Document Open Access Logo

Stable Hypergraph Matching in Unimodular Hypergraphs

Authors Péter Biró , Gergely Csáji , Ildikó Schlotter

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2025.31.pdf
  • Filesize: 0.89 MB
  • 20 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Algorithmic mechanism design
  • Mathematics of computing → Hypergraphs
Keywords
  • stable hypergraph matching
  • Scarf’s Lemma
  • unimodular hypergraphs
  • university dual admission

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We study the NP-hard Stable Hypergraph Matching (SHM) problem and its generalization allowing capacities, the Stable Hypergraph b-Matching (SHbM) problem, and investigate their computational properties under various structural constraints. Our study is motivated by the fact that Scarf’s Lemma [Scarf, 1967] together with a result of Lovász [Lovász, 1972] guarantees the existence of a stable matching whenever the underlying hypergraph is normal. Furthermore, if the hypergraph is unimodular (i.e., its incidence matrix is totally unimodular), then even a stable b-matching is guaranteed to exist. However, no polynomial-time algorithm is known for finding a stable matching or b-matching in unimodular hypergraphs.
We identify subclasses of unimodular hypergraphs where SHM and SHbM are tractable such as laminar hypergraphs or so-called subpath hypergraphs with bounded-size hyperedges; for the latter case, even a maximum-weight stable b-matching can be found efficiently. We complement our algorithms by showing that optimizing over stable matchings is NP-hard even in laminar hypergraphs. As a practically important special case of SHbM for unimodular hypergraphs, we investigate a tripartite stable matching problem with students, schools, and companies as agents, called the University Dual Admission problem, which models real-world scenarios in higher education admissions. 
Finally, we examine a superclass of subpath hypergraphs that are normal but not necessarily unimodular, namely subtree hypergraphs where hyperedges correspond to subtrees of a tree. We establish that for such hypergraphs, stable matchings can be found in polynomial time but, in the setting with capacities, finding a stable b-matching is NP-hard.

Cite As Get BibTex

Péter Biró, Gergely Csáji, and Ildikó Schlotter. Stable Hypergraph Matching in Unimodular Hypergraphs. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 31:1-31:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.31

Author Details

Péter Biró
  • HUN-REN Centre for Economic and Regional Studies, Budapest, Hungary
Gergely Csáji
  • HUN-REN Centre for Economic and Regional Studies, Budapest, Hungary
  • Eötvös Loránd University, Budapest, Hungary
Ildikó Schlotter
  • HUN-REN Centre for Economic and Regional Studies, Budapest, Hungary

Funding

Supported by the Hungarian Academy of Sciences under Momentum grant no. LP2021-2.
  • Biró, Péter: Supported by the Hungarian Scientific Research Fund (OTKA grant no. K143858).
  • Csáji, Gergely: Supported by the Hungarian Scientific Research Fund (OTKA grant no. K143858) and by the Ministry of Culture and Innovation of Hungary from the National Research, Development and Innovation fund, financed under the KDP-2023 funding scheme (grant no. C2258525).
  • Schlotter, Ildikó: Supported by the Hungarian Academy of Sciences under its János Bolyai Research Scholarship.

Acknowledgements

We are grateful for the reviewers of ICALP 2025 for their valuable comments.

References

  1. Ron Aharoni and Tamás Fleiner. On a lemma of Scarf. Journal of Combinatorial Theory, Series B, 87(1):72-80, 2003. URL: https://doi.org/10.1016/S0095-8956(02)00028-X.
  2. Haris Aziz and Florian Brandl. Existence of stability in hedonic coalition formation games. In Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2012), volume 2, pages 763-770, 2012. URL: http://dl.acm.org/citation.cfm?id=2343806.
  3. Suryapratim Banerjee, Hideo Konishi, and Tayfun Sönmez. Core in a simple coalition formation game. Social Choice and Welfare, 18(1):135-153, 2001. URL: https://doi.org/10.1007/s003550000067.
  4. Péter Biró, Gergely Csáji, and Ildikó Schlotter. Stable hypergraph matching in unimodular hypergraphs. arXiv:2502.08827 [cs.GT], 2025. URL: https://doi.org/10.48550/arXiv.2502.08827.
  5. Péter Biró and Tamás Fleiner. Fractional solutions for capacitated NTU-games, with applications to stable matchings. Discrete Optimization, 22:241-254, 2016. URL: https://doi.org/10.1016/j.disopt.2015.02.002.
  6. Alain Bretto. Hypergraph Theory: An Introduction. Mathematical Engineering. Springer Cham, 2013. Google Scholar
  7. Karthekeyan Chandrasekaran, Yuri Faenza, Chengyue He, and Jay Sethuraman. Scarf’s algorithm on arborescence hypergraphs. arXiv:2412.03397 [cs.DM], 2024. URL: https://doi.org/10.48550/arXiv.2412.03397.
  8. Gergely Csáji. On the complexity of stable hypergraph matching, stable multicommodity flow and related problems. Theoretical Computer Science, 931:1-16, 2022. URL: https://doi.org/10.1016/j.tcs.2022.07.025.
  9. Yuri Faenza, Chengyue He, and Jay Sethuraman. Scarf’s algorithm and stable marriages. arXiv:2303.00791 [math.CO], 2023. URL: https://doi.org/10.48550/arXiv.2303.00791.
  10. Rita Fleiner, András Ferkai, and Péter Biró. College admission problem for university dual education. In Proceedings of the IEEE 17th World Symposium on Applied Machine Intelligence and Informatics (SAMI 2019), pages 31-36. IEEE, 2019. URL: https://doi.org/10.1109/SAMI.2019.8782783.
  11. Tamás Fleiner. On the stable b-matching polytope. Mathematical Social Sciences, 46(2):149-158, 2003. URL: https://doi.org/10.1016/S0165-4896(03)00074-X.
  12. David Gale and Lloyd S. Shapley. College admissions and the stability of marriage. The American Mathematical Monthly, 69(1):9-15, 1962. URL: https://doi.org/10.2307/2312726.
  13. Guillaume Haeringer. Market Design: Auctions and Matching. MIT Press, 2018. Google Scholar
  14. Shiva Kintali, Laura J. Poplawski, Rajmohan Rajaraman, Ravi Sundaram, and Shang-Hua Teng. Reducibility among fractional stability problems. SIAM Journal on Computing, 42(6):2063-2113, 2013. URL: https://doi.org/10.1137/120874655.
  15. László Lovász. Normal hypergraphs and the perfect graph conjecture. Discrete Mathematics, 2(3):253-267, 1972. URL: https://doi.org/10.1016/0012-365X(72)90006-4.
  16. David F. Manlove. Algorithmics of Matching Under Preferences, volume 2 of Series on Theoretical Computer Science. World Scientific, 2013. URL: https://doi.org/10.1142/8591.
  17. Alvin E. Roth. The evolution of the labor market for medical interns and residents: A case study in game theory. Journal of Political Economy, 92(6):991-1016, 1984. URL: https://doi.org/10.1086/261272.
  18. Alvin E. Roth and Marilda A. O. Sotomayor. Two-sided Matching: a Study in Game-theoretic Modeling and Analysis. Econometric Society Monographs, Cambridge, 1990. Google Scholar
  19. Herbert E. Scarf. The core of an N person game. Econometrica, 35(1):50-69, 1967. URL: https://doi.org/10.2307/1909383.
  20. Alexander Schrijver. Combinatorial Optimization. Springer-Verlag Berlin Heidelberg, 2003. Google Scholar
  21. Gerhard J. Woeginger. Core stability in hedonic coalition formation. In Proceedings of the 39th International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM 2013), volume 7741 of Lecture Notes in Computer Science, pages 33-50. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-35843-2_4.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact