Abstract
We design quantum algorithms for maximum matching. Working in the query model, in both adjacency matrix and adjacency list settings, we improve on the best known algorithms for general graphs, matching previously obtained results for bipartite graphs. In particular, for a graph with n vertices and m edges, our algorithm makes \(O(n^{7/4})\) queries in the matrix model and \(O(n^{3/4}(m+n)^{1/2})\) queries in the list model. Our approach combines Gabow’s classical maximum matching algorithm [Gabow, Fundamenta Informaticae, ’17] with the guessing tree method of Beigi and Taghavi [Beigi and Taghavi, Quantum, ’20].
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Notes
- 1.
One can easily extend to the case that G is a subgraph of a multigraph; we consider complete graphs only for simplicity.
References
Ambainis, A.: Quantum lower bounds by quntum arguments. J. Comput. Syst. Sci. 64(4), 750–767 (2002). https://doi.org/10.1006/jcss.2002.1826, http://www.sciencedirect.com/science/article/pii/S002200000291826X
Ambainis, A., Špalek, R.: Quantum algorithms for matching and network flows. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 172–183. Springer, Heidelberg (2006). https://doi.org/10.1007/11672142_13
Beigi, S., Taghavi, L.: Quantum speedup based on classical decision trees. Quantum 4, 241 (2020). https://doi.org/10.22331/q-2020-03-02-241, https://quantum-journal.org/papers/q-2020-03-02-241/, publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
Berzina, A., Dubrovsky, A., Freivalds, R., Lace, L., Scegulnaja, O.: Quantum query complexity for some graph problems. In: Van Emde Boas, P., Pokorný, J., Bieliková, M., Štuller, J. (eds.) SOFSEM 2004. LNCS, vol. 2932, pp. 140–150. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24618-3_11
Dörn, S.: Quantum algorithms for matching problems. Theor. Comput. Syst. 45, 613–628 (2009). https://doi.org/10.1007/s00224-008-9118-x
Edmonds, J.: Paths, trees, and flowers. Can. J. Math. 17, 449–467 (1965). https://doi.org/10.4153/CJM-1965-045-4, https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/paths-trees-and-flowers/08B492B72322C4130AE800C0610E0E21
Fujii, M., Kasami, T., Ninomiya, K.: Optimal sequencing of two equivalent processors. SIAM J. Appl. Math. 17(4), 784–789 (1969). https://www.jstor.org/stable/2099319
Gabow, H.N.: The weighted matching approach to maximum cardinality matching. Fundamenta Informaticae 154(1–4), 109–130 (2017). https://doi.org/10.3233/FI-2017-1555, https://content.iospress.com/articles/fundamenta-informaticae/fi1555, publisher: IOS Press
Hopcroft, J.E., Karp, R.M.: An n\(^{\wedge }\)(5/2) algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. Philadelphia 2(4), 7 (1973). http://dx.doi.org.ezproxy.middlebury.edu/10.1137/0202019, http://search.proquest.com/docview/919736551/abstract/79AD5CB7D4BA4C4EPQ/1, num Pages: 7 Place: Philadelphia, United States, Philadelphia Publisher: Society for Industrial and Applied Mathematics
Hoyer, P., Lee, T., Spalek, R.: Negative weights make adversaries stronger. In: Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, pp. 526–535. STOC 2007, Association for Computing Machinery, San Diego, California, USA (2007). https://doi.org/10.1145/1250790.1250867, https://doi.org/10.1145/1250790.1250867
Jeffery, S., Kimmel, S.: Quantum algorithms for graph connectivity and formula evaluation. Quantum 1, 26 (2017). https://doi.org/10.22331/q-2017-08-17-26, https://quantum-journal.org/papers/q-2017-08-17-26/
Lee, T., Mittal, R., Reichardt, B.W., Spalek, R., Szegedy, M.: Quantum query complexity of state conversion. In: 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pp. 344–353 (2011). https://doi.org/10.1109/FOCS.2011.75, http://arxiv.org/abs/1011.3020, arXiv: 1011.3020
Lin, C., Lin, H.H.: Upper bounds on quantum query complexity inspired by the Elitzur-Vaidman bomb tester. Theor. Comput. 12(18), 1–35 (2016). https://doi.org/10.4086/toc.2016.v012a018
May, J.W.: Cheminformatics for genome-scale metabolic reconstructions. Ph.D. Thesis, Cambridge University (2015). https://doi.org/10.17863/CAM.15987
Micali, S., Vazirani, V.V.: An O(sqrt(|v|)|E|) algorithm for finding maximum matching in general graphs. In: 21st Annual Symposium on Foundations of Computer Science (sfcs 1980), pp. 17–27 (Oct 1980). https://doi.org/10.1109/SFCS.1980.12, ISSN: 0272-5428
Reichardt, B.W.: Span programs and quantum query complexity: the general adversary bound is nearly tight for every boolean function. In: 2009 50th Annual IEEE Symposium on Foundations of Computer Science, pp. 544–551 (2009). https://doi.org/10.1109/FOCS.2009.55, http://arxiv.org/abs/0904.2759, arXiv: 0904.2759
Roth, A.E., Sonmez, T., Unver, M.U.: Pairwise Kidney Exchange. J. Econ. Theor. 125(2), 151–188 (2005). https://www.hbs.edu/faculty/Pages/item.aspx?num=19520
Vazirani, V.V.: A simplification of the MV matching algorithm and its proof. arXiv:1210.4594 [cs] (2013). http://arxiv.org/abs/1210.4594, arXiv: 1210.4594
Zhang, S.: On the power of ambainis lower bounds. Theor. Comput. Sci. 339(2), 241–256 (2005). https://doi.org/10.1016/j.tcs.2005.01.019, http://www.sciencedirect.com/science/article/pii/S0304397505001234
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Kimmel, S., Witter, R.T. (2021). A Query-Efficient Quantum Algorithm for Maximum Matching on General Graphs. In: Lubiw, A., Salavatipour, M., He, M. (eds) Algorithms and Data Structures. WADS 2021. Lecture Notes in Computer Science(), vol 12808. Springer, Cham. https://doi.org/10.1007/978-3-030-83508-8_39
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