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Solving Linear Equations Parameterized by Hamming Weight

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Abstract

Given a system of linear equations \(Ax=b\) over the binary field \(\mathbb {F}_2\) and an integer \(t\ge 1\), we study the following three algorithmic problems:

  1. 1.

    Does \(Ax=b\) have a solution of weight at most t?

  2. 2.

    Does \(Ax=b\) have a solution of weight exactly t?

  3. 3.

    Does \(Ax=b\) have a solution of weight at least t?

We investigate the parameterized complexity of these problems with t as parameter. A special aspect of our study is to show how the maximum multiplicity k of variable occurrences in \(Ax=b\) influences the complexity of the problem. We show a sharp dichotomy: for each \(k\ge 3\) the first two problems are \(\textsf {W[1] }\)-hard [which strengthens and simplifies a result of Downey et al. (SIAM J Comput 29(2), 545–570, 1999)]. For \(k=2\), the problems turn out to be intimately connected to well-studied matching problems and can be efficiently solved using matching algorithms.

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References

  1. Alon, N., Yuster, R., Zwick, U.: Color coding. J. ACM 42(4), 844–856 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Buntrock, G., Damm, C., Hertrampf, U., Meinel, C.: Structure and importance of logspace-MOD class. Theory Comput. Syst. 25(3), 223–237 (1992)

    MathSciNet  MATH  Google Scholar 

  4. Berlekamp, E.R., McEliece, R.J., van Tilborg, H.C.A.: On the inherent intractability of certain coding problems. IEEE Trans. Inform. Theory 24, 384–386 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bruschi, D., Ravasio, F.: Random parallel algorithms for finding exact branchings, perfect matchings, and cycles. Algorithmica 13(4), 346–356 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cheng, Q., Wan, D.: A deterministic reduction for the gap minimum distance problem. IEEE Trans. Inform. Theory 58, 6935–6941 (2012)

    Article  MathSciNet  Google Scholar 

  7. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)

    Book  MATH  Google Scholar 

  8. Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer, Texts in Computer Science, New York (2013)

    Book  MATH  Google Scholar 

  9. Downey, R.G., Fellows, M.R., Vardy, A., Whittle, G.: The parametrized complexity of some fundamental problems in coding theory. SIAM J. Comput. 29(2), 545–570 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Elberfeld, M., Jakoby, A., Tantau, T.: Logspace versions of the theorems of Bodlaender and Courcelle. In: Proceedings of 51st FOCS, pp. 143–152 (2010)

  11. Jones, N.D., Edmund Lien, Y., Laaser, W.T.: New problems complete for nondeterministic log space. Math. Syst. Theory 10(1), 1–17 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  12. Johnson, D.S.: The NP-completeness column. ACM Trans. Algorithms 1(1), 160–176 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ntafos, S.C., Louis Hakimi, S.: On the complexity of some coding problems. IEEE Trans. Inf. Theory 27(6), 794–796 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  14. Marx, D.: Parameterized complexity of constraint satisfaction problems. Comput. Complex. 14(2), 153–183 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mulmuley, K., Vazirani, U., Vazirani, V.: Matching is as easy as matrix inversion. Combinatorica 7, 105–113 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  16. Papadimitriou, C., Yannakakis, M.: The complexity of restricted spanning tree problems. J. ACM 29, 285–309 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  17. Reingold, O.: Undirected connectivity in log-space. J. ACM 55(4), 17:1–17:24 (2008)

  18. Vardy, A.: The intractability of computing the minimum distance of a code. IEEE Trans. Inform. Theory 43, 1757–1766 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Vardy, A.: Algorithmic complexity in coding theory and the minimum distance problem. In: Proceedings of 29th ACM Symposium on Theory of Computing, pp. 92–109 (1997)

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Acknowledgments

Gaurav Rattan and Oleg Verbitsky suggested and proved Lemma 6.1, which allowed us to improve the fpt algorithm for \(\textsc {LinEq}_{\ge {,t}}\) of the conference version to the polynomial time kernelization of Lemma 6.2. We also thank the reviewers for their helpful remarks.

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Authors and Affiliations

  1. Institute of Mathematical Sciences, Chennai, India

    V. Arvind

  2. Institut für Informatik, Humboldt-Universität zu Berlin, Berlin, Germany

    Johannes Köbler & Sebastian Kuhnert

  3. Institut für Theoretische Informatik, Universität Ulm, Ulm, Germany

    Jacobo Torán

Authors
  1. V. Arvind
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  2. Johannes Köbler
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  3. Sebastian Kuhnert
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  4. Jacobo Torán
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Corresponding author

Correspondence to Sebastian Kuhnert.

Additional information

An extended abstract of this article appears in the proceedings of IPEC 2014. This work was supported by the Alexander von Humboldt Foundation in its research group linkage program. The third author was supported by DFG grant KO 1053/7-2.

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Arvind, V., Köbler, J., Kuhnert, S. et al. Solving Linear Equations Parameterized by Hamming Weight. Algorithmica 75, 322–338 (2016). https://doi.org/10.1007/s00453-015-0098-3

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  • DOI: https://doi.org/10.1007/s00453-015-0098-3

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