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Algorithmic Solutions for Envy-Free Cake Cutting

Author

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  • Xiaotie Deng

    (Department of Computer Science, University of Liverpool, Liverpool L69 3BX, United Kingdom; and Department of Computer Science, City University of Hong Kong, Hong Kong)

  • Qi Qi

    (Department of Industrial Engineering and Logistics Management, Hong Kong University of Science and Technology, Hong Kong)

  • Amin Saberi

    (Department of Management Science and Engineering, Stanford University, Stanford, California 94305)

Abstract

We study the problem of finding an envy-free allocation of a cake to d + 1 players using d cuts. Two models are considered, namely, the oracle-function model and the polynomial-time function model. In the oracle-function model, we are interested in the number of times an algorithm has to query the players about their preferences to find an allocation with the envy less than (epsilon). We derive a matching lower and upper bound of (theta)(1/(epsilon)) d - 1 for players with Lipschitz utilities and any d > 1. In the polynomial-time function model, where the utility functions are given explicitly by polynomial-time algorithms, we show that the envy-free cake-cutting problem has the same complexity as finding a Brouwer's fixed point, or, more formally, it is PPAD-complete. On the flip side, for monotone utility functions, we propose a fully polynomial-time algorithm (FPTAS) to find an approximate envy-free allocation of a cake among three people using two cuts.

Suggested Citation

  • Xiaotie Deng & Qi Qi & Amin Saberi, 2012. "Algorithmic Solutions for Envy-Free Cake Cutting," Operations Research, INFORMS, vol. 60(6), pages 1461-1476, December.
    • Handle: RePEc:inm:oropre:v:60:y:2012:i:6:p:1461-1476
    DOI: 10.1287/opre.1120.1116
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    References listed on IDEAS

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    1. Sagrario Lantarón & Mariló López & Susana Merchán & Javier Rodrigo & José Samuel Rodríguez, 2021. "Envy-Free Allocation by Sperner’s Lemma Adapted to Rotation Shifts in a Company," Mathematics, MDPI, vol. 9(9), pages 1-12, April.
    2. Lee, Myungho & Lee, Kangbok & Pinedo, Michael, 2025. "The circular balancing problem," European Journal of Operational Research, Elsevier, vol. 321(1), pages 41-56.
    3. Vittorio Bilò & Ioannis Caragiannis & Michele Flammini & Ayumi Igarashi & Gianpiero Monaco & Dominik Peters & Cosimo Vinci & William Zwicker, 2021. "Almost Envy-Free Allocations with Connected Bundles," Post-Print hal-03834506, HAL.
    4. Erel Segal-Halevi & Shmuel Nitzan & Avinatan Hassidim & Yonatan Aumann, 2020. "Envy-Free Division of Land," Mathematics of Operations Research, INFORMS, vol. 45(3), pages 896-922, August.
    5. Vittorio Bil`o & Ioannis Caragiannis & Michele Flammini & Ayumi Igarashi & Gianpiero Monaco & Dominik Peters & Cosimo Vinci & William S. Zwicker, 2018. "Almost Envy-Free Allocations with Connected Bundles," Papers 1808.09406, arXiv.org, revised May 2022.
    6. Bilò, Vittorio & Caragiannis, Ioannis & Flammini, Michele & Igarashi, Ayumi & Monaco, Gianpiero & Peters, Dominik & Vinci, Cosimo & Zwicker, William S., 2022. "Almost envy-free allocations with connected bundles," Games and Economic Behavior, Elsevier, vol. 131(C), pages 197-221.

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