Covering Pairs in Directed Acyclic Graphs

  • Niko Beerenwinkel
  • Stefano Beretta
  • Paola Bonizzoni
  • Riccardo Dondi
  • Yuri Pirola
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8370)

Abstract

The Minimum Path Cover problem on directed acyclic graphs (DAGs) is a classical problem that provides a clear and simple mathematical formulation for several applications in different areas and that has an efficient algorithmic solution. In this paper, we study the computational complexity of two constrained variants of Minimum Path Cover motivated by the recent introduction of next-generation sequencing technologies in bioinformatics. The first problem (MinPCRP), given a DAG and a set of pairs of vertices, asks for a minimum cardinality set of paths “covering” all the vertices such that both vertices of each pair belong to the same path. For this problem, we show that, while it is NP-hard to compute if there exists a solution consisting of at most three paths, it is possible to decide in polynomial time whether a solution consisting of at most two paths exists. The second problem (MaxRPSP), given a DAG and a set of pairs of vertices, asks for a single path containing the maximum number of the given pairs of vertices. We show its NP-hardness and also its W[1]-hardness when parametrized by the number of covered pairs. On the positive side, we give a fixed-parameter algorithm when the parameter is the maximum overlapping degree, a natural parameter in the bioinformatics applications of the problem.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Niko Beerenwinkel
    • 1
  • Stefano Beretta
    • 2
    • 4
  • Paola Bonizzoni
    • 2
  • Riccardo Dondi
    • 3
  • Yuri Pirola
    • 2
  1. 1.Dept. of Biosystems Science and EngineeringETH ZurichBaselSwitzerland
  2. 2.DISCoUniv. degli Studi di Milano-BicoccaMilanItaly
  3. 3.Dip. di Scienze Umane e SocialiUniv. degli Studi di BergamoBergamoItaly
  4. 4.Inst. for Biomedical TechnologiesNational Research CouncilSegrateItaly

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