的分配。对于我们的新概念的变体之一，我们分析了寻找完整且同时令人羡慕的分配的经典和参数化复杂性。为此，我们研究了代理商偏好和社交网络结构的不同限制。当比较经典时，我们确定了变得更容易的情况（从$ \ Sigma ^ \ textrm {p} _2 $ -hard或NP-hard到多项式时间可解性）和变得更困难的情况（从多项式时间可解性到NP-hard）令人羡慕的自由与我们的图羡慕。此外，我们发现了一些案例，在这些案例中，相比于强大的图羡慕度，更容易确定图羡慕度，反之亦然。在获得我们的固定参数易处理性结果之一的过程中，