Sharp separation and applications to exact and parameterized algorithms

  • Authors:

  • Fedor V. Fomin;Daniel Lokshtanov;Fabrizio Grandoni;Saket Saurabh

  • Affiliations:

  • Department of Informatics, University of Bergen, Bergen, Norway;Department of Informatics, University of Bergen, Bergen, Norway;Dipartimento di Informatica, Sistemi e Produzione, Università di Roma Tor Vergata, Roma, Italy;The Institute of Mathematical Sciences, Chennai, India

  • Venue:
  • LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
  • Year:
  • 2010

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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 parameterized algorithm with running time O(16k+o(k)+nO(1)) for the Maximum Internal Subtree problem in directed graphs. This is a significant improvement over the best previously known parameterized algorithm for the problem by [Cohen et al.’09], running in time O(49.4k+nO(1)). The second application is a O(2n+o(n)) time and space 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, Maximum Internal Spanning Tree and their edge weighted variants.