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.
Does \(Ax=b\) have a solution of weight at most t?
-
2.
Does \(Ax=b\) have a solution of weight exactly t?
-
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.
Similar content being viewed by others
References
Alon, N., Yuster, R., Zwick, U.: Color coding. J. ACM 42(4), 844–856 (1995)
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)
Buntrock, G., Damm, C., Hertrampf, U., Meinel, C.: Structure and importance of logspace-MOD class. Theory Comput. Syst. 25(3), 223–237 (1992)
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)
Bruschi, D., Ravasio, F.: Random parallel algorithms for finding exact branchings, perfect matchings, and cycles. Algorithmica 13(4), 346–356 (1995)
Cheng, Q., Wan, D.: A deterministic reduction for the gap minimum distance problem. IEEE Trans. Inform. Theory 58, 6935–6941 (2012)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)
Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer, Texts in Computer Science, New York (2013)
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)
Elberfeld, M., Jakoby, A., Tantau, T.: Logspace versions of the theorems of Bodlaender and Courcelle. In: Proceedings of 51st FOCS, pp. 143–152 (2010)
Jones, N.D., Edmund Lien, Y., Laaser, W.T.: New problems complete for nondeterministic log space. Math. Syst. Theory 10(1), 1–17 (1976)
Johnson, D.S.: The NP-completeness column. ACM Trans. Algorithms 1(1), 160–176 (2005)
Ntafos, S.C., Louis Hakimi, S.: On the complexity of some coding problems. IEEE Trans. Inf. Theory 27(6), 794–796 (1981)
Marx, D.: Parameterized complexity of constraint satisfaction problems. Comput. Complex. 14(2), 153–183 (2005)
Mulmuley, K., Vazirani, U., Vazirani, V.: Matching is as easy as matrix inversion. Combinatorica 7, 105–113 (1987)
Papadimitriou, C., Yannakakis, M.: The complexity of restricted spanning tree problems. J. ACM 29, 285–309 (1982)
Reingold, O.: Undirected connectivity in log-space. J. ACM 55(4), 17:1–17:24 (2008)
Vardy, A.: The intractability of computing the minimum distance of a code. IEEE Trans. Inform. Theory 43, 1757–1766 (1997)
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)
Author information
Authors and Affiliations
Corresponding author
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.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-015-0098-3