Abstract
The Vertex Separator Problem of a directed graph consists in finding all combinations of vertices which can disconnect the source and the terminal of the graph, these combinations are minimal if they contain only the minimal number of vertices. In this paper, we introduce a new quantum algorithm based on a movement strategy to find these separators in a quantum superposition with linear complexity. Our algorithm has been tested on small directed graphs using a real Quantum Computer made by IBM .
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ambainis, A.: Quantum walk algorithm for element distinctness. SIAM J. Comput. 37(1), 210–239 (2007)
Apers, S., Lee, T.: Quantum complexity of minimum cut. arXiv preprint arXiv:2011.09823 (2020)
Ashcraft, C., Liu, J.W.H.: A partition improvement algorithm for generalized nested dissection. Boeing Computer Services, Seattle, WA, Technical report, BCSTECH-94-020 (1994)
Bui, T.N., Jones, C.: Finding good approximate vertex and edge partitions is NP-hard. Inf. Process. Lett. 42(3), 153–159 (1992)
Cui, S.X., Freedman, M.H., Sattath, O., Stong, R., Minton, G.: Quantum max-flow/min-cut. J. Math. Phys. 57(6), 062206 (2016)
Davis, T.A., Hager, W.W., Kolodziej, S.P., Yeralan, S.N.: Algorithm 1003: mongoose, a graph coarsening and partitioning library. ACM Trans. Math. Softw. 46(1), 1–18 (2019)
Dörn, S.: Quantum algorithms for graph traversals and related problems. In: Proceedings of CIE, vol. 7, pp. 123–131. Citeseer (2007)
Eldredge, Z., et al.: Entanglement bounds on the performance of quantum computing architectures. Phys. Rev. Res. 2(3), 033316 (2020)
Fiduccia, C.M., Mattheyses, R.M.: A linear-time heuristic for improving network partitions. In: 19th Design Automation Conference, pp. 175–181. IEEE (1982)
Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, pp. 212–219 (1996)
Hager, W.W., Hungerford, J.T.: Continuous quadratic programming formulations of optimization problems on graphs. Eur. J. Oper. Res. 240(2), 328–337 (2015)
Hager, W.W., Hungerford, J.T., Safro, I.: A multilevel bilinear programming algorithm for the vertex separator problem. Comput. Optim. Appl. 69(1), 189–223 (2017). https://doi.org/10.1007/s10589-017-9945-2
Hendrickson, B., Rothberg, E.: Improving the run time and quality of nested dissection ordering. SIAM J. Sci. Comput. 20(2), 468–489 (1998)
Jeffery, S., Kimmel, S.: Quantum algorithms for graph connectivity and formula evaluation. Quantum 1, 26 (2017)
Kernighan, B.W., Lin, S.: An efficient heuristic procedure for partitioning graphs. Bell Syst. Tech. J. 49(2), 291–307 (1970)
Kloks, T., Kratsch, D.: Listing all minimal separators of a graph. SIAM J. Comput. 27(3), 605–613 (1998)
Kolodziej, S., Davis, T.: Vertex separators with mixed-integer linear optimization. In: 17th SIAM Conference on Parallel Processing for Scientific Computing (2016)
Kolodziej, S.P., Davis, T.A.: Generalized gains for hybrid vertex separator algorithms. In: 2020 Proceedings of the SIAM Workshop on Combinatorial Scientific Computing, pp. 96–105. SIAM (2020)
Kolodziej, S.P.: Computational optimization techniques for graph partitioning. Ph.D. thesis (2019)
Bellac, M.L.: Introduction à l’information quantique. Belin (2005)
Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Appl. Math. 36(2), 177–189 (1979)
Lipton, R.J., Tarjan, R.E.: Applications of a planar separator theorem. SIAM J. Comput. 9(3), 615–627 (1980)
Magniez, F., Santha, M., Szegedy, M.: Quantum algorithms for the triangle problem. SIAM J. Comput. 37(2), 413–424 (2007)
Nielsen, M.A., Chuang, I.: Quantum computation and quantum information (2002)
Pednault, E., Gunnels, J.A., Nannicini, G., Horesh, L., Wisnieff, R.: Leveraging secondary storage to simulate deep 54-qubit sycamore circuits. arXiv preprint arXiv:1910.09534 (2019)
Preskill, J.: Quantum computing and the entanglement frontier. arXiv preprint arXiv:1203.5813 (2012)
Shimizu, K., Mori, R.: Exponential-time quantum algorithms for graph coloring problems. In: Kohayakawa, Y., Miyazawa, F.K. (eds.) LATIN 2021. LNCS, vol. 12118, pp. 387–398. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-61792-9_31
Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev. 41(2), 303–332 (1999)
Zhong, H.-S., et al.: Quantum computational advantage using photons. Science 370(6523), 1460–1463 (2020)
Acknowledgements
This research work was partially funded by the European project NExt ApplicationS of Quantum Computing (NEASQC), supported by the Horizon 2020-FETFLAG (Grant Agreement 951821).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 IFIP International Federation for Information Processing
About this paper
Cite this paper
Zaiou, A., Bennani, Y., Hibti, M., Matei, B. (2022). Quantum Approach for Vertex Separator Problem in Directed Graphs. In: Maglogiannis, I., Iliadis, L., Macintyre, J., Cortez, P. (eds) Artificial Intelligence Applications and Innovations. AIAI 2022. IFIP Advances in Information and Communication Technology, vol 646. Springer, Cham. https://doi.org/10.1007/978-3-031-08333-4_40
Download citation
DOI: https://doi.org/10.1007/978-3-031-08333-4_40
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-08332-7
Online ISBN: 978-3-031-08333-4
eBook Packages: Computer ScienceComputer Science (R0)
