Parameterized Complexity of Incomplete Connected Fair Division

Authors Harmender Gahlawat , Meirav Zehavi

Thumbnail PDF


  • Filesize: 0.94 MB
  • 18 pages

Document Identifiers

Author Details

Harmender Gahlawat
  • Ben-Gurion University of the Negev, Beersheba, Israel
Meirav Zehavi
  • Ben-Gurion University of the Negev, Beersheba, Israel

Cite AsGet BibTex

Harmender Gahlawat and Meirav Zehavi. Parameterized Complexity of Incomplete Connected Fair Division. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Fair division of resources among competing agents is a fundamental problem in computational social choice and economic game theory. It has been intensively studied on various kinds of items (divisible and indivisible) and under various notions of fairness. We focus on Connected Fair Division (CFD), the variant of fair division on graphs, where the resources are modeled as an item graph. Here, each agent has to be assigned a connected subgraph of the item graph, and each item has to be assigned to some agent. We introduce a generalization of CFD, termed Incomplete CFD (ICFD), where exactly p vertices of the item graph should be assigned to the agents. This might be useful, in particular when the allocations are intended to be "economical" as well as fair. We consider four well-known notions of fairness: PROP, EF, EF1, EFX. First, we prove that EF-ICFD, EF1-ICFD, and EFX-ICFD are W[1]-hard parameterized by p plus the number of agents, even for graphs having constant vertex cover number (vcn). In contrast, we present a randomized FPT algorithm for PROP-ICFD parameterized only by p. Additionally, we prove both positive and negative results concerning the kernelization complexity of ICFD under all four fairness notions, parameterized by p, vcn, and the total number of different valuations in the item graph (val).

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Mathematics of computing → Graph algorithms
  • Fair Division
  • Kernelization
  • Connected Fair Allocation
  • Fixed parameter tractability


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


  1. Amir Abboud, Kevin Lewi, and Ryan Williams. Losing weight by gaining edges. In European Symposium on Algorithms, pages 1-12. Springer, 2014. Google Scholar
  2. Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. J. ACM, 42(4):844-856, July 1995. URL:
  3. Haris Aziz, Sylvain Bouveret, Ioannis Caragiannis, Ira Giagkousi, and Jérôme Lang. Knowledge, fairness, and social constraints. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018. Google Scholar
  4. Siddharth Barman and Sanath Kumar Krishnamurthy. Approximation algorithms for maximin fair division. ACM Transactions on Economics and Computation (TEAC), 8(1):1-28, 2020. Google Scholar
  5. Ben Berger, Avi Cohen, Michal Feldman, and Amos Fiat. Almost full efx exists for four agents. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, pages 4826-4833, 2022. Google Scholar
  6. Vittorio Bilò, Ioannis Caragiannis, Michele Flammini, Ayumi Igarashi, Gianpiero Monaco, Dominik Peters, Cosimo Vinci, and William S Zwicker. Almost envy-free allocations with connected bundles. Games and Economic Behavior, 131:197-221, 2022. Google Scholar
  7. Bernhard Bliem, Robert Bredereck, and Rolf Niedermeier. Complexity of efficient and envy-free resource allocation: Few agents, resources, or utility levels. In IJCAI, pages 102-108, 2016. Google Scholar
  8. Niclas Boehmer, Robert Bredereck, Klaus Heeger, Dusan Knop, and Junjie Luo. Multivariate algorithmics for eliminating envy by donating goods. In Piotr Faliszewski, Viviana Mascardi, Catherine Pelachaud, and Matthew E. Taylor, editors, 21st International Conference on Autonomous Agents and Multiagent Systems, AAMAS 2022, Auckland, New Zealand, May 9-13, 2022, pages 127-135. International Foundation for Autonomous Agents and Multiagent Systems (IFAAMAS), 2022. URL:
  9. Sylvain Bouveret, Katarína Cechlárová, Edith Elkind, Ayumi Igarashi, and Dominik Peters. Fair division of a graph. In Proceedings of the 26th International Joint Conference on Artificial Intelligence, IJCAI'17, pages 135-141. AAAI Press, 2017. Google Scholar
  10. Sylvain Bouveret and Jérôme Lang. Efficiency and envy-freeness in fair division of indivisible goods: Logical representation and complexity. J. Artif. Int. Res., 32(1):525-564, June 2008. Google Scholar
  11. Sylvain Bouveret and Michel Lemaître. Characterizing conflicts in fair division of indivisible goods using a scale of criteria. Autonomous Agents and Multi-Agent Systems, 30(2):259-290, 2016. Google Scholar
  12. Steven J Brams, Paul H Edelman, and Peter C Fishburn. Fair division of indivisible items. Theory and Decision, 55(2):147-180, 2003. Google Scholar
  13. Steven J Brams and Alan D Taylor. Fair Division: From cake-cutting to dispute resolution. Cambridge University Press, 1996. Google Scholar
  14. Eric Budish. The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. Journal of Political Economy, 119(6):1061-1103, 2011. Google Scholar
  15. Ioannis Caragiannis, Nick Gravin, and Xin Huang. Envy-freeness up to any item with high nash welfare: The virtue of donating items. In Proceedings of the 2019 ACM Conference on Economics and Computation, pages 527-545, 2019. Google Scholar
  16. Ioannis Caragiannis, David Kurokawa, Hervé Moulin, Ariel D Procaccia, Nisarg Shah, and Junxing Wang. The unreasonable fairness of maximum nash welfare. ACM Transactions on Economics and Computation (TEAC), 7(3):1-32, 2019. Google Scholar
  17. Bhaskar Ray Chaudhury, Jugal Garg, and Kurt Mehlhorn. Efx exists for three agents. In Proceedings of the 21st ACM Conference on Economics and Computation, pages 1-19, 2020. Google Scholar
  18. Bhaskar Ray Chaudhury, Telikepalli Kavitha, Kurt Mehlhorn, and Alkmini Sgouritsa. A little charity guarantees almost envy-freeness. SIAM Journal on Computing, 50(4):1336-1358, 2021. Google Scholar
  19. Marek Cygan, Fedor V Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized algorithms, volume 5. Springer, 2015. Google Scholar
  20. Argyrios Deligkas, Eduard Eiben, Robert Ganian, Thekla Hamm, and Sebastian Ordyniak. The parameterized complexity of connected fair division. In IJCAI, pages 139-145, 2021. Google Scholar
  21. Argyrios Deligkas, Eduard Eiben, Robert Ganian, Thekla Hamm, and Sebastian Ordyniak. The complexity of envy-free graph cutting. In Proceedings of the 31st International Joint Conference on Artificial Intelligence (IJCAI), pages 237-243, 2022. Google Scholar
  22. Reinhard Diestel. Graph theory. Springer Publishing Company, Incorporated, 2018. Google Scholar
  23. Michael Dom, Daniel Lokshtanov, and Saket Saurabh. Incompressibility through colors and ids. In International Colloquium on Automata, Languages, and Programming, pages 378-389. Springer, 2009. Google Scholar
  24. Eduard Eiben, Robert Ganian, Thekla Hamm, and Sebastian Ordyniak. Parameterized complexity of envy-free resource allocation in social networks. Artificial Intelligence, 315:103826, 2023. Google Scholar
  25. Fedor V Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: theory of parameterized preprocessing. Cambridge University Press, 2019. Google Scholar
  26. Harmender Gahlawat, , and Meirav Zehavi. Kernels for the disjoint paths problem on subclasses of chordal graphs. arXiv, 2023. URL:
  27. Gianluigi Greco and Francesco Scarcello. The complexity of computing maximin share allocations on graphs. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pages 2006-2013, 2020. Google Scholar
  28. Ayumi Igarashi and Dominik Peters. Pareto-optimal allocation of indivisible goods with connectivity constraints. In Proceedings of the AAAI conference on artificial intelligence, volume 33, pages 2045-2052, 2019. Google Scholar
  29. David Kurokawa, Ariel D Procaccia, and Junxing Wang. Fair enough: Guaranteeing approximate maximin shares. Journal of the ACM (JACM), 65(2):1-27, 2018. Google Scholar
  30. 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, 2004. Google Scholar
  31. Moni Naor, Leonard J Schulman, and Aravind Srinivasan. Splitters and near-optimal derandomization. In Proceedings of IEEE 36th Annual Foundations of Computer Science, pages 182-191. IEEE, 1995. Google Scholar
  32. Benjamin Plaut and Tim Roughgarden. Almost envy-freeness with general valuations. SIAM Journal on Discrete Mathematics, 34(2):1039-1068, 2020. Google Scholar
  33. Tayfun Sönmez and M Utku Ünver. Course bidding at business schools. International Economic Review, 51(1):99-123, 2010. Google Scholar
  34. David P Williamson and David B Shmoys. The design of approximation algorithms. Cambridge university press, 2011. Google Scholar