Fair Allocation of Indivisible Items With Externalities.

One of the important yet insufficiently studied subjects in fair allocation is the externality effect among agents. For a resource allocation problem, externalities imply that a bundle allocated to an agent may affect the utilities of other agents. In this paper, we conduct a study of fair allocation of indivisible goods when the externalities are not negligible. We present a simple and natural model, namely emph{network externalities}, to capture the externalities. To evaluate fairness in the network externalities model, we generalize the idea behind the notion of maximin-share ($MMS$) to achieve a new criterion, namely, emph{extended-maximin-share} ($EMMS$). we consider two problems concerning our model. First, we discuss the computational aspects of finding the value of $EMMS$ for every agent. For this, we introduce a generalized form of partitioning problem that includes many famous partitioning problems such as maximin, minimax, and leximin partitioning problems. We show that a $1/2$-approximation algorithm exists for this partitioning problem. Next, we investigate on finding approximately optimal $EMMS$ allocations. That is, allocations that guarantee every agent a utility of at least a fraction of his extended-maximin-share. We show that under a natural assumption that the agents are $alpha$-self-reliant, an $alpha/2$-$EMMS$ allocation always exists. The combination of this with the former result yields a polynomial-time $alpha/4$-$EMMS$ allocation algorithm.