Fixed-Parameter algorithms for cochromatic number and disjoint rectangle stabbing

  • Authors:

  • Pinar Heggernes;Dieter Kratsch;Daniel Lokshtanov;Venkatesh Raman;Saket Saurabh

  • Affiliations:

  • Department of Informatics, University of Bergen, Bergen, Norway;Université de Metz, France;Department of Informatics, University of Bergen, Bergen, Norway;The Institute of Mathematical Sciences, Chennai, India;The Institute of Mathematical Sciences, Chennai, India

  • Venue:
  • SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
  • Year:
  • 2010

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Abstract

Given a permutation π of {1,...,n} and a positive integer k, we give an algorithm with running time $2^{O(k^2 \log k)}n^{O(1)}$ that decides whether π can be partitioned into at most k increasing or decreasing subsequences. Thus we resolve affirmatively the open question of whether the problem is fixed parameter tractable. This NP-complete problem is equivalent to deciding whether the cochromatic number (the minimum number of cliques and independent sets the vertices of the graph can be partitioned into) of a given permutation graph on n vertices is at most k. In fact, we give a more general result: within the mentioned running time, one can decide whether the cochromatic number of a given perfect graph on n vertices is at most k. To obtain our result we use a combination of two well-known techniques within parameterized algorithms, namely greedy localization and iterative compression. We further demonstrate the power of this combination by giving a $2^{O(k^2 \log k)}n \log n$ time algorithm for deciding whether a given set of n non-overlapping axis-parallel rectangles can be stabbed by at most k of the given set of horizontal and vertical lines. Whether such an algorithm exists was mentioned as an open question in several papers.