Interval completion with few edges

  • Authors:

  • Pinar Heggernes;Christophe Paul;Jan Arne Telle;Yngve Villanger

  • Affiliations:

  • University of Bergen, Bergen, Norway;CNRS - LIRMM and School of Computer Science McGill Univ. Canada, Montpellier, France;University of Bergen, Bergen, Norway;University of Bergen, Bergen, Norway

  • Venue:
  • Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
  • Year:
  • 2007

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Abstract

We present an algorithm with runtime O(k(2k)n3 * m) for the following NP-complete problem: Given an arbitrary graph G on n vertices and m edges, can we obtain an interval graph by adding at most k new edges to G? This resolves the long-standing open question, first posed by Kaplan, Shamir and Tarjan, of whether this problem could be solved in time f(k) * n(O(1)).The problem has applications in Physical Mapping of DNA and in Profile Minimization for Sparse Matrix Computations. For the first application, our results show tractability for the case of a small number k of false negative errors, and for the second, a small number k of zero elements in the envelope. Our algorithm performs bounded search among possible ways of adding edges to a graph to obtain an interval graph, and combines this with a greedy algorithm when graphs of a certain structure are reached by the search. The presented result is surprising, as it was not believed that a bounded search tree algorithm would suffice to answer the open question affirmatively.