Feedback vertex set on AT-free graphs

Laboratoire d’Informatique Théorique et Appliquée, Université de Metz, 57045 Metz Cedex 01, France; School of Computing, University of Leeds, Leeds LS2 9JT, UK; Laboratoire d’Informatique Fondamentale d’Orléans (LIFO), Université d’Orléans, BP 6759, 45067 Orléans Cedex 2, France
Discrete Applied Mathematics 01/2008; DOI: 10.1016/j.dam.2007.10.006
Source: DBLP

ABSTRACT 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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    ABSTRACT: While loopy belief propagation (LBP) performs reasonably well for inference in some Gaussian graphical models with cycles, its performance is unsatisfactory for many others. In particular for some models LBP does not converge, and in general when it does converge, the computed variances are incorrect (except for cycle-free graphs for which belief propagation (BP) is non-iterative and exact). In this paper we propose {\em feedback message passing} (FMP), a message-passing algorithm that makes use of a special set of vertices (called a {\em feedback vertex set} or {\em FVS}) whose removal results in a cycle-free graph. In FMP, standard BP is employed several times on the cycle-free subgraph excluding the FVS while a special message-passing scheme is used for the nodes in the FVS. The computational complexity of exact inference is $O(k^2n)$, where $k$ is the number of feedback nodes, and $n$ is the total number of nodes. When the size of the FVS is very large, FMP is intractable. Hence we propose {\em approximate FMP}, where a pseudo-FVS is used instead of an FVS, and where inference in the non-cycle-free graph obtained by removing the pseudo-FVS is carried out approximately using LBP. We show that, when approximate FMP converges, it yields exact means and variances on the pseudo-FVS and exact means throughout the remainder of the graph. We also provide theoretical results on the convergence and accuracy of approximate FMP. In particular, we prove error bounds on variance computation. Based on these theoretical results, we design efficient algorithms to select a pseudo-FVS of bounded size. The choice of the pseudo-FVS allows us to explicitly trade off between efficiency and accuracy. Experimental results show that using a pseudo-FVS of size no larger than $\log(n)$, this procedure converges much more often, more quickly, and provides more accurate results than LBP on the entire graph.
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    ABSTRACT: Given a graph G=(V,E) and a set S⊆V, a set U⊆V is a subset feedback vertex set of (G,S) if no cycle in G[V∖U] contains a vertex of S. The Subset Feedback Vertex Set problem takes as input G, S, and an integer k, and the question is whether (G,S) has a subset feedback vertex set of cardinality or weight at most k. Both the weighted and the unweighted versions of this problem are NP-complete on chordal graphs, even on their subclass split graphs. We give an algorithm with running time O(1.6708n) that enumerates all minimal subset feedback vertex sets on chordal graphs with n vertices. As a consequence, Subset Feedback Vertex Set can be solved in time O(1.6708n) on chordal graphs, both in the weighted and in the unweighted case. On arbitrary graphs, the fastest known algorithm for the problems has O(1.8638n) running time.
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  • Conference Paper: Colouring AT-Free graphs
    Proceedings of the 20th Annual European conference on Algorithms; 09/2012


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