Matrix Rounding with Low Error in Small Submatrices

We show that any real valued matrix A can be rounded to an integer one B such that the error in all 2 x 2 (geometric) submatrices is less than 1.5, that is, we have vertical bar a(ij) - b(ij)vertical bar 1, and vertical bar Sigma(i+1)(k=i) Sigma(j+1)(l=j)(a(kl) - b(kl))vertical bar < 1.5 for all l, j. More precisely, an error of less than 1.5 - 3(-2mn) + 3(-d+1) can be achieved in time O(mnd).