Document Open Access Logo

Scarf’s Algorithm on Arborescence Hypergraphs

Authors Karthekeyan Chandrasekaran , Yuri Faenza , Chengyue He , Jay Sethuraman

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2025.45.pdf
  • Filesize: 0.81 MB
  • 18 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Scarf’s algorithm
  • Arborescence Hypergraphs
  • Stable Matchings

Metrics

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

Abstract

Scarf’s algorithm - a pivoting procedure that finds a dominating extreme point in a down-monotone polytope - can be used to show the existence of a fractional stable matching in hypergraphs. The problem of finding a fractional stable matching in hypergraphs, however, is PPAD-complete. In this work, we study the behavior of Scarf’s algorithm on arborescence hypergraphs, the family of hypergraphs in which hyperedges correspond to the paths of an arborescence. For arborescence hypergraphs, we prove that Scarf’s algorithm can be implemented to find an integral stable matching in polynomial time. En route to our result, we uncover novel structural properties of bases and pivots for the more general family of network hypergraphs. Our work provides the first proof of polynomial-time convergence of Scarf’s algorithm on hypergraphic stable matching problems, giving hope to the possibility of polynomial-time convergence of Scarf’s algorithm for other families of polytopes.

Cite As Get BibTex

Karthekeyan Chandrasekaran, Yuri Faenza, Chengyue He, and Jay Sethuraman. Scarf’s Algorithm on Arborescence Hypergraphs. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 45:1-45:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.45

Author Details

Karthekeyan Chandrasekaran
  • Grainger College of Engineering, University of Illinois, Urbana-Champaign, IL, USA
Yuri Faenza
  • Columbia University, New York, NY, USA
Chengyue He
  • Columbia University, New York, NY, USA
Jay Sethuraman
  • Columbia University, New York, NY, USA

Funding

  • Chandrasekaran, Karthekeyan: Karthekeyan Chandrasekaran is supported in part by NSF grant CCF-2402667.
  • Faenza, Yuri: Yuri Faenza is supported by NSF Grant 2046146 and by a Meta Research Award.
  • He, Chengyue: Chengyue He is supported by NSF Grant 2046146 and by a Meta Research Award.

Acknowledgements

The first three authors gratefully acknowledge the organizers of the ICERM Spring 2023 semester program on Discrete Optimization: Mathematics, Algorithms, and Computation, where this project was initiated.

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. Ron Aharoni and Ron Holzman. Fractional kernels in digraphs. J. Combinatorial Theory, 73(1):1-6, 1998. URL: https://doi.org/10.1006/JCTB.1997.1731.
  3. Ravindra K Ahuja, James B Orlin, Prabha Sharma, and PT Sokkalingam. A network simplex algorithm with o (n) consecutive degenerate pivots. Operations research letters, 30(3):141-148, 2002. URL: https://doi.org/10.1016/S0167-6377(02)00114-1.
  4. Mourad Baïou and Michel Balinski. The stable admissions polytope. Mathematical programming, 87:427-439, 2000. URL: https://doi.org/10.1007/S101070050004.
  5. Mourad Baïou and Michel Balinski. The stable allocation (or ordinal transportation) problem. Mathematics of Operations Research, 27(3):485-503, 2002. URL: https://doi.org/10.1287/MOOR.27.3.485.310.
  6. Dimitris Bertsimas and John N Tsitsiklis. Introduction to linear optimization, volume 6. Athena Scientific Belmont, MA, 1997. Google Scholar
  7. Péter Biró, Gergely Csáji, and Ildikó Schlotter. Stable hypergraph matching in unimodular hypergraphs. arXiv preprint arXiv:2502.08827, 2025. URL: https://doi.org/10.48550/arXiv.2502.08827.
  8. 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.
  9. Péter Biró, Tamás Fleiner, and Robert W Irving. Matching couples with scarf’s algorithm. Annals of Mathematics and Artificial Intelligence, 77:303-316, 2016. URL: https://doi.org/10.1007/S10472-015-9491-5.
  10. Péter Biró, David F Manlove, and Iain McBride. The hospitals/residents problem with couples: Complexity and integer programming models. In Experimental Algorithms: 13th International Symposium, SEA 2014, Copenhagen, Denmark, June 29-July 1, 2014. Proceedings 13, pages 10-21. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-07959-2_2.
  11. Karthekeyan Chandrasekaran, Yuri Faenza, Chengyue He, and Jay Sethuraman. Scarf’s algorithm on arborescence hypergraphs. arXiv preprint arXiv:2412.03397, 2024. URL: https://doi.org/10.48550/arXiv.2412.03397.
  12. Thomas H Cormen, Charles E Leiserson, Ronald L Rivest, and Clifford Stein. Introduction to algorithms. MIT press, 2022. Google Scholar
  13. 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.
  14. William H Cunningham. A network simplex method. Mathematical Programming, 11:105-116, 1976. URL: https://doi.org/10.1007/BF01580379.
  15. Piotr Dworczak. Deferred acceptance with compensation chains. In Proceedings of the 2016 ACM Conference on Economics and Computation, pages 65-66, 2016. URL: https://doi.org/10.1145/2940716.2940727.
  16. Yuri Faenza, Chengyue He, and Jay Sethuraman. Scarf’s algorithm and stable marriages. Mathematics of Operations Research, 2025. Accepted for publication. URL: https://doi.org/10.1287/moor.2023.0055.
  17. Tamás Fleiner. A fixed-point approach to stable matchings and some applications. Mathematics of Operations research, 28(1):103-126, 2003. URL: https://doi.org/10.1287/MOOR.28.1.103.14256.
  18. Delbert Fulkerson and Oliver Gross. Incidence matrices and interval graphs. Pacific journal of mathematics, 15(3):835-855, 1965. Google Scholar
  19. David Gale and Lloyd S Shapley. College admissions and the stability of marriage. The American Mathematical Monthly, 69(1):9-15, 1962. Google Scholar
  20. Penny E Haxell and Gordon T Wilfong. A fractional model of the border gateway protocol (bgp). In Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms, pages 193-199. Citeseer, 2008. URL: http://dl.acm.org/citation.cfm?id=1347082.1347104.
  21. Takashi Ishizuka and Naoyuki Kamiyama. On the complexity of stable fractional hypergraph matching. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Schloss-Dagstuhl-Leibniz Zentrum für Informatik, 2018. Google Scholar
  22. 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.
  23. Donald Ervin Knuth. Marriages stables. Technical report, 1976. Google Scholar
  24. Cheng Ng and Daniel S Hirschberg. Three-dimensional stabl matching problems. SIAM Journal on Discrete Mathematics, 4(2):245-252, 1991. URL: https://doi.org/10.1137/0404023.
  25. Hai Nguyen, Thành Nguyen, and Alexander Teytelboym. Stability in matching markets with complex constraints. Management Science, 67(12):7438-7454, 2021. URL: https://doi.org/10.1287/MNSC.2020.3869.
  26. Thanh Nguyen and Rakesh Vohra. Near-feasible stable matchings with couples. American Economic Review, 108(11):3154-3169, 2018. Google Scholar
  27. Thành Nguyen and Rakesh Vohra. Stable matching with proportionality constraints. Operations Research, 67(6):1503-1519, 2019. URL: https://doi.org/10.1287/OPRE.2019.1909.
  28. Adèle Pass-Lanneau, Ayumi Igarashi, and Frédéric Meunier. Perfect graphs with polynomially computable kernels. Discrete Applied Mathematics, 272:69-74, 2020. URL: https://doi.org/10.1016/J.DAM.2018.09.027.
  29. Alvin E Roth. Online and matching-based market design. Cambridge University Press, 2023. Google Scholar
  30. Alvin E Roth, Uriel G Rothblum, and John H Vande Vate. Stable matchings, optimal assignments, and linear programming. Mathematics of operations research, 18(4):803-828, 1993. URL: https://doi.org/10.1287/MOOR.18.4.803.
  31. Uriel G Rothblum. Characterization of stable matchings as extreme points of a polytope. Mathematical Programming, 54:57-67, 1992. URL: https://doi.org/10.1007/BF01586041.
  32. Herbert E Scarf. The core of an n person game. Econometrica: Journal of the Econometric Society, pages 50-69, 1967. Google Scholar
  33. Alexander Schrijver. Theory of linear and integer programming. John Wiley & Sons, 1998. Google Scholar
  34. Alexander Schrijver et al. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer, 2003. Google Scholar
  35. Chung-Piaw Teo and Jay Sethuraman. The geometry of fractional stable matchings and its applications. Mathematics of Operations Research, 23(4):874-891, 1998. URL: https://doi.org/10.1287/MOOR.23.4.874.
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