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High-Multiplicity Fair Allocation Using Parametric Integer Linear Programming

Robert Bredereck, Andrzej Kaczmarczyk, Dušan Knop, Rolf Niedermeier

TL;DR

This paper tackles the problem of finding envy-free Pareto-efficient allocations of indivisible items when item types appear in multiple copies (high-multiplicity) and utilities are additive. It introduces a parametric ILP approach to achieve fixed-parameter tractability with respect to the combined parameter $n+m$ while allowing binary encoding of multiplicities and utilities, improving upon unary-encoded results. The method constructs a domination matrix $A$ and a polyhedron $Q$ to encode envy-freeness and domination, and reduces the existence (or nonexistence) of a suitable allocation to a PILP instance solvable within FPT bounds, yielding a constructive allocation when one exists. The framework extends to a broad class of fairness and efficiency notions and points to empirical evaluation and broader applicability in fair division problems.

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.

High-Multiplicity Fair Allocation Using Parametric Integer Linear Programming

TL;DR

This paper tackles the problem of finding envy-free Pareto-efficient allocations of indivisible items when item types appear in multiple copies (high-multiplicity) and utilities are additive. It introduces a parametric ILP approach to achieve fixed-parameter tractability with respect to the combined parameter while allowing binary encoding of multiplicities and utilities, improving upon unary-encoded results. The method constructs a domination matrix and a polyhedron to encode envy-freeness and domination, and reduces the existence (or nonexistence) of a suitable allocation to a PILP instance solvable within FPT bounds, yielding a constructive allocation when one exists. The framework extends to a broad class of fairness and efficiency notions and points to empirical evaluation and broader applicability in fair division problems.

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.

Paper Structure

This paper contains 18 sections, 4 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work
  4. Efficient and Envy-free Allocations of Indivisible Resources.
  5. Parametric ILP Aplications.
  6. Organization
  7. Preliminaries
  8. Allocations, Envy-Freeness, and Pareto-Efficiency
  9. Parameterized Complexity
  10. Parametric Integer Programming
  11. Preparation for Main Result
  12. Finding EEF--Allocations via PILP
  13. Proving the result
  14. Describing Domination of Allocation with Matrix $A$.
  15. The Polyhedron $Q$.
  16. ...and 3 more sections

Key Result

Proposition 1

For every two computable functions $g\colon \mathbb{N} \to \mathbb{N}$ and $h\colon \mathbb{N} \to \mathbb{N}$ with $g(n) = o(\log (n))$, there exists a computable function $f\colon \mathbb{N} \to \mathbb{N}$ such that for every $k$ and $n$ we have $2^{g(n)h(k)} \le f(k) \cdot n$.

Theorems & Definitions (6)

  • Definition 1
  • Definition 2
  • Proposition 1: JonesLRSS17
  • Proposition 2
  • Theorem 1
  • Lemma 1