Bandwidth on AT-free graphs

  • Authors:

  • Petr Golovach;Pinar Heggernes;Dieter Kratsch;Daniel Lokshtanov;Daniel Meister;Saket Saurabh

  • Affiliations:

  • Department of Informatics, University of Bergen, 5020 Bergen, Norway;Department of Informatics, University of Bergen, 5020 Bergen, Norway;Laboratoire dInformatique Théorique et Appliquée, Université Paul VerlaineMetz, 57045 Metz Cedex 01, France;Department of Informatics, University of Bergen, 5020 Bergen, Norway;Department of Informatics, University of Bergen, 5020 Bergen, Norway;Department of Informatics, University of Bergen, 5020 Bergen, Norway

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2011

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Abstract

We study the classical Bandwidth problem from the viewpoint of parametrised algorithms. Given a graph G=(V,E) and a positive integer k, the Bandwidth problem asks whether there exists a bijective function @b:{1,...,|V|}-V such that for every edge uv@?E, |@b^-^1(u)-@b^-^1(v)|@?k. It is known that under standard complexity assumptions, no algorithm for Bandwidth with running time of the form f(k)n^O^(^1^) exists, even when the input is restricted to trees. We initiate the search for classes of graphs where such algorithms do exist. We present an algorithm with running time n@?2^O^(^k^l^o^g^k^) for Bandwidth on AT-free graphs, a well-studied graph class that contains interval, permutation, and cocomparability graphs. Our result is the first non-trivial algorithm that shows fixed-parameter tractability of Bandwidth on a graph class on which the problem remains NP-complete.