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Primal-dual approximation algorithms for feedback problems in planar graphs

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Integer Programming and Combinatorial Optimization (IPCO 1996)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1084))

Abstract

Given a subset of cycles of a graph, we consider the problem of finding a minimum-weight set of vertices that meets all cycles in the subset. This problem generalizes a number of problems, including the minimum-weight feedback vertex set problem in both directed and undirected graphs, the subset feedback vertex set problem, and the graph bipartization problem, in which one must remove a minimum-weight set of vertices so that the remaining graph is bipartite. We give a 9/4-approximation algorithm for the general problem in planar graphs, given that the subset of cycles obeys certain properties. This results in 9/4-approximation algorithms for the aforementioned feedback and bipartization problems in planar graphs. Our algorithms use the primaldual method for approximation algorithms as given in Goemans and Williamson [14]. We also show that our results have an interesting bearing on a conjecture of Akiyama and Watanabe [2] on the cardinality of feedback vertex sets in planar graphs.

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Author information

Authors and Affiliations

  1. Dept. of Mathematics, M.I.T., Room 2-382, 02139, Cambridge, MA

    Michel X. Goemans

  2. IBM T.J. Watson Research Center, P.O. Box 218, 10598, Yorktown Heights, NY

    David P. Williamson

Authors
  1. Michel X. Goemans
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  2. David P. Williamson
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William H. Cunningham S. Thomas McCormick Maurice Queyranne

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© 1996 Springer-Verlag Berlin Heidelberg

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Goemans, M.X., Williamson, D.P. (1996). Primal-dual approximation algorithms for feedback problems in planar graphs. In: Cunningham, W.H., McCormick, S.T., Queyranne, M. (eds) Integer Programming and Combinatorial Optimization. IPCO 1996. Lecture Notes in Computer Science, vol 1084. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61310-2_12

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  • DOI: https://doi.org/10.1007/3-540-61310-2_12

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61310-7

  • Online ISBN: 978-3-540-68453-4

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