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Quantum Algorithms for One-Sided Crossing Minimization

Authors Susanna Caroppo , Giordano Da Lozzo , Giuseppe Di Battista

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Graph theory
  • Theory of computation → Quantum complexity theory
Keywords
  • One-sided crossing minimization
  • quantum graph drawing
  • quantum dynamic programming
  • quantum divide and conquer
  • exact exponential algorithms

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Abstract

We present singly-exponential quantum algorithms for the One-Sided Crossing Minimization (OSCM) problem. We show that OSCM can be viewed as a set problem amenable for exact algorithms with a quantum speedup with respect to their classical counterparts. First, we exploit the quantum dynamic programming framework of Ambainis et al. [Quantum Speedups for Exponential-Time Dynamic Programming Algorithms. SODA 2019] to devise a QRAM-based algorithm that solves OSCM in 𝒪^*(1.728ⁿ) time and space. Second, we use quantum divide and conquer to obtain an algorithm that solves OSCM without using QRAM in 𝒪^*(2ⁿ) time and polynomial space.

Cite As Get BibTex

Susanna Caroppo, Giordano Da Lozzo, and Giuseppe Di Battista. Quantum Algorithms for One-Sided Crossing Minimization. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 20:1-20:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.GD.2024.20

Author Details

Susanna Caroppo
  • Roma Tre University, Rome, Italy
Giordano Da Lozzo
  • Roma Tre University, Rome, Italy
Giuseppe Di Battista
  • Roma Tre University, Rome, Italy

Funding

This research was supported, in part, by MUR of Italy (PRIN Project no. 2022ME9Z78 - NextGRAAL and PRIN Project no. 2022TS4Y3N - EXPAND).

Acknowledgements

We acknowledge the CINECA award under the ISCRA initiative, for the availability of high-performance computing resources and support

References

  1. Noga Alon, Daniel Lokshtanov, and Saket Saurabh. Fast FAST. In Susanne Albers, Alberto Marchetti-Spaccamela, Yossi Matias, Sotiris E. Nikoletseas, and Wolfgang Thomas, editors, Automata, Languages and Programming, 36th International Colloquium, ICALP 2009, Rhodes, Greece, July 5-12, 2009, Proceedings, Part I, volume 5555 of Lecture Notes in Computer Science, pages 49-58. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-02927-1_6.
  2. Andris Ambainis, Kaspars Balodis, Janis Iraids, Martins Kokainis, Krisjanis Prusis, and Jevgenijs Vihrovs. Quantum speedups for exponential-time dynamic programming algorithms. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 1783-1793. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.107.
  3. Susanna Caroppo, Giordano Da Lozzo, and Giuseppe Di Battista. Quantum algorithms for one-sided crossing minimization, 2024. URL: https://arxiv.org/abs/2409.01942.
  4. Susanna Caroppo, Giordano Da Lozzo, and Giuseppe Di Battista. Quantum graph drawing. In Ryuhei Uehara, Katsuhisa Yamanaka, and Hsu-Chun Yen, editors, WALCOM: Algorithms and Computation - 18th International Conference and Workshops on Algorithms and Computation, WALCOM 2024, Kanazawa, Japan, March 18-20, 2024, Proceedings, volume 14549 of Lecture Notes in Computer Science, pages 32-46. Springer, 2024. URL: https://doi.org/10.1007/978-981-97-0566-5_4.
  5. Vida Dujmovic, Henning Fernau, and Michael Kaufmann. Fixed parameter algorithms for one-sided crossing minimization revisited. In Giuseppe Liotta, editor, Graph Drawing, 11th International Symposium, GD 2003, Perugia, Italy, September 21-24, 2003, Revised Papers, volume 2912 of Lecture Notes in Computer Science, pages 332-344. Springer, 2003. URL: https://doi.org/10.1007/978-3-540-24595-7_31.
  6. Vida Dujmovic, Henning Fernau, and Michael Kaufmann. Fixed parameter algorithms for one-sided crossing minimization revisited. J. Discrete Algorithms, 6(2):313-323, 2008. URL: https://doi.org/10.1016/J.JDA.2006.12.008.
  7. Vida Dujmovic and Sue Whitesides. An efficient fixed parameter tractable algorithm for 1-sided crossing minimization. In Stephen G. Kobourov and Michael T. Goodrich, editors, Graph Drawing, 10th International Symposium, GD 2002, Irvine, CA, USA, August 26-28, 2002, Revised Papers, volume 2528 of Lecture Notes in Computer Science, pages 118-129. Springer, 2002. URL: https://doi.org/10.1007/3-540-36151-0_12.
  8. Vida Dujmovic and Sue Whitesides. An efficient fixed parameter tractable algorithm for 1-sided crossing minimization. Algorithmica, 40(1):15-31, 2004. URL: https://doi.org/10.1007/S00453-004-1093-2.
  9. Christoph Dürr and Peter Høyer. A quantum algorithm for finding the minimum. CoRR, quant-ph/9607014, 1996. URL: https://arxiv.org/abs/quant-ph/9607014.
  10. Peter Eades and Nicholas C. Wormald. Edge crossings in drawings of bipartite graphs. Algorithmica, 11(4):379-403, 1994. URL: https://doi.org/10.1007/BF01187020.
  11. Henning Fernau, Fedor V. Fomin, Daniel Lokshtanov, Matthias Mnich, Geevarghese Philip, and Saket Saurabh. Ranking and drawing in subexponential time. In Costas S. Iliopoulos and William F. Smyth, editors, Combinatorial Algorithms - 21st International Workshop, IWOCA 2010, London, UK, July 26-28, 2010, Revised Selected Papers, volume 6460 of Lecture Notes in Computer Science, pages 337-348. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-19222-7_34.
  12. Shion Fukuzawa, Michael T. Goodrich, and Sandy Irani. Quantum Tutte embeddings. CoRR, abs/2307.08851, 2023. URL: https://doi.org/10.48550/arXiv.2307.08851.
  13. Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Quantum random access memory. Phys. Rev. Lett., 100:160501, April 2008. URL: https://doi.org/10.1103/PhysRevLett.100.160501.
  14. Lov K. Grover. A fast quantum mechanical algorithm for database search. In Gary L. Miller, editor, STOC 1996, pages 212-219. ACM, 1996. URL: https://doi.org/10.1145/237814.237866.
  15. Aram W. Harrow. Quantum algorithms for systems of linear equations. In Encyclopedia of Algorithms, pages 1680-1683. Springer New York, 2016. URL: https://doi.org/10.1007/978-1-4939-2864-4_771.
  16. Michael Jünger and Petra Mutzel. Exact and heuristic algorithms for 2-layer straightline crossing minimization. In Franz-Josef Brandenburg, editor, Graph Drawing, Symposium on Graph Drawing, GD '95, Passau, Germany, September 20-22, 1995, Proceedings, volume 1027 of Lecture Notes in Computer Science, pages 337-348. Springer, 1995. URL: https://doi.org/10.1007/BFB0021817.
  17. Claire Kenyon-Mathieu and Warren Schudy. How to rank with few errors. In David S. Johnson and Uriel Feige, editors, Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, California, USA, June 11-13, 2007, pages 95-103. ACM, 2007. URL: https://doi.org/10.1145/1250790.1250806.
  18. Yasuaki Kobayashi and Hisao Tamaki. A fast and simple subexponential fixed parameter algorithm for one-sided crossing minimization. Algorithmica, 72(3):778-790, 2015. URL: https://doi.org/10.1007/S00453-014-9872-X.
  19. Xavier Muñoz, Walter Unger, and Imrich Vrto. One sided crossing minimization is NP-hard for sparse graphs. In Petra Mutzel, Michael Jünger, and Sebastian Leipert, editors, Graph Drawing, 9th International Symposium, GD 2001 Vienna, Austria, September 23-26, 2001, Revised Papers, volume 2265 of Lecture Notes in Computer Science, pages 115-123. Springer, 2001. URL: https://doi.org/10.1007/3-540-45848-4_10.
  20. Petra Mutzel and René Weiskircher. Two-layer planarization in graph drawing. In Kyung-Yong Chwa and Oscar H. Ibarra, editors, Algorithms and Computation, 9th International Symposium, ISAAC '98, Taejon, Korea, December 14-16, 1998, Proceedings, volume 1533 of Lecture Notes in Computer Science, pages 69-78. Springer, 1998. URL: https://doi.org/10.1007/3-540-49381-6_9.
  21. Kazuya Shimizu and Ryuhei Mori. Exponential-time quantum algorithms for graph coloring problems. Algorithmica, 84(12):3603-3621, 2022. URL: https://doi.org/10.1007/S00453-022-00976-2.
  22. Kozo Sugiyama, Shojiro Tagawa, and Mitsuhiko Toda. Methods for visual understanding of hierarchical system structures. IEEE Trans. Syst. Man Cybern., 11(2):109-125, 1981. URL: https://doi.org/10.1109/TSMC.1981.4308636.
  23. William Thomas Tutte. How to draw a graph. Proceedings of the London Mathematical Society, 3(1):743-767, 1963. Google Scholar
  24. Vicente Valls, Rafael Martí, and Pilar Lino. A branch and bound algorithm for minimizing the number of crossing arcs in bipartite graphs. European journal of operational research, 90(2):303-319, 1996. URL: https://doi.org/10.1016/0377-2217(95)00356-8.
  25. Gerhard J. Woeginger. Open problems around exact algorithms. Discret. Appl. Math., 156(3):397-405, 2008. URL: https://doi.org/10.1016/J.DAM.2007.03.023.
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