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Finding perfect matchings in bipartite hypergraphs

  • Chidambaram Annamalai
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Abstract

Haxell’s condition [14] is a natural hypergraph analog of Hall’s condition, which is a wellknown necessary and sufficient condition for a bipartite graph to admit a perfect matching. That is, when Haxell’s condition holds it forces the existence of a perfect matching in the bipartite hypergraph. Unlike in graphs, however, there is no known polynomial time algorithm to find the hypergraph perfect matching that is guaranteed to exist when Haxell’s condition is satisfied.

We prove the existence of an efficient algorithm to find perfect matchings in bipartite hypergraphs whenever a stronger version of Haxell’s condition holds. Our algorithm can be seen as a generalization of the classical Hungarian algorithm for finding perfect matchings in bipartite graphs. The techniques we use to achieve this result could be of use more generally in other combinatorial problems on hypergraphs where disjointness structure is crucial, e.g., Set Packing.

Mathematics Subject Classification (2000)

05C65 05C70 05C85 

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References

  1. [1]
    N. Alon, P. Frankl, H. Huang, V. Rödl, A. Ruciński and B. Sudakov: Large matchings in uniform hypergraphs and the conjectures of Erdős and Samuels, Journal of Combinatorial Theory, Series A 119 (2012), 1200–1215.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    A. Asadpour, U. Feige and A. Saberi: Santa claus meets hypergraph matchings, ACM Transactions on Algorithms (TALG) 8 (2012), 24.MathSciNetzbMATHGoogle Scholar
  3. [3]
    C. Annamalai, C. Kalaitzis and O. Svensson: Combinatorial algorithm for restricted max-min fair allocation, in: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, 1357–1372, 2015.CrossRefGoogle Scholar
  4. [4]
    N. Bansal and M. Sviridenko: The santa claus problem, in: Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, 31–40. ACM, 2006.Google Scholar
  5. [5]
    M. Conforti, G. Cornuéjols, A. Kapoor and K. Vušković: Perfect matchings in balanced hypergraphs, Combinatorica 16 (1996), 325–329.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    M. Cygan, F. Grandoni and M. Mastrolilli: How to sell hyperedges: The hyper-matching assignment problem, in: Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, 342–351. SIAM, 2013.CrossRefGoogle Scholar
  7. [7]
    Y. H. Chan and L. C. Lau: On linear and semidefinite programming relaxations for hypergraph matching, Mathematical Programming 135 (2012), 123–148.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    M. Cygan: Improved approximation for 3-dimensional matching via bounded path-width local search, in: Proceedings of the Fifty-Fourth Annual Symposium on Foundations of Computer Science, 509–518. IEEE, 2013.Google Scholar
  9. [9]
    G. A. Dirac: Some theorems on abstract graphs, Proceedings of the London Mathematical Society 3 (1952), 69–81.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    U. Feige: On allocations that maximize fairness, in: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 287–293. Society for Industrial and Applied Mathematics, 2008.Google Scholar
  11. [11]
    M. Fürer and H. Yu: Approximating the k-set packing problem by local improvements, in: Combinatorial Optimization, 408–420. Springer, 2014.Google Scholar
  12. [12]
    P. Hall: On representatives of subsets, J. London Math. Soc 10 (1935), 26–30.CrossRefzbMATHGoogle Scholar
  13. [13]
    M. M. Halldórsson: Approximating discrete collections via local improvements, in: Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, volume 95, 160–169. SIAM, 1995.MathSciNetzbMATHGoogle Scholar
  14. [14]
    P. E. Haxell: A condition for matchability in hypergraphs, Graphs and Combinatorics 11 (1995), 245–248.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    J. E. Hopcroft and R. M. Karp: An n 5/2 algorithm for maximum matchings in bipartite graphs, SIAM Journal on Computing 2 (1973), 225–231.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    G. H. Hardy and S. Ramanujan: Asymptotic formula in combinatory analysis, Proceedings of the London Mathematical Society, s2 17 (1918), 75.CrossRefzbMATHGoogle Scholar
  17. [17]
    C. A. J. Hurkens and A. Schrijver: On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems, SIAM Journal on Discrete Mathematics 2 (1989), 68–72.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    B. Haeupler, B. Saha and A. Srinivasan: New constructive aspects of the lovasz local lemma, Journal of the ACM (JACM) 58 (2011), 28.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    R. M. Karp: Reducibility among combinatorial problems, Springer, 1972.CrossRefzbMATHGoogle Scholar
  20. [20]
    A. V. Karzanov: O nakhozhdenii maksimal’nogo potoka v setyakh spetsial’nogo vida i nekotorykh prilozheniyakh, Matematicheskie Voprosy Upravleniya Proizvodstvom 5 (1973), 81.Google Scholar
  21. [21]
    L. Lovász: On determinants, matchings, and random algorithms, in: FCT, volume 79, 565–574, 1979.MathSciNetzbMATHGoogle Scholar
  22. [22]
    A. Madry: Navigating central path with electrical flows: From flows to matchings, and back, in: Proceedings of the Fifty-Fourth Annual Symposium on Foundations of Computer Science, 253–262. IEEE, 2013.Google Scholar
  23. [23]
    M. Mucha and P. Sankowski: Maximum matchings via gaussian elimination, in: Proceedings of the Forty-Fifth Annual Symposium on Foundations of Computer Science, 248–255. IEEE, 2004.Google Scholar
  24. [24]
    L. Polacek and O. Svensson: Quasi-polynomial local search for restricted maxmin fair allocation, in: Automata, Languages, and Programming, 726–737. Springer, 2012.CrossRefGoogle Scholar
  25. [25]
    M. Singh and K. Talwar: Improving integrality gaps via Chvátal-Gomory rounding, in: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 366–379. Springer, 2010.CrossRefGoogle Scholar
  26. [26]
    O. Svensson: Santa Claus schedules jobs on unrelated machines, SIAM Journal on Computing 41 (2012), 1318–1341.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    M. Sviridenko and J. Ward: Large neighborhood local search for the maximum set packing problem, in: Automata, Languages, and sProgramming, 792–803. Springer, 2013.CrossRefGoogle Scholar
  28. [28]
    D. B. West: Introduction to graph theory, volume 2, Prentice hall Upper Saddle River, 2001.Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceETH ZurichZurichSwitzerland

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