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An O(2O(k) n 3) FPT Algorithm for the Undirected Feedback Vertex Set Problem

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Computing and Combinatorics (COCOON 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3595))

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Abstract

We describe an algorithm for the Feedback Vertex Set problem on undirected graphs, parameterized by the size k of the feedback vertex set, that runs in time O(c k n 3) where c=10.567 and n is the number of vertices in the graph. The best previous algorithms were based on the method of bounded search trees, branching on short cycles. The best previous running time of an FPT algorithm for this problem, due to Raman, Saurabh and Subramanian, has a parameter function of the form 2O( klogk / loglogk). Whether an exponentially linear in k FPT algorithm for this problem is possible has been previously noted as a significant challenge. Our algorithm is based on the new FPT technique of iterative compression. Our result holds for a more general “annotated” form of the problem, where a subset of the vertices may be marked as not to belong to the feedback set. We also establish “exponential optimality” for our algorithm by proving that no FPT algorithm with a parameter function of the form O(2o(k)) is possible, unless there is an unlikely collapse of parameterized complexity classes, namely FPT =M[1].

This research has been supported in part by the U.S. National Science Foundation under grant CCR–0075792, by the U.S. Office of Naval Research under grant N00014–01–1–0608, by the U.S. Department of Energy under contract DE–AC05–00OR22725, by the Australian Research Council and by the Australian Centre for Bioinformatics

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Author information

Authors and Affiliations

  1. Carleton University, Ottawa, Canada

    Frank Dehne

  2. University of Newcastle, Callaghan, NSW 2308, Australia

    Michael Fellows & Frances Rosamond

  3. University of Tennessee, Knoxville, TN, 37996-3450, USA

    Michael A. Langston

  4. The Mechanics Institute, Bob’s Farm, NSW 2316, Australia

    Kim Stevens

Authors
  1. Frank Dehne
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  2. Michael Fellows
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  3. Michael A. Langston
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  4. Frances Rosamond
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  5. Kim Stevens
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Editor information

Editors and Affiliations

  1. Department of Computer Science, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

    Lusheng Wang

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Dehne, F., Fellows, M., Langston, M.A., Rosamond, F., Stevens, K. (2005). An O(2O(k) n 3) FPT Algorithm for the Undirected Feedback Vertex Set Problem. In: Wang, L. (eds) Computing and Combinatorics. COCOON 2005. Lecture Notes in Computer Science, vol 3595. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11533719_87

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  • DOI: https://doi.org/10.1007/11533719_87

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-28061-3

  • Online ISBN: 978-3-540-31806-4

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