Sparse Solutions of Sparse Linear Systems: Fixed-Parameter Tractability and an Application of Complex Group Testing

  • Peter Damaschke
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7112)


A vector with at most k nonzeros is called k-sparse. We show that enumerating the support vectors of k-sparse solutions to a system Ax = b of r-sparse linear equations (i.e., where the rows of A are r-sparse) is fixed-parameter tractable (FPT) in the combined parameter r,k. For r = 2 the problem is simple. For 0,1-matrices A we can also compute an O(rk r ) kernel. For systems of linear inequalities we get an FPT result in the combined parameter d,k, where d is the total number of minimal solutions. This is achieved by interpeting the problem as a case of group testing in the complex model. The problems stem from the reconstruction of chemical mixtures by observable reaction products.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bruckstein, A.M., Elad, M., Zibulevsky, M.: On the Uniqueness of Nonnegative Sparse Solutions to Underdetermined Systems of Equations. IEEE Trans. on Info. Theory 54, 4813–4820 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Candés, E.J., Wakin, M.B.: An Introduction to Compressive Sampling. IEEE Signal Proc. Magazine, 21–30 (March 2008)Google Scholar
  3. 3.
    Chen, T., Hwang, F.K.: A Competitive Algorithm in Searching for Many Edges in a Hypergraph. Discr. Appl. Math. 155, 566–571 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Damaschke, P.: Parameterized Enumeration, Transversals, and Imperfect Phylogeny Reconstruction. Theor. Comput. Sci. 351, 337–350 (2006)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Damaschke, P., Molokov, L.: The Union of Minimal Hitting Sets: Parameterized Combinatorial Bounds and Counting. J. Discr. Algor. 7, 391–401 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Donoho, D.L., Tanner, J.: Sparse Nonnegative Solution of Underdetermined Linear Equations by Linear Programming. Proc. Nat. Acad. of Sciences 102, 9446–9451 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dost, B., Bandeira, N., Li, X., Shen, Z., Briggs, S., Bafna, V.: Shared Peptides in Mass Spectrometry Based Protein Quantification. In: Batzoglou, S. (ed.) RECOMB 2009. LNCS, vol. 5541, pp. 356–371. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. 8.
    Fernau, H.: Parameterized Algorithms for d-Hitting Set: The Weighted Case. Theor. Comput. Sci. 411, 1698–1713 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fernau, H.: A Top-Down Approach to Search-Trees: Improved Algorithmics for 3-Hitting Set. Algorithmica 57, 97–118 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of NP-Completeness. Freeman and Company, New York (1979)MATHGoogle Scholar
  11. 11.
    Johann, P.: A Group Testing Problem for Graphs with Several Defective Edges. Discr. Appl. Math. 117, 99–108 (2002)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lacroix, V., Sammeth, M., Guigo, R., Bergeron, A.: Exact Transcriptome Reconstruction from Short Sequence Reads. In: Crandall, K.A., Lagergren, J. (eds.) WABI 2008. LNCS (LNBI), vol. 5251, pp. 50–63. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  13. 13.
    Lai, M.J.: On Sparse Solutions of Underdetermined Linear Systems. J. Concr. Applic. Math. 8, 296–327 (2010)MathSciNetMATHGoogle Scholar
  14. 14.
    Natarajan, B.K.: Sparse Approximate Solutions to Linear Systems. SIAM J. Comput. 24, 227–234 (1995)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Nesvizhskii, A.I., Aebersold, R.: Interpretation of Shotgun Proteomic Data: The Protein Inference Problem. Mol. Cellular Proteomics 4, 1419–1440 (2005)CrossRefGoogle Scholar
  16. 16.
    Triesch, E.: A Group Testing Problem for Hypergraphs of Bounded Rank. Discr. Appl. Math. 66, 185–188 (1996)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Wang, M., Xu, W., Tang, A.: A Unique “Nonnegative” Solution to an Underdetermined System: From Vectors to Matrices. arXiv:1003.4778v1 (2010) (manuscript)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Peter Damaschke
    • 1
  1. 1.Department of Computer Science and EngineeringChalmers UniversityGöteborgSweden

Personalised recommendations