Exact and Approximate Algorithms for Computing a Second Hamiltonian Cycle

Authors Argyrios Deligkas, George B. Mertzios , Paul G. Spirakis , Viktor Zamaraev

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Argyrios Deligkas
  • Department of Computer Science, Royal Holloway, University of London, Egham, UK
George B. Mertzios
  • Department of Computer Science, Durham University, UK
Paul G. Spirakis
  • Department of Computer Science, University of Liverpool, UK
  • Computer Engineering & Informatics Department, University of Patras, Greece
Viktor Zamaraev
  • Department of Computer Science, University of Liverpool, UK

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Argyrios Deligkas, George B. Mertzios, Paul G. Spirakis, and Viktor Zamaraev. Exact and Approximate Algorithms for Computing a Second Hamiltonian Cycle. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 27:1-27:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


In this paper we consider the following total functional problem: Given a cubic Hamiltonian graph G and a Hamiltonian cycle C₀ of G, how can we compute a second Hamiltonian cycle C₁ ≠ C₀ of G? Cedric Smith and William Tutte proved in 1946, using a non-constructive parity argument, that such a second Hamiltonian cycle always exists. Our main result is a deterministic algorithm which computes the second Hamiltonian cycle in O(n⋅2^0.299862744n) = O(1.23103ⁿ) time and in linear space, thus improving the state of the art running time of O*(2^0.3n) = O(1.2312ⁿ) for solving this problem (among deterministic algorithms running in polynomial space). Whenever the input graph G does not contain any induced cycle C₆ on 6 vertices, the running time becomes O(n⋅ 2^0.2971925n) = O(1.22876ⁿ). Our algorithm is based on a fundamental structural property of Thomason’s lollipop algorithm, which we prove here for the first time. In the direction of approximating the length of a second cycle in a (not necessarily cubic) Hamiltonian graph G with a given Hamiltonian cycle C₀ (where we may not have guarantees on the existence of a second Hamiltonian cycle), we provide a linear-time algorithm computing a second cycle with length at least n - 4α (√n+2α)+8, where α = (Δ-2)/(δ-2) and δ,Δ are the minimum and the maximum degree of the graph, respectively. This approximation result also improves the state of the art.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Graph theory
  • Hamiltonian cycle
  • cubic graph
  • exact algorithm
  • approximation algorithm


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  1. Cristina Bazgan, Miklos Santha, and Zsolt Tuza. On the approximation of finding a(nother) hamiltonian cycle in cubic hamiltonian graphs. Journal of Algorithms, 31(1):249-268, 1999. Google Scholar
  2. Richard Bellman. Dynamic programming treatment of the Travelling Salesman Problem. Journal of the ACM, 9(1):61-63, 1962. Google Scholar
  3. Andreas Bjorklund. Determinant sums for undirected hamiltonicity. SIAM Journal on Computing, 43(1):280-299, 2014. Google Scholar
  4. Andreas Björklund and Thore Husfeldt. The parity of directed Hamiltonian cycles. In Profeecings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 727-735, 2013. Google Scholar
  5. Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. The traveling salesman problem in bounded degree graphs. ACM Transactions on Algorithms, 8(2):18:1-18:13, 2012. Google Scholar
  6. Hans L. Bodlaender, Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Information and Computation, 243:86-111, 2015. Google Scholar
  7. Kathie Cameron. Thomason’s algorithm for finding a second Hamiltonian circuit through a given edge in a cubic graph is exponential on Krawczyk’s graphs. Discrete Mathematics, 235:69-77, 2001. Google Scholar
  8. Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Fast hamiltonicity checking via bases of perfect matchings. In Proceedings of the 45th ACM Symposium on Theory of Computing Conference (STOC), pages 301-310, 2013. Google Scholar
  9. Argyrios Deligkas, George B. Mertzios, Paul G. Spirakis, and Viktor Zamaraev. Exact and approximate algorithms for computing a second hamiltonian cycle. CoRR, 2020. Available online at URL: https://arxiv.org/abs/2004.06036.
  10. R.C. Entringer and Henda Swart. Spanning cycles of nearly cubic graphs. Journal of Combinatorial Theory, Series B, 29(3):303-309, 1980. Google Scholar
  11. David Eppstein. The Traveling Salesman Problem for cubic graphs. Journal of Graph Algorithms and Applications, 11(1):61-81, 2007. Google Scholar
  12. Herbert Fleischner. Uniqueness of maximal dominating cycles in 3-regular graphs and of Hamiltonian cycles in 4-regular graphs. Journal of Graph Theory, 18(5):449-459, 1994. Google Scholar
  13. António Girão, Teeradej Kittipassorn, and Bhargav Narayanan. Long cycles in Hamiltonian graphs. Israel Journal of Mathematics, 229(1):269-285, 2019. Google Scholar
  14. Ronald J. Gould. Recent advances on the Hamiltonian problem: Survey III. Graphs and Combinatorics, 30(1):1-46, 2014. Google Scholar
  15. Michael Held and Richard M. Karp. A dynamic programming approach to sequencing problems. Journal of the Society for Industrial and Applied Mathematics, 10(1):196-210, 1962. Google Scholar
  16. Richard M. Karp. Reducibility among combinatorial problems. In Complexity of Computer Computations, pages 85-103. Springer, 1972. Google Scholar
  17. Adam Krawczyk. The complexity of finding a second Hamiltonian cycle in cubic graphs. Journal of Computer and System Sciences, 58(3):641-647, 2001. Google Scholar
  18. Maciej Liśkiewicz and Martin R Schuster. A new upper bound for the Traveling Salesman Problem in cubic graphs. Journal of Discrete Algorithms, 27:1-20, 2014. Google Scholar
  19. Christos H. Papadimitriou. On the complexity of the parity argument and other inefficient proofs of existence. Journal of Computer and system Sciences, 48(3):498-532, 1994. Google Scholar
  20. Andrew G. Thomason. Hamiltonian cycles and uniquely edge colourable graphs. Advances in Graph Theory, 3:259-268, 1978. Google Scholar
  21. William T. Tutte. On Hamiltonian circuits. Journal of the London Mathematical Society, 1(2):98-101, 1946. Google Scholar
  22. Douglas West. Introduction to graph theory. Prentice hall Upper Saddle River, 2 edition, 2001. Google Scholar
  23. Mingyu Xiao and Hiroshi Nagamochi. An exact algorithm for TSP in degree-3 graphs via circuit procedure and amortization on connectivity structure. Algorithmica, 74(2):713-741, 2016. Google Scholar