Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
Candés, E.J., Wakin, M.B.: An Introduction to Compressive Sampling. IEEE Signal Proc. Magazine, 21–30 (March 2008)
Chen, T., Hwang, F.K.: A Competitive Algorithm in Searching for Many Edges in a Hypergraph. Discr. Appl. Math. 155, 566–571 (2007)
Damaschke, P.: Parameterized Enumeration, Transversals, and Imperfect Phylogeny Reconstruction. Theor. Comput. Sci. 351, 337–350 (2006)
Damaschke, P., Molokov, L.: The Union of Minimal Hitting Sets: Parameterized Combinatorial Bounds and Counting. J. Discr. Algor. 7, 391–401 (2009)
Donoho, D.L., Tanner, J.: Sparse Nonnegative Solution of Underdetermined Linear Equations by Linear Programming. Proc. Nat. Acad. of Sciences 102, 9446–9451 (2005)
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)
Fernau, H.: Parameterized Algorithms for d-Hitting Set: The Weighted Case. Theor. Comput. Sci. 411, 1698–1713 (2010)
Fernau, H.: A Top-Down Approach to Search-Trees: Improved Algorithmics for 3-Hitting Set. Algorithmica 57, 97–118 (2010)
Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of NP-Completeness. Freeman and Company, New York (1979)
Johann, P.: A Group Testing Problem for Graphs with Several Defective Edges. Discr. Appl. Math. 117, 99–108 (2002)
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)
Lai, M.J.: On Sparse Solutions of Underdetermined Linear Systems. J. Concr. Applic. Math. 8, 296–327 (2010)
Natarajan, B.K.: Sparse Approximate Solutions to Linear Systems. SIAM J. Comput. 24, 227–234 (1995)
Nesvizhskii, A.I., Aebersold, R.: Interpretation of Shotgun Proteomic Data: The Protein Inference Problem. Mol. Cellular Proteomics 4, 1419–1440 (2005)
Triesch, E.: A Group Testing Problem for Hypergraphs of Bounded Rank. Discr. Appl. Math. 66, 185–188 (1996)
Wang, M., Xu, W., Tang, A.: A Unique “Nonnegative” Solution to an Underdetermined System: From Vectors to Matrices. arXiv:1003.4778v1 (2010) (manuscript)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Damaschke, P. (2012). Sparse Solutions of Sparse Linear Systems: Fixed-Parameter Tractability and an Application of Complex Group Testing. In: Marx, D., Rossmanith, P. (eds) Parameterized and Exact Computation. IPEC 2011. Lecture Notes in Computer Science, vol 7112. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28050-4_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-28050-4_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-28049-8
Online ISBN: 978-3-642-28050-4
eBook Packages: Computer ScienceComputer Science (R0)