Efficient Almost-Egalitarian Allocation of Goods and Bads

We consider the allocation of indivisible objects among agents with different valuations, which can be positive or negative. An egalitarian allocation is an allocation that maximizes the smallest value given to an agent; finding such an allocation is NP-hard. We present a simple polynomial-time algorithm that finds an allocation that is Pareto-efficient and almost-egalitarian: each agent's value is at least his value in an egalitarian allocation, minus the absolute value of a single object. The main tool is an algorithm for rounding a fractional allocation to a discrete allocation, by which each agent loses at most one good or gains at most one chore. Our algorithm generalizes and simplifies three previous algorithms. We discuss several aspects and observations about the algorithm and the problem at hand that open doors for efficient and robust implementations.