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)


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bao, E., Jiang, T., Girke, T.: BRANCH: Boosting RNA-Seq assemblies with partial or related genomic sequences. Bioinformatics 29(10), 1250–1259 (2013)CrossRefGoogle Scholar
  2. 2.
    Bonizzoni, P., Dondi, R., Pirola, Y.: Maximum disjoint paths on edge-colored graphs: Approximability and tractability. Algorithms 6(1), 1–11 (2013)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Downey, R., Fellows, M.: Parameterized complexity. Springer (1999)Google Scholar
  4. 4.
    Downey, R., Fellows, M.: Fixed-parameter tractability and completeness II: On completeness for W[1]. Theoretical Computer Science 141(1&2), 109–131 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Eriksson, N., Pachter, L., Mitsuya, Y., Rhee, S., Wang, C., Gharizadeh, B., Ronaghi, M., Shafer, R., Beerenwinkel, N.: Viral population estimation using pyrosequencing. PLoS Computational Biology 4(5), e1000074 (2008)Google Scholar
  6. 6.
    Fulkerson, D.R.: Note on Dilworth’s decomposition theorem for partially ordered sets. Proc. American Mathematical Society 7, 701–702 (1956)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Gabow, H., Maheshwari, S., Osterweil, L.: On two problems in the generation of program test paths. IEEE Trans.on Software Engineering 2(3), 227–231 (1976)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Garey, M., Johnson, D.: Computer and intractability: A guide to the theory of NP-completeness. W. H. Freeman (1979)Google Scholar
  9. 9.
    Niedermeier, R.: Invitation to fixed-parameter algorithms. Oxford Univ. Press (2006)Google Scholar
  10. 10.
    Ntafos, S., Hakimi, S.: On path cover problems in digraphs and applications to program testing. IEEE Trans. on Software Engineering 5(5), 520–529 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Trapnell, C., Williams, B., Pertea, G., Mortazavi, A., Kwan, G., van Baren, M., Salzberg, S., Wold, B., Pachter, L.: Transcript assembly and quantification by RNA-Seq reveals unannotated transcripts and isoform switching during cell differentiation. Nature Biotechnology 28(5), 516–520 (2010)CrossRefGoogle Scholar
  12. 12.
    Wu, B.Y.: On the maximum disjoint paths problem on edge-colored graphs. Discrete Optimization 9(1), 50–57 (2012)CrossRefzbMATHMathSciNetGoogle Scholar

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

Personalised recommendations