Fair division has emerged as a very hot topic in AI, and envy-freeness is arguably the most compelling fairness concept. An allocation of indivisible items to agents is envy-free if no agent prefers the bundle of any other agent to his own in terms of value. As envy-freeness is rarely a feasible goal, there is a recent focus on relaxations of its definition. An approach in this direction is to complement allocations with payments (or subsidies) to the agents. A feasible goal then is to achieve envy-freeness in terms of the total value an agent gets from the allocation and the subsidies. We consider the natural optimization problem of computing allocations that are envy-freeable using the minimum amount of subsidies. As the problem is NP-hard, we focus on the design of approximation algorithms. On the positive side, we present an algorithm which, for a constant number of agents, approximates the minimum amount of subsidies within any required accuracy, at the expense of a graceful increase in the running time. On the negative side, we show that, for a superconstant number of agents, the problem of minimizing subsidies for envy-freeness is not only hard to compute exactly (as a folklore argument shows) but also, more importantly, hard to approximate. * In this paper, we follow an optimization approach. We define and study the optimization problem SMEF (standing for Subsidy Minimization for Envy-Freeness). Given an allocation problem consisting of items and agents with valuations for the items, SMEF asks for an allocation that is envy-freeable using the minimum total amount of subsidies. SMEF is NP-hard; this follows by the NP-hardness of deciding whether a given allocation problem has an envy-free allocation or not. Thus, we resort to approximation algorithms for SMEF. As multiplicative approximation guarantees are hopeless, our aim is to design algorithms that run in polynomial-time and compute an allocation that is envy-freeable with an amount of subsidies that does not exceed the minimum possible amount of subsidies (denoted χ) by much. In particular, we use the total valuation of all agents for all goods (denoted by sum v) as benchmark and seek allocations that are envy-freeable with an amount of at most χ + ρ · sum v as subsidies. The goal for the approximation guarantee ρ of an algorithm is to be as small as possible. We initiate the study of SMEF and present two results. On the positive side, we design an algorithm that achieves an arbitrary low approximation guarantee of ǫ > 0. When applied to allocation instances with a constant number of agents, the algorithm uses dynamic programming and runs in time that is polynomial in the number of items and 1/ǫ. On the negative side, we show that, in general, SMEF is not only hard to solve exactly, but also hard to approximate within a small constant. Unlike the folklore reduction 1 for proving hardness of envy-freeness, our proof uses a novel approximation-preserving reduction. Besides separating the general case from that with constantly many agents, our negative result indicates that achieving good approximation guarantees will be a challenging goal.
Moreover, REF considered the optimization problem of minimizing amount of subsidy to obtain envy-freeness for an allocation instance.
Computing envy-freeable allocations with limited subsidies
ArXiv
ArXiv
Mathematics Computer Science