Balancing matrices with line shifts

Abstract

We give a purely deterministic proof of the following theorem of J. Komlós and M. Sulyok. LetA=(a ij ),a ij =±1 be ann×n matrix. One can multiply some rows and columns by −1 such that the absolute value of the sum of the elements of the matrix is ≦2 ifn is even and 1 ifn is odd. Note that Komlós and Sulyok applied probabilistic ideas and so their method worked only forn>n 0.

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References

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    T. A. Brown andJ. H. Spencer, Minimization of ±1 matrices under line shifts,Colloquium Mathematicum (Poland)23 (1971), 165–171.

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    J. Beck andT. Fiala, Integer Making Theorems,Discrete Applied Math. 3 (1981), 1–8.

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    J. Komlós andM. Sulyok. On the sum of elements of ±1 matrices, in:Combinatorial Theory and Its Applications (Erdős et al., eds.), North-Holland 1970, 721–728.

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Dedicated to Paul Erdős on his seventieth birthday

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Beck, J., Spencer, J. Balancing matrices with line shifts. Combinatorica 3, 299–304 (1983). https://doi.org/10.1007/BF02579185

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AMS subject classification (1980)

  • 05 B 20