High-Multiplicity Fair Allocation: Lenstra Empowered by N-fold Integer Programming

We study the (parameterized) computational complexity of problems in the context of fair allocations of indivisible goods. More specifically, we show fixed-parameter tractability results for a broad set of problems concerned with envy-free, Pareto-efficient allocations of items (with agent-specific utility functions) to agents. In principle, this implies efficient exact algorithms for these in general computationally intractable problems whenever we face instances with few agents and low maximum (absolute) utility values. This holds true also in high-multiplicity settings where we may have high numbers of identical items. On the technical side, our approach provides algorithmic meta-theorems covering a rich set of fair allocation problems in the additive preferences model. To achieve this, our main technical contribution is to make an elaborate use of tools from integer linear programming. More specifically, we exploit results originally going back to a famous theorem of Lenstra [Math. Oper. Res. 1983] concerning (the fixed-parameter tractability of) Integer Linear Programs (ILPs) with bounded dimension (that is, the dimension shall be considered as a (small) parameter) and the more recent framework of (combinatorial) N-fold ILPs. We reveal and exploit a fruitful interaction between these two cornerstones in the theory of integer linear programming, which may be of independent interest in applications going beyond fair allocations.