Complexity and Polynomial-Time Approximation Algorithms around the Scaffolding Problem

  • Annie Chateau
  • Rodolphe Giroudeau
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8542)


We explore in this paper some complexity issues inspired by the contig scaffolding problem in bioinformatics. We focus on the following problem: given an undirected graph with no loop, and a perfect matching on this graph, find a set of cycles and paths covering every vertex of the graph, with edges alternatively in the matching and outside the matching, and satisfying a given constraint on the numbers of cycles and paths. We show that this problem is \(\mathcal{NP}\)-complete, even in bipartite graphs. We also exhibit non-approximability and polynomial-time approximation results, in the optimization versions of the problem.


Complexity Polynomial-Time Approximation Scaffolding 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Burton, J.N., Adey, A., Patwardhan, R.P., Qiu, R., Kitzman, J.O., Shendure, J.: Chromosome-scale scaffolding of de novo genome assemblies based on chromatin interactions. Nature Biotechnology, 1119–1125 (November 2013)Google Scholar
  2. 2.
    Chauve, C., Patterson, M., Rajaraman, A.: Hypergraph covering problems motivated by genome assembly questions. In: Lecroq, T., Mouchard, L. (eds.) IWOCA 2013. LNCS, vol. 8288, pp. 428–432. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  3. 3.
    Chiba, S., Fujita, S.: Covering vertices by a specified number of disjoint cycles, edges and isolated vertices. Discrete Mathematics 313(3), 269–277 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Dayarian, A., Michael, T.P., Sengupta, A.M.: SOPRA: scaffolding algorithm for paired reads via statistical optimization. BMC Bioinformatics 11, 345 (2010)CrossRefGoogle Scholar
  5. 5.
    Donmez, N., Brudno, M.: SCARPA: scaffolding reads with practical algorithms. Bioinformatics 29(4), 428–434 (2013)CrossRefGoogle Scholar
  6. 6.
    Gabow, H.N.: An efficient implementation of Edmonds’ algorithm for maximum matching on graphs. Journal of the ACM 23(2), 221–234 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Gao, S., Sung, W., Nagarajan, N.: Opera: reconstructing optimal genomic scaffolds with high-throughput paired-end sequences. Journal of Computational Biology 18(11), 1681–1691 (2011)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)Google Scholar
  9. 9.
    Gritsenko, A.A., Nijkamp, J.F., Reinders, M.J., Ridder, D.D.: GRASS: a generic algorithm for scaffolding next-generation sequencing assemblies. Bioinformatics, 1429–1437 (2012)Google Scholar
  10. 10.
    Huson, D.H., Reinert, K., Myers, E.W.: The greedy path-merging algorithm for contig scaffolding. Journal of ACM 49(5), 603–615 (2002)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Krivelevich, M., Nutov, Z., Salavatipour, M.R., Yuster, J.V., Yuster, R.: Approximation algorithms and hardness results for cycle packing problems. ACM Transaction on Algorithms 3(4) (November 2007)Google Scholar
  12. 12.
    Sahni, S., Gonzalez, T.: P-complete approximation problems. Journal of ACM 23(3), 555–565 (1976), CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Shmoys, D.B., Lenstra, J.K., Kan, A.H.G.R., Lawler, E.L.: The Traveling Salesman Problem: a guided tour of combinatorial optimization. John Wiley & Sons (1985)Google Scholar
  14. 14.
    Steiner, G.: On the k-path partition of graphs. Theoretical Computer Science 290, 2147–2155 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Tutte, W.T.: A short proof of the factor theorem for finite graphs. Canadian Journal of Mathematics 6, 347–352 (1954)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Annie Chateau
    • 1
    • 2
  • Rodolphe Giroudeau
    • 1
  1. 1.LIRMM - CNRS UMR 5506MontpellierFrance
  2. 2.Institut de Biologie ComputationnelleMontpellierFrance

Personalised recommendations