Algorithmica

, Volume 65, Issue 1, pp 129–145

Exact Algorithms for Finding Longest Cycles in Claw-Free Graphs

Authors

  • Hajo Broersma
    • School of Engineering and Computing Sciences, Science LaboratoriesDurham University
  • Fedor V. Fomin
    • Department of InformaticsUniversity of Bergen
  • Pim van ’t Hof
    • School of Engineering and Computing Sciences, Science LaboratoriesDurham University
    • School of Engineering and Computing Sciences, Science LaboratoriesDurham University
Article

DOI: 10.1007/s00453-011-9576-4

Cite this article as:
Broersma, H., Fomin, F.V., van ’t Hof, P. et al. Algorithmica (2013) 65: 129. doi:10.1007/s00453-011-9576-4

Abstract

The Hamiltonian Cycle problem is the problem of deciding whether an n-vertex graph G has a cycle passing through all vertices of G. This problem is a classic NP-complete problem. Finding an exact algorithm that solves it in \({\mathcal {O}}^{*}(\alpha^{n})\) time for some constant α<2 was a notorious open problem until very recently, when Björklund presented a randomized algorithm that uses \({\mathcal {O}}^{*}(1.657^{n})\) time and polynomial space. The Longest Cycle problem, in which the task is to find a cycle of maximum length, is a natural generalization of the Hamiltonian Cycle problem. For a claw-free graph G, finding a longest cycle is equivalent to finding a closed trail (i.e., a connected even subgraph, possibly consisting of a single vertex) that dominates the largest number of edges of some associated graph H. Using this translation we obtain two deterministic algorithms that solve the Longest Cycle problem, and consequently the Hamiltonian Cycle problem, for claw-free graphs: one algorithm that uses \({\mathcal {O}}^{*}(1.6818^{n})\) time and exponential space, and one algorithm that uses \({\mathcal {O}}^{*}(1.8878^{n})\) time and polynomial space.

Copyright information

© Springer Science+Business Media, LLC 2011