Parameterized Complexity of Fair Bisection: (FPT-Approximation meets Unbreakability)

Authors Tanmay Inamdar , Daniel Lokshtanov, Saket Saurabh , Vaishali Surianarayanan

Thumbnail PDF


  • Filesize: 1.04 MB
  • 17 pages

Document Identifiers

Author Details

Tanmay Inamdar
  • University of Bergen, Norway
Daniel Lokshtanov
  • University of California Santa Barbara, CA, USA
Saket Saurabh
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
  • University of Bergen, Norway
Vaishali Surianarayanan
  • University of California Santa Barbara, CA, USA

Cite AsGet BibTex

Tanmay Inamdar, Daniel Lokshtanov, Saket Saurabh, and Vaishali Surianarayanan. Parameterized Complexity of Fair Bisection: (FPT-Approximation meets Unbreakability). In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 63:1-63:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


In the Minimum Bisection problem input is a graph G and the goal is to partition the vertex set into two parts A and B, such that ||A|-|B|| ≤ 1 and the number k of edges between A and B is minimized. The problem is known to be NP-hard, and assuming the Unique Games Conjecture even NP-hard to approximate within a constant factor [Khot and Vishnoi, J.ACM'15]. On the other hand, a 𝒪(log n)-approximation algorithm [Räcke, STOC'08] and a parameterized algorithm [Cygan et al., ACM Transactions on Algorithms'20] running in time k^𝒪(k) n^𝒪(1) is known. The Minimum Bisection problem can be viewed as a clustering problem where edges represent similarity and the task is to partition the vertices into two equally sized clusters while minimizing the number of pairs of similar objects that end up in different clusters. Motivated by a number of egregious examples of unfair bias in AI systems, many fundamental clustering problems have been revisited and re-formulated to incorporate fairness constraints. In this paper we initiate the study of the Minimum Bisection problem with fairness constraints. Here the input is a graph G, positive integers c and k, a function χ:V(G) → {1, …, c} that assigns a color χ(v) to each vertex v in G, and c integers r_1,r_2,⋯,r_c. The goal is to partition the vertex set of G into two almost-equal sized parts A and B with at most k edges between them, such that for each color i ∈ {1, …, c}, A has exactly r_i vertices of color i. Each color class corresponds to a group which we require the partition (A, B) to treat fairly, and the constraints that A has exactly r_i vertices of color i can be used to encode that no group is over- or under-represented in either of the two clusters. We first show that introducing fairness constraints appears to make the Minimum Bisection problem qualitatively harder. Specifically we show that unless FPT=W[1] the problem admits no f(c)n^𝒪(1) time algorithm even when k = 0. On the other hand, our main technical contribution shows that is that this hardness result is simply a consequence of the very strict requirement that each color class i has exactly r_i vertices in A. In particular we give an f(k,c,ε)n^𝒪(1) time algorithm that finds a balanced partition (A, B) with at most k edges between them, such that for each color i ∈ [c], there are at most (1±ε)r_i vertices of color i in A. Our approximation algorithm is best viewed as a proof of concept that the technique introduced by [Lampis, ICALP'18] for obtaining FPT-approximation algorithms for problems of bounded tree-width or clique-width can be efficiently exploited even on graphs of unbounded width. The key insight is that the technique of Lampis is applicable on tree decompositions with unbreakable bags (as introduced in [Cygan et al., SIAM Journal on Computing'14]). An important ingredient of our approximation scheme is a combinatorial result that may be of independent interest, namely that for every k, every graph G admits a tree decomposition with adhesions of size at most 𝒪(k), unbreakable bags, and logarithmic depth.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • FPT Approximation
  • Minimum Bisection
  • Unbreakable Tree Decomposition
  • Treewidth


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


  1. Sayan Bandyapadhyay, Fedor V. Fomin, and Kirill Simonov. On coresets for fair clustering in metric and euclidean spaces and their applications. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 23:1-23:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL:
  2. Sayan Bandyapadhyay, Tanmay Inamdar, Shreyas Pai, and Kasturi R. Varadarajan. A constant approximation for colorful k-center. In Michael A. Bender, Ola Svensson, and Grzegorz Herman, editors, 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany, volume 144 of LIPIcs, pages 12:1-12:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL:
  3. Johannes Blömer, Christiane Lammersen, Melanie Schmidt, and Christian Sohler. Theoretical analysis of the k-means algorithm-a survey. In Algorithm Engineering, pages 81-116. Springer, 2016. Google Scholar
  4. Hans L. Bodlaender and Torben Hagerup. Parallel algorithms with optimal speedup for bounded treewidth. SIAM J. Comput., 27(6):1725-1746, 1998. URL:
  5. Mikolaj Bojanczyk and Michal Pilipczuk. Definability equals recognizability for graphs of bounded treewidth. In Martin Grohe, Eric Koskinen, and Natarajan Shankar, editors, Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '16, New York, NY, USA, July 5-8, 2016, pages 407-416. ACM, 2016. URL:
  6. Xingyu Chen, Brandon Fain, Liang Lyu, and Kamesh Munagala. Proportionally fair clustering. In Kamalika Chaudhuri and Ruslan Salakhutdinov, editors, Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA, volume 97 of Proceedings of Machine Learning Research, pages 1032-1041. PMLR, 2019. URL:
  7. Yixin Chen, Ya Zhang, and Xiang Ji. Size regularized cut for data clustering. In Advances in Neural Information Processing Systems 18 [Neural Information Processing Systems, NIPS 2005, December 5-8, 2005, Vancouver, British Columbia, Canada], pages 211-218, 2005. URL:
  8. Rajesh Chitnis, Marek Cygan, MohammadTaghi Hajiaghayi, Marcin Pilipczuk, and Michal Pilipczuk. Designing FPT algorithms for cut problems using randomized contractions. SIAM J. Comput., 45(4):1171-1229, 2016. Google Scholar
  9. Marek Cygan, Pawel Komosa, Daniel Lokshtanov, Marcin Pilipczuk, Michal Pilipczuk, Saket Saurabh, and Magnus Wahlström. Randomized contractions meet lean decompositions. ACM Trans. Algorithms, 17(1):6:1-6:30, 2021. URL:
  10. Marek Cygan, Daniel Lokshtanov, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Minimum bisection is fixed-parameter tractable. SIAM J. Comput., 48(2):417-450, 2019. URL:
  11. Jefferey Dastin. Amazon scraps secret ai recruiting tool that showed bias against women. Reuters, 2018. URL:
  12. Sorelle A. Friedler, Carlos Scheidegger, and Suresh Venkatasubramanian. The (im)possibility of fairness: different value systems require different mechanisms for fair decision making. Commun. ACM, 64(4):136-143, 2021. URL:
  13. M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  14. Mehrdad Ghadiri, Samira Samadi, and Santosh S. Vempala. Socially fair k-means clustering. In Madeleine Clare Elish, William Isaac, and Richard S. Zemel, editors, FAccT '21: 2021 ACM Conference on Fairness, Accountability, and Transparency, Virtual Event / Toronto, Canada, March 3-10, 2021, pages 438-448. ACM, 2021. URL:
  15. Patrick J Grother, Patrick J Grother, Mei Ngan, and K Hanaoka. Face recognition vendor test (FRVT). US Department of Commerce, National Institute of Standards and Technology, 2014. Google Scholar
  16. Rudolf Halin. Tree-partitions of infinite graphs. Discret. Math., 97(1-3):203-217, 1991. URL:
  17. Lingxiao Huang, Shaofeng H.-C. Jiang, and Nisheeth K. Vishnoi. Coresets for clustering with fairness constraints. In Hanna M. Wallach, Hugo Larochelle, Alina Beygelzimer, Florence d'Alché-Buc, Emily B. Fox, and Roman Garnett, editors, Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, pages 7587-7598, 2019. URL:
  18. Xinrui Jia, Kshiteej Sheth, and Ola Svensson. Fair colorful k-center clustering. In Daniel Bienstock and Giacomo Zambelli, editors, Integer Programming and Combinatorial Optimization - 21st International Conference, IPCO 2020, London, UK, June 8-10, 2020, Proceedings, volume 12125 of Lecture Notes in Computer Science, pages 209-222. Springer, 2020. URL:
  19. Subhash Khot and Nisheeth K. Vishnoi. The unique games conjecture, integrality gap for cut problems and embeddability of negative-type metrics into l_1. J. ACM, 62(1):8:1-8:39, 2015. URL:
  20. Michael Lampis. Parameterized approximation schemes using graph widths. In Javier Esparza, Pierre Fraigniaud, Thore Husfeldt, and Elias Koutsoupias, editors, Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, volume 8572 of Lecture Notes in Computer Science, pages 775-786. Springer, 2014. URL:
  21. Daniel Lokshtanov, M. S. Ramanujan, Saket Saurabh, and Meirav Zehavi. Parameterized complexity and approximability of directed odd cycle transversal. In Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 2181-2200. SIAM, 2020. URL:
  22. Yury Makarychev and Ali Vakilian. Approximation algorithms for socially fair clustering. In Mikhail Belkin and Samory Kpotufe, editors, Conference on Learning Theory, COLT 2021, 15-19 August 2021, Boulder, Colorado, USA, volume 134 of Proceedings of Machine Learning Research, pages 3246-3264. PMLR, 2021. URL:
  23. Dániel Marx. Can you beat treewidth? Theory Comput., 6(1):85-112, 2010. URL:
  24. Ziad Obermeyer, Brian Powers, Christine Vogeli, and Sendhil Mullainathan. Dissecting racial bias in an algorithm used to manage the health of populations. Science, 366(6464):447-453, 2019. Google Scholar
  25. Evelien Otte and Ronald Rousseau. Social network analysis: a powerful strategy, also for the information sciences. Journal of information Science, 28(6):441-453, 2002. Google Scholar
  26. Harald Räcke. Optimal hierarchical decompositions for congestion minimization in networks. In Cynthia Dwork, editor, Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17-20, 2008, pages 255-264. ACM, 2008. URL:
  27. Lior Rokach. A survey of clustering algorithms. In Data mining and knowledge discovery handbook, pages 269-298. Springer, 2009. Google Scholar
  28. Melanie Schmidt, Chris Schwiegelshohn, and Christian Sohler. Fair coresets and streaming algorithms for fair k-means. In Evripidis Bampis and Nicole Megow, editors, Approximation and Online Algorithms - 17th International Workshop, WAOA 2019, Munich, Germany, September 12-13, 2019, Revised Selected Papers, volume 11926 of Lecture Notes in Computer Science, pages 232-251. Springer, 2019. URL:
  29. Detlef Seese. Tree-partite graphs and the complexity of algorithms. In Lothar Budach, editor, Fundamentals of Computation Theory, FCT '85, Cottbus, GDR, September 9-13, 1985, volume 199 of Lecture Notes in Computer Science, pages 412-421. Springer, 1985. URL:
  30. Dongkuan Xu and Yingjie Tian. A comprehensive survey of clustering algorithms. Annals of Data Science, 2(2):165-193, 2015. Google Scholar
  31. Jin-Tai Yan and Pei-Yung Hsiao. A fuzzy clustering algorithm for graph bisection. Inf. Process. Lett., 52(5):259-263, 1994. URL:
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