This article deals with the allocation of objects where each agent receives a single item. Starting from an initial endowment, the agents can be better off by exchanging their objects. However, not all trades are likely because some participants are unable to communicate. By considering that the agents are embedded in a social network, we propose to study the possible allocations emerging from a sequence of simple swaps between pairs of neighbors in the network. This model raises natural questions regarding (i) the reachability of a given full allocation, (ii) the ability of an agent to obtain a given object, and (iii) the search of Paretoefficient allocations. We investigate the complexity of these problems by providing, according to the structure of the social network, polynomial and NP-complete cases.

Reallocating resources to get mutually beneficial outcomes is a fundamental problem in various multi-agent settings. While finding an arbitrary Pareto optimal allocation is generally easy, checking whether a particular allocation is Pareto optimal can be much more difficult. This problem is equivalent to checking that the allocated objects cannot be reallocated in such a way that at least one agent prefers her new share to his old one, and no agent prefers her old share to her new one. We consider the problem for two related types of preference relations over sets of objects. In the first part of the paper we focus on the setting in which agents express additive cardinal utilities over objects. We present computational hardness results as well as polynomial-time algorithms for testing Pareto optimality under different restrictions such as two utility values or lexicographic utilities. In the second part of the paper we assume that agents express only their (ordinal) preferences over single objects, and that their preferences are additively separable. In this setting, we present characterizations and polynomial-time algorithms for possible and necessary Pareto optimality.

We study the fair division problem consisting in allocating one item per agent so as to avoid (or minimize) envy, in a setting where only agents connected in a given social network may experience envy. In a variant of the problem, agents themselves can be located on the network by the central authority. These problems turn out to be difficult even on very simple graph structures, but we identify several tractable cases. We further provide practical algorithms and experimental insights.