Approximation Algorithms for Envy-Free Cake Division with Connected Pieces

Authors Siddharth Barman, Pooja Kulkarni

Thumbnail PDF


  • Filesize: 0.79 MB
  • 19 pages

Document Identifiers

Author Details

Siddharth Barman
  • Indian Institute of Science, Bangalore, India
Pooja Kulkarni
  • University of Illinois at Urbana-Champaign, IL, USA

Cite AsGet BibTex

Siddharth Barman and Pooja Kulkarni. Approximation Algorithms for Envy-Free Cake Division with Connected Pieces. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 16:1-16:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Cake cutting is a classic model for studying fair division of a heterogeneous, divisible resource among agents with individual preferences. Addressing cake division under a typical requirement that each agent must receive a connected piece of the cake, we develop approximation algorithms for finding envy-free (fair) cake divisions. In particular, this work improves the state-of-the-art additive approximation bound for this fundamental problem. Our results hold for general cake division instances in which the agents' valuations satisfy basic assumptions and are normalized (to have value 1 for the cake). Furthermore, the developed algorithms execute in polynomial time under the standard Robertson-Webb query model. Prior work has shown that one can efficiently compute a cake division (with connected pieces) in which the additive envy of any agent is at most 1/3. An efficient algorithm is also known for finding connected cake divisions that are (almost) 1/2-multiplicatively envy-free. Improving the additive approximation guarantee and maintaining the multiplicative one, we develop a polynomial-time algorithm that computes a connected cake division that is both (1/4 +o(1))-additively envy-free and (1/2 - o(1))-multiplicatively envy-free. Our algorithm is based on the ideas of interval growing and envy-cycle elimination. In addition, we study cake division instances in which the number of distinct valuations across the agents is parametrically bounded. We show that such cake division instances admit a fully polynomial-time approximation scheme for connected envy-free cake division.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithmic game theory
  • Theory of computation → Algorithmic game theory and mechanism design
  • Fair Division
  • Envy-Freeness
  • Envy-Cycle Elimination


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Reza Alijani, Majid Farhadi, Mohammad Ghodsi, Masoud Seddighin, and Ahmad Tajik. Envy-free mechanisms with minimum number of cuts. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 31(1), 2017. Google Scholar
  2. Georgios Amanatidis, Georgios Birmpas, Aris Filos-Ratsikas, and Alexandros A Voudouris. Fair division of indivisible goods: A survey. arXiv preprint, 2022. URL:
  3. Eshwar Ram Arunachaleswaran, Siddharth Barman, Rachitesh Kumar, and Nidhi Rathi. Fair and efficient cake division with connected pieces. In International Conference on Web and Internet Economics, pages 57-70. Springer, 2019. Google Scholar
  4. Haris Aziz and Simon Mackenzie. A discrete and bounded envy-free cake cutting protocol for any number of agents. In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pages 416-427. IEEE, 2016. Google Scholar
  5. Dinesh Kumar Baghel, Vadim E Levit, and Erel Segal-Halevi. Fair division algorithms for electricity distribution. arXiv preprint, 2022. URL:
  6. Siddharth Barman and Pooja Kulkarni. Approximation algorithms for envy-free cake division with connected pieces, 2022. URL:
  7. Siddharth Barman and Nidhi Rathi. Fair cake division under monotone likelihood ratios. Mathematics of Operations Research, 2021. Google Scholar
  8. Steven J Brams and Alan D Taylor. An envy-free cake division protocol. The American Mathematical Monthly, 102(1):9-18, 1995. Google Scholar
  9. Steven J Brams and Alan D Taylor. Fair Division: From cake-cutting to dispute resolution. Cambridge University Press, 1996. Google Scholar
  10. Simina Brânzei and Noam Nisan. The query complexity of cake cutting. In Advances in Neural Information Processing Systems, 2022. Google Scholar
  11. Constantinos Daskalakis, Aranyak Mehta, and Christos Papadimitriou. Progress in approximate nash equilibria. In Proceedings of the 8th ACM Conference on Electronic Commerce, pages 355-358, 2007. Google Scholar
  12. Xiaotie Deng, Qi Qi, and Amin Saberi. Algorithmic solutions for envy-free cake cutting. Operations Research, 60(6):1461-1476, 2012. Google Scholar
  13. Aris Filos-Ratsikas, Alexandros Hollender, Katerina Sotiraki, and Manolis Zampetakis. Consensus-halving: Does it ever get easier? In Proceedings of the 21st ACM Conference on Economics and Computation, pages 381-399, 2020. Google Scholar
  14. Duncan K Foley. Resource allocation and the public sector, 1967. Google Scholar
  15. Paul Goldberg, Alexandros Hollender, and Warut Suksompong. Contiguous cake cutting: Hardness results and approximation algorithms. Journal of Artificial Intelligence Research, 69:109-141, 2020. Google Scholar
  16. Hadi Hosseini, Ayumi Igarashi, and Andrew Searns. Fair division of time: Multi-layered cake cutting. In 29th International Joint Conference on Artificial Intelligence, IJCAI 2020, pages 182-188. International Joint Conferences on Artificial Intelligence, 2020. Google Scholar
  17. Duško Jojić, Gaiane Panina, and Rade Živaljević. Splitting necklaces, with constraints. SIAM Journal on Discrete Mathematics, 35(2):1268-1286, 2021. Google Scholar
  18. Spyros C Kontogiannis, Panagiota N Panagopoulou, and Paul G Spirakis. Polynomial algorithms for approximating nash equilibria of bimatrix games. In International Workshop on Internet and Network Economics, pages 286-296. Springer, 2006. Google Scholar
  19. Richard J Lipton, Evangelos Markakis, Elchanan Mossel, and Amin Saberi. On approximately fair allocations of indivisible goods. In Proceedings of the 5th ACM conference on Electronic commerce, pages 125-131. ACM, 2004. Google Scholar
  20. Gaiane Panina and Rade Živaljević. Envy-free division via configuration spaces. arXiv preprint, 2021. URL:
  21. Ariel D Procaccia. Cake cutting algorithms. In Handbook of Computational Social Choice, chapter 13. Citeseer, 2015. Google Scholar
  22. Jack Robertson and William Webb. Cake-cutting algorithms: Be fair if you can. AK Peters/CRC Press, 1998. Google Scholar
  23. FW Simmons. Private communication to Michael Starbird, 1980. Google Scholar
  24. Hugo Steinhaus. The Problem of Fair Division. Econometrica, 16:101-104, 1948. Google Scholar
  25. Walter Stromquist. How to cut a cake fairly. The American Mathematical Monthly, 87(8):640-644, 1980. Google Scholar
  26. Walter Stromquist. Envy-free cake divisions cannot be found by finite protocols. the electronic journal of combinatorics, 15(1):11, 2008. Google Scholar
  27. Francis Edward Su. Rental harmony: Sperner’s lemma in fair division. The American mathematical monthly, 106(10):930-942, 1999. Google Scholar
  28. Haralampos Tsaknakis and Paul G Spirakis. An optimization approach for approximate nash equilibria. In International Workshop on Web and Internet Economics, pages 42-56. Springer, 2007. Google Scholar
  29. Chenhao Wang and Xiaoying Wu. Cake cutting with single-peaked valuations. In Combinatorial Optimization and Applications: 13th International Conference, COCOA 2019, Xiamen, China, December 13-15, 2019, Proceedings 13, pages 507-516. Springer, 2019. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail