Feedback vertex set on AT-free graphs

School of Computing, University of Leeds, Leeds LS2 9JT, UK
Discrete Applied Mathematics (Impact Factor: 0.8). 05/2008; 156(10):1936-1947. DOI: 10.1016/j.dam.2007.10.006
Source: DBLP


We present a polynomial time algorithm to compute a minimum (weight) feedback vertex set for AT-free graphs, and extending this approach we obtain a polynomial time algorithm for graphs of bounded asteroidal number.

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    • "This parameterization exploits the fact that FEEDBACK VERTEX SET is polynomial-time solvable on subcubic graphs [55] . A similar question is whether FEEDBACK VERTEX SET is FPT parameterized by the asteroidal number of a graph; see Kratsch et al. [2] for details. An XP algorithm follows from their work, generalizing the fact that FEEDBACK VERTEX SET is polynomial-time solvable on AT-free graphs. "
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    ABSTRACT: This paper deals with the Feedback Vertex Set problem on undirected graphs, which asks for the existence of a vertex set of bounded size that intersects all cycles. Due it is theoretical and practical importance, the problem has been the subject of intensive study. Motivated by the parameter ecology program we attempt to classify the parameterized and kernelization complexity of Feedback Vertex Set for a wide range of parameters. We survey known results and present several new complexity classifications. For example, we prove that Feedback Vertex Set is fixed-parameter tractable parameterized by the vertex-deletion distance to a chordal graph. We also prove that the problem admits a polynomial kernel when parameterized by the vertex-deletion distance to a pseudo forest, a graph in which every connected component has at most one cycle. In contrast, we prove that a slightly smaller parameterization does not allow for a polynomial kernel unless NP ⊆ coNP/poly and the polynomial-time hierarchy collapses.
    Full-text · Article · Aug 2014 · Tsinghua Science & Technology

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    ABSTRACT: For Gaussian graphical models with cycles, loopy belief propagation often performs reasonably well, but its convergence is not guaranteed and the computation of variances is generally incorrect. In this paper, we identify a set of special vertices called a feedback vertex set whose removal results in a cycle-free graph. We propose a feedback message passing algorithm in which non-feedback nodes send out one set of messages while the feedback nodes use a different message update scheme. Exact inference results can be obtained in O(k<sup>2</sup>n), where k is the number of feedback nodes and n is the total number of nodes. For graphs with large feedback vertex sets, we describe a tractable approximate feedback message passing algorithm. Experimental results show that this procedure converges more often, faster, and provides better results than loopy belief propagation.
    Preview · Conference Paper · Jul 2010
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