We consider the problem of fairly allocating indivisible public goods. We model the public goods as elements with feasibility constraints on what subsets of elements can be chosen, and assume that agents have additive utilities across elements. Our model generalizes existing frameworks such as fair public decision making and participatory budgeting. We study a groupwise fairness notion called the core, which generalizes well-studied notions of proportionality and Pareto efficiency, and requires that each subset of agents must receive an outcome that is fair relative to its size. In contrast to the case of divisible public goods (where fractional allocations are permitted), the core is not guaranteed to exist when allocating indivisible public goods. Our primary contributions are the notion of an additive approximation to the core (with a tiny multiplicative loss), and polynomial time algorithms that achieve a small additive approximation, where the additive factor is relative to the largest utility of an agent for an element. If the feasibility constraints define a matroid, we show an additive approximation of 2. A similar approach yields a constant additive bound when the feasibility constraints define a matching. For feasibility constraints defining an arbitrary packing polytope with mild restrictions, we show an additive guarantee that is logarithmic in the width of the polytope. Our algorithms are based on the convex program for maximizing the Nash social welfare, but differ significantly from previous work in how it is used. As far as we are aware, our work is the first to approximate the core in indivisible settings.
For integer allocations with packing constraints, we extend our techniques to provide approximations to the notion of core with endowments in the full version REF .
Fair Allocation of Indivisible Public Goods
EC '18
Mathematics Computer Science