Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fair Societies: Algorithms for House Allocations

Hadi Hosseini, Sanjukta Roy, Aditi Sethia

TL;DR

This paper addresses fairness in one-sided house allocation by targeting the minimization of envy, a problem known to be hard even for simple valuations. It introduces a fixed-parameter tractable refinement framework that, given an initial allocation, reduces envy using at most $q$ reallocations, with running time depending on $q$ and the maximum degree $d$ of the preference graph. It also provides polynomial-time algorithms for restricted domains, specifically single-peaked and single-dipped preferences, including Pareto-efficiency considerations in the single-peaked case. An empirical study illustrates the fairness-welfare trade-offs and shows that a small number of targeted reallocations can substantially reduce envy with limited welfare loss. Overall, the work offers a versatile approach to achieving near-optimal fairness in practice while accommodating computational constraints and domain restrictions.

Abstract

House Allocations concern with matchings involving one-sided preferences, where houses serve as a proxy encoding valuable indivisible resources (e.g. organs, course seats, subsidized public housing units) to be allocated among the agents. Every agent must receive exactly one resource. We study algorithmic approaches towards ensuring fairness in such settings. Minimizing the number of envious agents is known to be NP-complete (Kamiyama et al. 2021). We present two tractable approaches to deal with the computational hardness. When the agents are presented with an initial allocation of houses, we aim to refine this allocation by reallocating a bounded number of houses to reduce the number of envious agents. We show an efficient algorithm when the agents express preference for a bounded number of houses. Next, we consider single peaked preference domain and present a polynomial time algorithm for finding an allocation that minimize the number of envious agents. We further extend it to satisfy Pareto efficiency. Our former algorithm works for other measures of envy such as total envy, or maximum envy, with suitable modifications. Finally, we present an empirical analysis recording the fairness-welfare trade-off of our algorithms.

Fair Societies: Algorithms for House Allocations

TL;DR

This paper addresses fairness in one-sided house allocation by targeting the minimization of envy, a problem known to be hard even for simple valuations. It introduces a fixed-parameter tractable refinement framework that, given an initial allocation, reduces envy using at most reallocations, with running time depending on and the maximum degree of the preference graph. It also provides polynomial-time algorithms for restricted domains, specifically single-peaked and single-dipped preferences, including Pareto-efficiency considerations in the single-peaked case. An empirical study illustrates the fairness-welfare trade-offs and shows that a small number of targeted reallocations can substantially reduce envy with limited welfare loss. Overall, the work offers a versatile approach to achieving near-optimal fairness in practice while accommodating computational constraints and domain restrictions.

Abstract

House Allocations concern with matchings involving one-sided preferences, where houses serve as a proxy encoding valuable indivisible resources (e.g. organs, course seats, subsidized public housing units) to be allocated among the agents. Every agent must receive exactly one resource. We study algorithmic approaches towards ensuring fairness in such settings. Minimizing the number of envious agents is known to be NP-complete (Kamiyama et al. 2021). We present two tractable approaches to deal with the computational hardness. When the agents are presented with an initial allocation of houses, we aim to refine this allocation by reallocating a bounded number of houses to reduce the number of envious agents. We show an efficient algorithm when the agents express preference for a bounded number of houses. Next, we consider single peaked preference domain and present a polynomial time algorithm for finding an allocation that minimize the number of envious agents. We further extend it to satisfy Pareto efficiency. Our former algorithm works for other measures of envy such as total envy, or maximum envy, with suitable modifications. Finally, we present an empirical analysis recording the fairness-welfare trade-off of our algorithms.

Paper Structure

This paper contains 34 sections, 20 theorems, 2 equations, 17 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Fixed Parameter Tractability.
  4. Polynomial Time Algorithms.
  5. Experiments.
  6. Additional Related Work
  7. Preliminaries
  8. Allocations.
  9. Fairness.
  10. Efficiency.
  11. Refining Fairness: A Parameterized Algorithm
  12. Intuition and Challenges.
  13. Notations.
  14. Overview of \ref{['alg:minEnvylocal']}.
  15. Separation of Paths / Cycles.
  16. ...and 19 more sections

Key Result

Theorem 3.1

Given an instance $\mathcal{I} = (N, H, \succeq)$ of house allocation, a complete allocation $\hat{A}$, and two positive integers $k$ and $q$, deciding if there is an allocation $A$ such that $\mathsf{Envy}(A) \leq \mathsf{Envy}(\hat{A})-k$ and $|A~\Delta~\hat{A}| \leq q$ admits an algorithm that ru

Figures (17)

  • Figure 1: Single-Peaked preferences
  • Figure 3: Welfare loss incurred starting from a Nash (blue) and Egalitarian (green) welfare-maximizing allocation and performing at most $q$ reallocations, where $1 \leq q \leq n$.
  • Figure 4: The drop in $\mathsf{Envy}$ starting from Nash (blue) and Egalitarian (green) welfare-maximizing allocations and performing at most $q$ reallocations, where $1 \leq q \leq n$.
  • Figure 6: Welfare loss incurred starting from a Utilitarian (blue) and Egalitarian (green) welfare-maximizing allocation and performing at most $q$ reallocations, where $1 \leq q \leq n$.
  • Figure 7: The drop in the number of envious agents starting from Utilitarian (blue) and Egalitarian (green) welfare-maximizing allocation and performing at most $q$ reallocations, where $1 \leq q \leq n$.
  • ...and 12 more figures

Theorems & Definitions (47)

  • Theorem 3.1
  • Example 3.2
  • Definition 3.3: Dependent Set
  • Definition 3.4: Minimal Improvement Set
  • Lemma 3.5
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.5
  • proof
  • ...and 37 more