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Multiple hypernode hitting sets and smallest two-cores with targets

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Abstract

The multiple weighted hitting set problem is to find a subset of nodes in a hypergraph that hits every hyperedge in at least m nodes. We extend the problem to a notion of hypergraphs with so-called hypernodes and show that, for m=2, it remains fixed-parameter tractable (FPT), parameterized by the number of hyperedges. This is accomplished by a nontrivial extension of the dynamic programming algorithm for hypergraphs. The algorithm might be interesting for certain assignment problems, but here we need it as a tool to solve another problem motivated by network analysis: A d-core of a graph is a subgraph in which every vertex has at least d neighbors. We give an FPT algorithm that computes a smallest 2-core including a given set of target vertices, where the number of targets is the parameter. This FPT result is best possible in the sense that no FPT algorithm for 3-cores can be expected.

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References

  • Amini O, Sau I, Saurabh S (2008) Parameterized complexity of the smallest degree-constrained subgraph problem. In: 3rd int workshop on parameterized and exact computation, IWPEC 2008. LNCS, vol 5018. Springer, Berlin, pp 13–29

    Chapter  Google Scholar 

  • Bader GD, Hogue CWV (2003) An automated method for finding molecular complexes in large protein interaction networks. BMC Bioinformatics 4:2

    Article  Google Scholar 

  • Downey RG, Fellows MR (1999) Parameterized complexity. Springer, Berlin

    Google Scholar 

  • Dreyfus S, Wagner R (1971) The Steiner problem in graphs. Networks 1:195–207

    Article  MathSciNet  Google Scholar 

  • Eiter T, Makino K, Gottlob G (2008) Computational aspects of monotone dualization: a brief survey. Discrete Appl Math 156:2035–2049

    Article  MATH  MathSciNet  Google Scholar 

  • Elbassioni K, Hagen M, Rauf I (2008) Some fixed-parameter tractable classes of hypergraph duality and related problems. In: 3rd int workshop on parameterized and exact computation, IWPEC 2008. LNCS, vol 5018. Springer, Berlin, pp 91–102

    Chapter  Google Scholar 

  • Fernau H (2006) Parameterized algorithms for hitting set: the weighted case. In: 6th Italian conf on algorithms and complexity, CIAC 2006. LNCS, vol 3998. Springer, Berlin, pp 332–343

    Google Scholar 

  • Fomin FV, Kratsch D, Woeginger GJ (2004) Exact (exponential) algorithms for the dominating set problem. In: 30th int workshop on graph-theoretic concepts in computer science, WG 2004. LNCS, vol 3353. Springer, Berlin, pp 245–256

    Google Scholar 

  • Haas R, Hoffmann M (2006) Chordless paths through three vertices. Theor Comput Sci 351:360–371

    Article  MATH  MathSciNet  Google Scholar 

  • Niedermeier R (2006) Invitation to fixed-parameter algorithms. Oxford lecture series in mathematics and its applications. Oxford University Press, Oxford

    Book  MATH  Google Scholar 

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Correspondence to Peter Damaschke.

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Damaschke, P. Multiple hypernode hitting sets and smallest two-cores with targets. J Comb Optim 18, 294–306 (2009). https://doi.org/10.1007/s10878-009-9234-9

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