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Exponential-Time Quantum Algorithms for Graph Coloring Problems

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LATIN 2020: Theoretical Informatics (LATIN 2021)

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Abstract

The fastest known classical algorithm deciding the k-colorability of n-vertex graph requires running time \(\Omega (2^n)\) for \(k\ge 5\). In this work, we present an exponential-space quantum algorithm computing the chromatic number with running time \(O(1.9140^n)\) using quantum random access memory (QRAM). Our approach is based on Ambainis et al.’s quantum dynamic programming with applications of Grover’s search to branching algorithms. We also present a polynomial-space quantum algorithm not using QRAM for the graph 20-coloring problem with running time \(O(1.9575^n)\). For the polynomial-space quantum algorithm, we essentially show \((4-\epsilon )^n\)-time classical algorithms that can be improved quadratically by Grover’s search.

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Notes

  1. 1.

    In this paper, \(O^*(f(n))\) means \(O(\mathrm {poly}(n)f(n))\).

References

  1. Ambainis, A., Balodis, K., Iraids, J., Kokainis, M., Prūsis, K., Vihrovs, J.: Quantum speedups for exponential-time dynamic programming algorithms. In: Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2019). pp. 1783–1793. SIAM (2019)

    Google Scholar 

  2. Beigel, R., Eppstein, D.: 3-coloring in time \(O(1.3289^n)\). J. Alg. 54(2), 168–204 (2005)

    Article  Google Scholar 

  3. Björklund, A., Husfeldt, T.: Inclusion-exclusion algorithms for counting set partitions. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006). pp. 575–582. IEEE (2006)

    Google Scholar 

  4. Björklund, A., Husfeldt, T.: Exact algorithms for exact satisfiability and number of perfect matchings. Algorithmica 52(2), 226–249 (2008)

    Article  MathSciNet  Google Scholar 

  5. Björklund, A., Husfeldt, T., Koivisto, M.: Set partitioning via inclusion-exclusion. SIAM J. Comput. 39(2), 546–563 (2009)

    Article  MathSciNet  Google Scholar 

  6. Boyer, M., Brassard, G., Høyer, P., Tapp, A.: Tight bounds on quantum searching. Fortschritte der Physik: Progress of Physics 46(4–5), 493–505 (1998)

    Article  Google Scholar 

  7. Byskov, J.M.: Enumerating maximal independent sets with applications to graph colouring. Oper. Res. Lett. 32(6), 547–556 (2004)

    Article  MathSciNet  Google Scholar 

  8. Cygan, M., et al.: On problems as hard as CNF-SAT. ACM Trans. Alg. (TALG) 12(3), 41 (2016)

    MathSciNet  MATH  Google Scholar 

  9. Dürr, C., Høyer, P.: A quantum algorithm for finding the minimum. arXiv preprint quant-ph/9607014 (1996)

    Google Scholar 

  10. Fomin, F.V., Gaspers, S., Saurabh, S.: Improved exact algorithms for counting 3- and 4-colorings. In: Lin, G. (ed.) COCOON 2007. LNCS, vol. 4598, pp. 65–74. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-73545-8_9

    Chapter  MATH  Google Scholar 

  11. Fomin, F.V., Kratsch, D.: Split and List. Exact Exponential Algorithms. TTCSAES, pp. 153–160. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-16533-7_9

    Chapter  MATH  Google Scholar 

  12. Fürer, M.: Solving np-complete problems with quantum search. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 784–792. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78773-0_67

    Chapter  Google Scholar 

  13. Gaspers, S., Lee, E.J.: Faster graph coloring in polynomial space. In: Cao, Y., Chen, J. (eds.) COCOON 2017. LNCS, vol. 10392, pp. 371–383. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-62389-4_31

    Chapter  Google Scholar 

  14. Giovannetti, V., Lloyd, S., Maccone, L.: Quantum random access memory. Phys. Rev. Lett. 100(16), 160501 (2008)

    Article  MathSciNet  Google Scholar 

  15. Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC 1996). pp. 212–219. ACM (1996)

    Google Scholar 

  16. Koivisto, M.: An \(O^*(2^{n})\) algorithm for graph coloring and other partitioning problems via inclusion-exclusion. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006). pp. 583–590. IEEE (2006)

    Google Scholar 

  17. Lawler, E.L.: A note on the complexity of the chromatic number problem. Inf. Proc. Lett. 5, 66–67 (1976)

    Article  MathSciNet  Google Scholar 

  18. Shimizu, K., Mori, R.: Exponential-time quantum algorithms for graph coloring problems. arXiv e-prints p. 1907.00529 (2019), https://arxiv.org/abs/1907.00529

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Acknowledgment

This work was supported by JST PRESTO Grant Number JPMJPR1867 and JSPS KAKENHI Grant Numbers JP17K17711 and JP18H04090. The authors thank François Le Gall for the insightful comments.

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Authors and Affiliations

  1. School of Computing, Tokyo Institute of Technology, Tokyo, Japan

    Kazuya Shimizu & Ryuhei Mori

  2. Japan Science and Technology Agency, PRESTO, Tokyo, Japan

    Ryuhei Mori

Authors
  1. Kazuya Shimizu
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  2. Ryuhei Mori
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Correspondence to Ryuhei Mori .

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Editors and Affiliations

  1. University of São Paulo, São Paulo, Brazil

    Yoshiharu Kohayakawa

  2. University of Campinas, Campinas, Brazil

    Flávio Keidi Miyazawa

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Shimizu, K., Mori, R. (2020). Exponential-Time Quantum Algorithms for Graph Coloring Problems. In: Kohayakawa, Y., Miyazawa, F.K. (eds) LATIN 2020: Theoretical Informatics. LATIN 2021. Lecture Notes in Computer Science(), vol 12118. Springer, Cham. https://doi.org/10.1007/978-3-030-61792-9_31

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  • DOI: https://doi.org/10.1007/978-3-030-61792-9_31

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  • Online ISBN: 978-3-030-61792-9

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