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Sharp Separation and Applications to Exact and Parameterized Algorithms

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Abstract

Many divide-and-conquer algorithms employ the fact that the vertex set of a graph of bounded treewidth can be separated in two roughly balanced subsets by removing a small subset of vertices, referred to as a separator. In this paper we prove a trade-off between the size of the separator and the sharpness with which we can fix the size of the two sides of the partition. Our result appears to be a handy and powerful tool for the design of exact and parameterized algorithms for NP-hard problems. We illustrate that by presenting two applications.

Our first application is a O(2n+o(n))-time algorithm for the Degree Constrained Spanning Tree problem: find a spanning tree of a graph with the maximum number of nodes satisfying given degree constraints. This problem generalizes some well-studied problems, among them Hamiltonian Path, Full Degree Spanning Tree, Bounded Degree Spanning Tree, and Maximum Internal Spanning Tree.

The second application is a parameterized algorithm with running time O(16k+o(k)+n O(1)) for the k-Internal Out-Branching problem: here the goal is to compute an out-branching of a digraph with at least k internal nodes. This is a significant improvement over the best previously known parameterized algorithm for the problem by Cohen et al. (J. Comput. Syst. Sci. 76:650–662, 2010), running in time O(49.4k+n O(1)).

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Authors and Affiliations

  1. Department of Informatics, University of Bergen, Bergen, Norway

    Fedor V. Fomin & Daniel Lokshtanov

  2. Computer Science Department, University of Rome Tor Vergata, Roma, Italy

    Fabrizio Grandoni

  3. The Institute of Mathematical Sciences, Chennai, India

    Saket Saurabh

Authors
  1. Fedor V. Fomin
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  2. Fabrizio Grandoni
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  3. Daniel Lokshtanov
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  4. Saket Saurabh
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Corresponding author

Correspondence to Saket Saurabh.

Additional information

A preliminary version of this paper appeared in the proceedings of LATIN 2010 [20].

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Fomin, F.V., Grandoni, F., Lokshtanov, D. et al. Sharp Separation and Applications to Exact and Parameterized Algorithms. Algorithmica 63, 692–706 (2012). https://doi.org/10.1007/s00453-011-9555-9

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