IDEAS home Printed from
   My bibliography  Save this article

Algorithmic Solutions for Envy-Free Cake Cutting


  • 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)


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

    Download full text from publisher

    File URL:
    Download Restriction: no

    References listed on IDEAS

    1. Abdelghani A. Elimam & Maurice Girgis & Samir Kotob, 1996. "The Use of Linear Programming in Disentangling the Bankruptcies of Al-Manakh Stock Market Crash," Operations Research, INFORMS, vol. 44(5), pages 665-676, October.
    2. Cloutier, John & Nyman, Kathryn L. & Su, Francis Edward, 2010. "Two-player envy-free multi-cake division," Mathematical Social Sciences, Elsevier, vol. 59(1), pages 26-37, January.
    3. Katerina Sherstyuk, 1998. "How to gerrymander: A formal analysis," Public Choice, Springer, vol. 95(1), pages 27-49, April.
    4. Scarf, Herbert E., 1993. "The computation of equilibrium prices: An exposition," Handbook of Mathematical Economics, in: K. J. Arrow & M.D. Intriligator (ed.),Handbook of Mathematical Economics, edition 4, volume 2, chapter 21, pages 1007-1061, Elsevier.
    5. John Winsor Pratt & Richard Jay Zeckhauser, 1990. "The Fair and Efficient Division of the Winsor Family Silver," Management Science, INFORMS, vol. 36(11), pages 1293-1301, November.
    6. Xiaotie Deng & Qi Qi & Amin Saberi & Jie Zhang, 2011. "Discrete Fixed Points: Models, Complexities, and Applications," Mathematics of Operations Research, INFORMS, vol. 36(4), pages 636-652, November.
    7. Frédéric Meunier, 2008. "Discrete Splittings of the Necklace," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 678-688, August.
    8. Shahar Dobzinski & Noam Nisan & Michael Schapira, 2010. "Approximation Algorithms for Combinatorial Auctions with Complement-Free Bidders," Mathematics of Operations Research, INFORMS, vol. 35(1), pages 1-13, February.
    9. Simmons, Forest W. & Su, Francis Edward, 2003. "Consensus-halving via theorems of Borsuk-Ulam and Tucker," Mathematical Social Sciences, Elsevier, vol. 45(1), pages 15-25, February.
    10. Hylland, Aanund & Zeckhauser, Richard, 1979. "The Efficient Allocation of Individuals to Positions," Journal of Political Economy, University of Chicago Press, vol. 87(2), pages 293-314, April.
    11. William Thomson, 2007. "Children Crying at Birthday Parties. Why?," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 31(3), pages 501-521, June.
    12. Hervé Moulin, 2007. "Minimizing the Worst Slowdown: Offline, Online," Operations Research, INFORMS, vol. 55(5), pages 876-889, October.
    Full references (including those not matched with items on IDEAS)


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    Cited by:

    1. 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,


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:oropre:v:60:y:2012:i:6:p:1461-1476. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Matthew Walls). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.