Minimum leaf out-branching and related problems

  • Authors:

  • Gregory Gutin;Igor Razgon;Eun Jung Kim

  • Affiliations:

  • Department of Computer Science, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK;Department of Computer Science, University College Cork, Ireland;Department of Computer Science, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2009

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Abstract

Given a digraph D, the Minimum Leaf Out-Branching problem (MinLOB) is the problem of finding in D an out-branching with the minimum possible number of leaves, i.e., vertices of out-degree 0. We prove that MinLOB is polynomial-time solvable for acyclic digraphs. In general, MinLOB is NP-hard and we consider three parameterizations of MinLOB. We prove that two of them are NP-complete for every value of the parameter, but the third one is fixed-parameter tractable (FPT). The FPT parameterization is as follows: given a digraph D of order n and a positive integral parameter k, check whether D contains an out-branching with at most n-k leaves (and find such an out-branching if it exists). We find a problem kernel of order O(k^2) and construct an algorithm of running time O(2^O^(^k^l^o^g^k^)+n^6), which is an 'additive' FPT algorithm. We also consider transformations from two related problems, the minimum path covering and the maximum internal out-tree problems into MinLOB, which imply that some parameterizations of the two problems are FPT as well.