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Finding Long Paths, Cycles and Circuits

  • Harold N. Gabow
  • Shuxin Nie
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5369)

Abstract

We present a polynomial-time algorithm to find a cycle of length \(\exp(\Omega(\sqrt{\log \ell}))\) in an undirected graph having a cycle of length ≥ ℓ. This is a slight improvement over previously known bounds. In addition the algorithm is more general, since it can similarly approximate the longest circuit, as well as the longest S-circuit (i.e., for S an arbitrary subset of vertices, a circuit that can visit each vertex in S at most once). We also show that any algorithm for approximating the longest cycle can approximate the longest circuit, with a square root reduction in length. For digraphs, we show that the long cycle and long circuit problems have the same approximation ratio up to a constant factor. We also give an algorithm to find a vw-path of length ≥ logn/loglogn if one exists; this is within a loglogn factor of a hardness result.

Keywords

Undirected Graph Approximation Ratio Recursive Call Hardness Result Hamiltonian Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Harold N. Gabow
    • 1
  • Shuxin Nie
    • 1
  1. 1.Department of Computer ScienceUniversity of ColoradoBoulder

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