Haxell’s condition  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
This is a preview of subscription content, log in to check access.
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 A119 (2012), 1200–1215.MathSciNetCrossRefMATHGoogle Scholar
A. Asadpour, U. Feige and A. Saberi: Santa claus meets hypergraph matchings, ACM Transactions on Algorithms (TALG)8 (2012), 24.MathSciNetMATHGoogle Scholar
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
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
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
Y. H. Chan and L. C. Lau: On linear and semidefinite programming relaxations for hypergraph matching, Mathematical Programming135 (2012), 123–148.MathSciNetCrossRefMATHGoogle Scholar
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
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
M. Fürer and H. Yu: Approximating the k-set packing problem by local improvements, in: Combinatorial Optimization, 408–420. Springer, 2014.Google Scholar
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.MathSciNetMATHGoogle Scholar
G. H. Hardy and S. Ramanujan: Asymptotic formula in combinatory analysis, Proceedings of the London Mathematical Society, s217 (1918), 75.CrossRefMATHGoogle Scholar
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 Mathematics2 (1989), 68–72.MathSciNetCrossRefMATHGoogle Scholar
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
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
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
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