High-Multiplicity Fair Allocation Using Parametric Integer Linear Programming

TL;DR

This paper advances high-multiplicity fair allocation by leveraging parametric integer linear programming to encode utility and multiplicity values in binary, significantly broadening the problem's practical applicability.

Contribution

It improves previous fixed-parameter tractability results by allowing binary encoding of utility and multiplicity values, expanding feasible problem instances.

Findings

Fixed-parameter tractability maintained with binary encoding

Expanded range of utility and multiplicity values feasible

Improved computational efficiency in high-multiplicity fair allocation

Abstract

Using insights from parametric integer linear programming, we significantly improve on our previous work [Proc. ACM EC 2019] on high-multiplicity fair allocation. Therein, answering an open question from previous work, we proved that the problem of finding envy-free Pareto-efficient allocations of indivisible items is fixed-parameter tractable with respect to the combined parameter "number of agents" plus "number of item types." Our central improvement, compared to this result, is to break the condition that the corresponding utility and multiplicity values have to be encoded in unary required there. Concretely, we show that, while preserving fixed-parameter tractability, these values can be encoded in binary, thus greatly expanding the range of feasible values.