Multivariate Algorithmics for Eliminating Envy by Donating Goods

TL;DR

This paper investigates the computational complexity of modifying resource allocations through donations to achieve envy-freeness, considering various parameters and preference models, and provides a comprehensive analysis of tractability and intractability results.

Contribution

It offers a detailed parameterized complexity analysis of resource donation strategies to eliminate envy, highlighting surprising contrasts between different fairness notions and parameters.

Findings

Identifies tractable cases for envy-free allocations after donations.

Shows intractability results under certain parameters and preference models.

Discovers contrasts between envy-freeness and envy-freeness up to one good.

Abstract

Fairly dividing a set of indivisible resources to a set of agents is of utmost importance in some applications. However, after an allocation has been implemented the preferences of agents might change and envy might arise. We study the following problem to cope with such situations: Given an allocation of indivisible resources to agents with additive utility-based preferences, is it possible to socially donate some of the resources (which means removing these resources from the allocation instance) such that the resulting modified allocation is envy-free (up to one good). We require that the number of deleted resources and/or the caused utilitarian welfare loss of the allocation are bounded. We conduct a thorough study of the (parameterized) computational complexity of this problem considering various natural and problem-specific parameters (e.g., the number of agents, the number of deleted resources, or the maximum number of resources assigned to an agent in the initial allocation) and different preference models, including unary and 0/1-valuations. In our studies, we obtain a rich set of (parameterized) tractability and intractability results and discover several surprising contrasts, for instance, between the two closely related fairness concepts envy-freeness and envy-freeness up to one good and between the influence of the parameters maximum number and welfare of the deleted resources.