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On Approximate Envy-Freeness for Indivisible Chores and Mixed Resources

Authors Umang Bhaskar, A. R. Sricharan, Rohit Vaish

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Author Details

Umang Bhaskar
  • Tata Institute of Fundamental Research, Mumbai, India
A. R. Sricharan
  • Chennai Mathematical Institute, India
Rohit Vaish
  • Tata Institute of Fundamental Research, Mumbai, India

Funding

UB and RV acknowledge support from project no. RTI4001 of the Department of Atomic Energy, Government of India. RV also acknowledges support from the Prof. R Narasimhan postdoctoral award.

Acknowledgements

We thank the anonymous reviewers for helpful comments.

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Umang Bhaskar, A. R. Sricharan, and Rohit Vaish. On Approximate Envy-Freeness for Indivisible Chores and Mixed Resources. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 1:1-1:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.1

Abstract

We study the fair allocation of undesirable indivisible items, or chores. While the case of desirable indivisible items (or goods) is extensively studied, with many results known for different notions of fairness, less is known about the fair division of chores. We study envy-free allocation of chores and make three contributions. First, we show that determining the existence of an envy-free allocation is NP-complete even in the simple case when agents have binary additive valuations. Second, we provide a polynomial-time algorithm for computing an allocation that satisfies envy-freeness up to one chore (EF1), correcting a claim in the existing literature. A modification of our algorithm can be used to compute an EF1 allocation for doubly monotone instances (where each agent can partition the set of items into objective goods and objective chores). Our third result applies to a mixed resources model consisting of indivisible items and a divisible, undesirable heterogeneous resource (i.e., a bad cake). We show that there always exists an allocation that satisfies envy-freeness for mixed resources (EFM) in this setting, complementing a recent result of Bei et al. [Bei et al., 2021] for indivisible goods and divisible cake.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithmic game theory
  • Theory of computation → Exact and approximate computation of equilibria
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • Fair Division
  • Indivisible Chores
  • Approximate Envy-Freeness

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