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Computing Welfare-Maximizing Fair Allocations of Indivisible Goods

Haris Aziz, Xin Huang, Nicholas Mattei, Erel Segal-Halevi

TL;DR

This work analyzes the run-time complexity of computing allocations that are both fair and maximize the utilitarian social welfare, defined as the sum of agents' utilities and designs pseudopolynomial-time algorithms for both problems.

Abstract

We analyze the run-time complexity of computing allocations that are both fair and maximize the utilitarian social welfare, defined as the sum of agents' utilities. We focus on two tractable fairness concepts: envy-freeness up to one item (EF1) and proportionality up to one item (PROP1). We consider two computational problems: (1) Among the utilitarian-maximal allocations, decide whether there exists one that is also fair; (2) among the fair allocations, compute one that maximizes the utilitarian welfare. We show that both problems are strongly NP-hard when the number of agents is variable, and remain NP-hard for a fixed number of agents greater than two. For the special case of two agents, we find that problem (1) is polynomial-time solvable, while problem (2) remains NP-hard. Finally, with a fixed number of agents, we design pseudopolynomial-time algorithms for both problems. We extend our results to the stronger fairness notions envy-freeness up to any item (EFx) and proportionality up to any item (PROPx).

Computing Welfare-Maximizing Fair Allocations of Indivisible Goods

TL;DR

This work analyzes the run-time complexity of computing allocations that are both fair and maximize the utilitarian social welfare, defined as the sum of agents' utilities and designs pseudopolynomial-time algorithms for both problems.

Abstract

We analyze the run-time complexity of computing allocations that are both fair and maximize the utilitarian social welfare, defined as the sum of agents' utilities. We focus on two tractable fairness concepts: envy-freeness up to one item (EF1) and proportionality up to one item (PROP1). We consider two computational problems: (1) Among the utilitarian-maximal allocations, decide whether there exists one that is also fair; (2) among the fair allocations, compute one that maximizes the utilitarian welfare. We show that both problems are strongly NP-hard when the number of agents is variable, and remain NP-hard for a fixed number of agents greater than two. For the special case of two agents, we find that problem (1) is polynomial-time solvable, while problem (2) remains NP-hard. Finally, with a fixed number of agents, we design pseudopolynomial-time algorithms for both problems. We extend our results to the stronger fairness notions envy-freeness up to any item (EFx) and proportionality up to any item (PROPx).

Paper Structure

This paper contains 14 sections, 19 theorems, 13 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Setting
  3. Fairness
  4. Welfare
  5. Related work
  6. UM and EF1
  7. UM and PROP1
  8. UM and Fairness for Two Agents
  9. UM within Fairness for Few Agents
  10. Empirical Evaluation
  11. Mixed Integer Linear Programs for Finding Utilitarian Maximal Allocations with EF/EF1 and PROP/PROP1
  12. Experimental Details and Results
  13. Conclusions and Future Work
  14. EF1 vs. PROP1

Key Result

Theorem 4.1

The problem ExistsUMandEF1 --- deciding whether there exists an allocation that is both UM and EF1 --- is strongly NP-complete.

Figures (2)

  • Figure 1: Run-time (on a logarithmic scale) of the Dynamic Programming algorithms (DP) and Mixed Integer Linear Programming (GRB) versions of algorithms to find UM within EF or PROP allocations. We plotted the run-times for three different settings of the Mallows model dispersion parameter $\phi$; the plots for these three settings are almost completely overlapping, though if we do not display on a log scale y-axis, we can see a small difference in runtime. The MILP implementations are able to scale much better as we increase the number of agents and objects. At $n=7$, the bottom plot is GRB-PROP (denoting the fastest performance), the next one is GRB-EF, then DP-PROP, and finally DP-EF.
  • Figure 2: Run-time (on a logarithmic scale) of the Dynamic Programming algorithms (DP) and Mixed Integer Linear Programming (GRB) versions of algorithms to find UM within EF1 or PROP1 allocations. We plotted the run-times for three different settings of the Mallows model dispersion parameter $\phi$; the plots for these three settings are almost completely overlapping, though if we do not display on a log scale y-axis, we can see a small difference in runtime. The MILP implementations are able to scale much better as we increase the number of agents and objects. At $n=7$, the bottom plot is GRB-PROP1 (denoting the fastest performance), the next one is GRB-EF1, then DP-PROP1, and finally DP-EF1.

Theorems & Definitions (43)

  • Theorem 4.1
  • proof
  • Remark 4.2
  • Corollary 4.3
  • proof
  • Theorem 4.4
  • proof
  • Remark 4.5
  • Corollary 4.6
  • Theorem 5.1
  • ...and 33 more