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Sparse 0−1 Matrices and Forbidden Hypergraphs

Published online by Cambridge University Press:  01 September 1999

CLAUDIA BERTRAM-KRETZBERG
Affiliation:
Lehrstuhl Informatik 2, Universität Dortmund, D–44221 Dortmund, Germany (e-mail: bertram@Ls2.cs.uni-dortmund.de, hofmeist@Ls2.cs.uni-dortmund.de, lefmann@Ls2.cs.uni-dortmund.de)
THOMAS HOFMEISTER
Affiliation:
Lehrstuhl Informatik 2, Universität Dortmund, D–44221 Dortmund, Germany (e-mail: bertram@Ls2.cs.uni-dortmund.de, hofmeist@Ls2.cs.uni-dortmund.de, lefmann@Ls2.cs.uni-dortmund.de)
HANNO LEFMANN
Affiliation:
Lehrstuhl Informatik 2, Universität Dortmund, D–44221 Dortmund, Germany (e-mail: bertram@Ls2.cs.uni-dortmund.de, hofmeist@Ls2.cs.uni-dortmund.de, lefmann@Ls2.cs.uni-dortmund.de)

Abstract

We consider the problem of determining the maximum number N(m, k, r) of columns of a 0−1 matrix with m rows and exactly r ones in each column such that every k columns are linearly independent over ℤ2. For fixed integers k[ges ]4 and r[ges ]2, where k is even and gcd(k−1, r) = 1, we prove the lower bound N(m, k, r) = Ω(mkr/2(k−1) ·(ln m)1/k−1). This improves on earlier results from [14] by a factor Θ((ln m)1/k−1). Moreover, we describe a polynomial time algorithm achieving this new lower bound.

Type
Research Article
Copyright
1999 Cambridge University Press

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Footnotes

This research was supported by the Deutsche Forschungsgemeinschaft as part of the Collaborative Research Center ‘Computational Intelligence’ (SFB 531).This paper was submitted to the Special Issue (Volume 8, Number 4) devoted to papers from the Oberwolfach meeting on Random Graphs and Combinatorial Structures, but was inadvertently omitted from that Issue.