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Parameterized Quantum Query Algorithms for Graph Problems

Authors Tatsuya Terao , Ryuhei Mori

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Author Details

Tatsuya Terao
  • Research Institute for Mathematical Sciences, Kyoto University, Japan
Ryuhei Mori
  • Graduate School of Mathematics, Nagoya University, Japan

Funding

  • Terao, Tatsuya: KAKENHI Grant Number JP24KJ1494.
  • Mori, Ryuhei: JST FOREST Program Grant Number JPMJFR216V and KAKENHI Grant Numbers JP20H04138, JP20H05966, and JP22H00522.

Acknowledgements

The authors thank Yusuke Kobayashi and Kazuhisa Makino for their insightful comments.

Cite As Get BibTex

Tatsuya Terao and Ryuhei Mori. Parameterized Quantum Query Algorithms for Graph Problems. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 99:1-99:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.99

Abstract

In this paper, we consider the parameterized quantum query complexity for graph problems. We design parameterized quantum query algorithms for k-vertex cover and k-matching problems, and present lower bounds on the parameterized quantum query complexity. Then, we show that our quantum query algorithms are optimal up to a constant factor when the parameters are small. Our main results are as follows.

Parameterized quantum query complexity of vertex cover. In the k-vertex cover problem, we are given an undirected graph G with n vertices and an integer k, and the objective is to determine whether G has a vertex cover of size at most k. We show that the quantum query complexity of the k-vertex cover problem is O(√kn + k^{3/2}√n) in the adjacency matrix model. For the design of the quantum query algorithm, we use the method of kernelization, a well-known tool for the design of parameterized classical algorithms, combined with Grover’s search.

Parameterized quantum query complexity of matching. In the k-matching problem, we are given an undirected graph G with n vertices and an integer k, and the objective is to determine whether G has a matching of size at least k. We show that the quantum query complexity of the k-matching problem is O(√kn + k²) in the adjacency matrix model. We obtain this upper bound by using Grover’s search carefully and analyzing the number of Grover’s searches by making use of potential functions. We also show that the quantum query complexity of the maximum matching problem is O(√pn + p²) where p is the size of the maximum matching. For small p, it improves known bounds Õ(n^{3/2}) for bipartite graphs [Blikstad-v.d.Brand-Efron-Mukhopadhyay-Nanongkai, FOCS 2022] and O(n^{7/4}) for general graphs [Kimmel-Witter, WADS 2021].

Lower bounds on parameterized quantum query complexity. We also present lower bounds on the quantum query complexities of the k-vertex cover and k-matching problems. The lower bounds prove the optimality of the above parameterized quantum query algorithms up to a constant factor when k is small. Indeed, the quantum query complexities of the k-vertex cover and k-matching problems are both Θ(√k n) when k = O(√n) and k = O(n^{2/3}), respectively.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum query complexity
Keywords
  • Quantum query complexity
  • parameterized algorithms
  • vertex cover
  • matching
  • kernelization

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References

  1. Andris Ambainis. Quantum lower bounds by quantum arguments. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (STOC 2000), pages 636-643, 2000. URL: https://doi.org/10.1145/335305.335394.
  2. Andris Ambainis, Kaspars Balodis, Jānis Iraids, Martins Kokainis, Krišjānis Prūsis, and Jevgēnijs Vihrovs. Quantum speedups for exponential-time dynamic programming algorithms. In Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2019), pages 1783-1793. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.107.
  3. Andris Ambainis, Kaspars Balodis, Jānis Iraids, Raitis Ozols, and Juris Smotrovs. Parameterized quantum query complexity of graph collision. arXiv preprint arXiv:1305.1021, 2013. URL: https://doi.org/10.48550/arXiv.1305.1021.
  4. Andris Ambainis, Andrew M Childs, and Yi-Kai Liu. Quantum property testing for bounded-degree graphs. In Proceedings of the 14th International Workshop on Approximation Algorithms for Combinatorial Optimization (APPROX 2011), pages 365-376. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-22935-0_31.
  5. Andris Ambainis, Kazuo Iwama, Masaki Nakanishi, Harumichi Nishimura, Rudy Raymond, Seiichiro Tani, and Shigeru Yamashita. Quantum query complexity of boolean functions with small on-sets. In Proceedings of the 19th International Symposium on Algorithms and Computation (ISAAC 2008), pages 907-918. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-92182-0_79.
  6. Noel T Anderson, Jay-U Chung, Shelby Kimmel, Da-Yeon Koh, and Xiaohan Ye. Improved quantum query complexity on easier inputs. Quantum, 8:1309, 2024. URL: https://doi.org/10.22331/q-2024-04-08-1309.
  7. Simon Apers and Ronald de Wolf. Quantum speedup for graph sparsification, cut approximation, and Laplacian solving. SIAM Journal on Computing, 51(6):1703-1742, 2022. URL: https://doi.org/10.1137/21m1391018.
  8. Simon Apers and Troy Lee. Quantum complexity of minimum cut. In Proceedings of the 36th Computational Complexity Conference (CCC 2021), volume 200, pages 28:1-28:33, 2021. URL: https://doi.org/10.4230/LIPIcs.CCC.2021.28.
  9. Agnis Āriņš. Span-program-based quantum algorithms for graph bipartiteness and connectivity. In International Doctoral Workshop on Mathematical and Engineering Methods in Computer Science, pages 35-41. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-29817-7_4.
  10. R Balasubramanian, Michael R Fellows, and Venkatesh Raman. An improved fixed-parameter algorithm for vertex cover. Information Processing Letters, 65(3):163-168, 1998. URL: https://doi.org/10.1016/S0020-0190(97)00213-5.
  11. Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum lower bounds by polynomials. Journal of the ACM (JACM), 48(4):778-797, 2001. URL: https://doi.org/10.1145/502090.502097.
  12. Salman Beigi and Leila Taghavi. Quantum speedup based on classical decision trees. Quantum, 4:241, 2020. URL: https://doi.org/10.22331/q-2020-03-02-241.
  13. Aleksandrs Belovs. Learning-graph-based quantum algorithm for k-distinctness. In Proceedings of the 53rd Annual Symposium on Foundations of Computer Science (FOCS 2012), pages 207-216. IEEE, 2012. URL: https://doi.org/10.1109/FOCS.2012.18.
  14. Aleksandrs Belovs. Span programs for functions with constant-sized 1-certificates. In Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC 2012), pages 77-84, 2012. URL: https://doi.org/10.1145/2213977.2213985.
  15. Aleksandrs Belovs and Ben W Reichardt. Span programs and quantum algorithms for st-connectivity and claw detection. In Proceedings of the 20th Annual European Symposium (ESA 2012), pages 193-204. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-33090-2_18.
  16. Ran Ben-Basat, Ken-ichi Kawarabayashi, and Gregory Schwartzman. Parameterized distributed algorithms. In Proceedings of the 33rd International Symposium on Distributed Computing (DISC 2019), volume 146, pages 6:1-6:16, 2019. URL: https://doi.org/10.4230/LIPIcs.DISC.2019.6.
  17. Aija Berzina, Andrej Dubrovsky, Rusins Freivalds, Lelde Lace, and Oksana Scegulnaja. Quantum query complexity for some graph problems. In Proceedings of the 30th Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM 2004), pages 140-150. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-24618-3_11.
  18. Joakim Blikstad, Jan Van Den Brand, Yuval Efron, Sagnik Mukhopadhyay, and Danupon Nanongkai. Nearly optimal communication and query complexity of bipartite matching. In Proceedings of the 63rd Annual Symposium on Foundations of Computer Science (FOCS 2022), pages 1174-1185. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00113.
  19. Michel Boyer, Gilles Brassard, Peter Høyer, and Alain Tapp. Tight bounds on quantum searching. Fortschritte der Physik: Progress of Physics, 46(4-5):493-505, 1998. URL: https://doi.org/10.1002/(SICI)1521-3978(199806)46:4/5<493::AID-PROP493>3.0.CO;2-P.
  20. Gilles Brassard, Peter Hoyer, Michele Mosca, and Alain Tapp. Quantum amplitude amplification and estimation. Contemporary Mathematics, 305:53-74, 2002. URL: https://doi.org/10.1090/conm/305.
  21. Jonathan F Buss and Judy Goldsmith. Nondeterminism within P^*. SIAM Journal on Computing, 22(3):560-572, 1993. URL: https://doi.org/10.1137/0222038.
  22. Chris Cade, Ashley Montanaro, and Aleksandrs Belovs. Time and space efficient quantum algorithms for detecting cycles and testing bipartiteness. Quantum Information and Computation, 18(1-2):18-50, 2018. URL: https://doi.org/10.26421/QIC18.1-2-2.
  23. Titouan Carette, Mathieu Laurière, and Frédéric Magniez. Extended learning graphs for triangle finding. Algorithmica, 82(4):980-1005, 2020. URL: https://doi.org/10.1007/s00453-019-00627-z.
  24. L Sunil Chandran and Fabrizio Grandoni. Refined memorisation for vertex cover. In Proceedings of the 1st International Workshop on Parameterized and Exact Computation (IWPEC 2004), pages 61-70. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-28639-4_6.
  25. Jianer Chen, Iyad A Kanj, and Weijia Jia. Vertex cover: Further observations and further improvements. Journal of Algorithms, 41(2):280-301, 2001. URL: https://doi.org/10.1006/jagm.2001.1186.
  26. Jianer Chen, Iyad A Kanj, and Ge Xia. Improved upper bounds for vertex cover. Theoretical Computer Science, 411(40-42):3736-3756, 2010. URL: https://doi.org/10.1016/j.tcs.2010.06.026.
  27. Andrew M Childs and Jason M Eisenberg. Quantum algorithms for subset finding. Quantum Information and Computation, 5(7):593-604, 2005. URL: https://doi.org/10.26421/QIC5.7-7.
  28. Andrew M Childs and Robin Kothari. Quantum query complexity of minor-closed graph properties. SIAM Journal on Computing, 41(6):1426-1450, 2012. URL: https://doi.org/10.1137/110833026.
  29. Rajesh Chitnis and Graham Cormode. Towards a theory of parameterized streaming algorithm. In Proceedings of the 14th International Symposium on Parameterized and Exact Computation (IPEC 2019), volume 148, pages 7:1-7:15, 2019. URL: https://doi.org/10.4230/LIPIcs.IPEC.2019.7.
  30. Rajesh Chitnis, Graham Cormode, Hossein Esfandiari, MohammadTaghi Hajiaghayi, Andrew McGregor, Morteza Monemizadeh, and Sofya Vorotnikova. Kernelization via sampling with applications to finding matchings and related problems in dynamic graph streams. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2016), pages 1326-1344. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch92.
  31. Rajesh Chitnis, Graham Cormode, MohammadTaghi Hajiaghayi, and Morteza Monemizadeh. Parameterized streaming: Maximal matching and vertex cover. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2014), pages 1234-1251. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973730.82.
  32. Marek Cygan, Fedor V Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms, volume 4. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  33. Holger Dell and Dieter Van Melkebeek. Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. Journal of the ACM (JACM), 61(4):1-27, 2014. URL: https://doi.org/10.1145/2629620.
  34. Michael J Dinneen and Rongwei Lai. Properties of vertex cover obstructions. Discrete Mathematics, 307(21):2484-2500, 2007. URL: https://doi.org/10.1016/j.disc.2007.01.003.
  35. Sebastian Dörn. Quantum complexity bounds for independent set problems. arXiv preprint quant-ph/0510084, 2005. URL: https://doi.org/10.48550/arXiv.quant-ph/0510084.
  36. Sebastian Dörn. Quantum algorithms for matching problems. Theory of Computing Systems, 45(3):613-628, 2009. URL: https://doi.org/10.1007/s00224-008-9118-x.
  37. Rodney G Downey and Michael R Fellows. Fixed-parameter tractability and completeness. In Complexity Theory: Current Research, pages 191-225, 1992. URL: https://doi.org/10.1109/SCT.1992.215379.
  38. Christoph Dürr, Mark Heiligman, Peter Høyer, and Mehdi Mhalla. Quantum query complexity of some graph problems. SIAM Journal on Computing, 35(6):1310-1328, 2006. URL: https://doi.org/10.1137/050644719.
  39. Stefan Fafianie and Stefan Kratsch. Streaming kernelization. In Proceedings of the 39th International Symposium on Mathematical Foundations of Computer Science (MFCS 2014), pages 275-286. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-44465-8_24.
  40. Fedor V Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: Theory of Parameterized Preprocessing. Cambridge University Press, 2019. URL: https://doi.org/10.1017/9781107415157.
  41. Lov K Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC 1996), pages 212-219, 1996. URL: https://doi.org/10.1145/237814.237866.
  42. Jiong Guo and Rolf Niedermeier. Invitation to data reduction and problem kernelization. ACM SIGACT News, 38(1):31-45, 2007. URL: https://doi.org/10.1145/1233481.1233493.
  43. David G. Harris and N. S. Narayanaswamy. A faster algorithm for vertex cover parameterized by solution size. In Proceedings of the 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024), volume 289, pages 40:1-40:18, 2024. URL: https://doi.org/10.4230/LIPIcs.STACS.2024.40.
  44. John E Hopcroft and Richard M Karp. An n^5/2 algorithm for maximum matchings in bipartite graphs. SIAM Journal on Computing, 2(4):225-231, 1973. URL: https://doi.org/10.1137/0202019.
  45. Michael Jarret, Stacey Jeffery, Shelby Kimmel, and Alvaro Piedrafita. Quantum algorithms for connectivity and related problems. In Proceedings of the 26th Annual European Symposium on Algorithms (ESA 2018), volume 112, pages 49:1-49:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.ESA.2018.49.
  46. Stacey Jeffery, Shelby Kimmel, and Alvaro Piedrafita. Quantum algorithm for path-edge sampling. In Proceedings of the 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023), volume 266, pages 5:1-5:28, 2023. URL: https://doi.org/10.4230/LIPIcs.TQC.2023.5.
  47. Stacey Jeffery, Robin Kothari, and Frédéric Magniez. Nested quantum walks with quantum data structures. In Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2013), pages 1474-1485. SIAM, 2013. URL: https://doi.org/10.1137/1.9781611973105.106.
  48. Richard M Karp. Reducibility Among Combinatorial Problems. Springer, 2010. URL: https://doi.org/10.1007/978-3-540-68279-0_8.
  49. Shelby Kimmel and R Teal Witter. A query-efficient quantum algorithm for maximum matching on general graphs. In Proceedings of the 17th International Symposium on Algorithms and Data Structures (WADS 2021), pages 543-555, 2021. URL: https://doi.org/10.1007/978-3-030-83508-8_39.
  50. Bernhard H Korte and Jens Vygen. Combinatorial Optimization. Springer, sixth edition, 2006. URL: https://doi.org/10.1007/978-3-662-56039-6.
  51. François Le Gall. Improved quantum algorithm for triangle finding via combinatorial arguments. In Proceedings of the 55th Annual Symposium on Foundations of Computer Science (FOCS 2014), pages 216-225. IEEE, 2014. URL: https://doi.org/10.1109/FOCS.2014.31.
  52. François Le Gall and Shogo Nakajima. Quantum algorithm for triangle finding in sparse graphs. Algorithmica, 79:941-959, 2017. URL: https://doi.org/10.1007/s00453-016-0267-z.
  53. François Le Gall and Saeed Seddighin. Quantum meets fine-grained complexity: Sublinear time quantum algorithms for string problems. Algorithmica, 85(5):1251-1286, 2023. URL: https://doi.org/10.1007/s00453-022-01066-z.
  54. Troy Lee, Frédéric Magniez, and Miklos Santha. Learning graph based quantum query algorithms for finding constant-size subgraphs. Chicago Journal of Theoretical Computer Science, 10:1-21, 2012. URL: https://doi.org/10.4086/cjtcs.2012.010.
  55. Troy Lee, Frédéric Magniez, and Miklos Santha. Improved quantum query algorithms for triangle finding and associativity testing. In Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2013), pages 1486-1502. SIAM, 2013. URL: https://doi.org/10.1137/1.9781611973105.107.
  56. Cedric Yen-Yu Lin and Han-Hsuan Lin. Upper bounds on quantum query complexity inspired by the Elitzur-Vaidman bomb tester. Theory of Computing, 12(18):1-35, 2016. URL: https://doi.org/10.4086/toc.2016.v012a018.
  57. Frédéric Magniez, Miklos Santha, and Mario Szegedy. Quantum algorithms for the triangle problem. SIAM Journal on Computing, 37(2):413-424, 2007. URL: https://doi.org/10.1137/050643684.
  58. Rolf Niedermeier and Peter Rossmanith. Upper bounds for vertex cover further improved. In Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science (STACS 1999), pages 561-570. Springer, 1999. URL: https://doi.org/10.1007/3-540-49116-3_53.
  59. Rolf Niedermeier and Peter Rossmanith. On efficient fixed-parameter algorithms for weighted vertex cover. Journal of Algorithms, 47(2):63-77, 2003. URL: https://doi.org/10.1016/S0196-6774(03)00005-1.
  60. Alexander Schrijver. Combinatorial Optimization: Polyhedra and Efficiency, volume 24. Springer, 2003. Google Scholar
  61. Ulrike Stege and Michael Ralph Fellows. An improved fixed parameter tractable algorithm for vertex cover. Technical report/Departement Informatik, ETH Zürich, 318, 1999. URL: https://doi.org/10.3929/ethz-a-006653305.
  62. Xiaoming Sun, Andrew C Yao, and Shengyu Zhang. Graph properties and circular functions: How low can quantum query complexity go? In Proceedings of the 19th IEEE Annual Conference on Computational Complexity (CCC 2004), pages 286-293. IEEE, 2004. URL: https://doi.org/10.1109/CCC.2004.1313851.
  63. Shengyu Zhang. On the power of Ambainis’s lower bounds. In Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP 2004), pages 1238-1250. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-27836-8_102.
  64. Yechao Zhu. Quantum query complexity of constant-sized subgraph containment. International Journal of Quantum Information, 10(03):1250019, 2012. URL: https://doi.org/10.1142/S0219749912500190.
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