Nonindependent Randomized Rounding and an Application to Digital Halftoning

Abstract

We investigate the problem of rounding a given [0,1]-valued matrix to a 0,1 matrix such that the rounding error with respect to $2 \times 2$ boxes is small. Such roundings yield good solutions for the digital halftoning problem, as shown by Asano et al. [Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, 2002, SIAM, Philadelphia, 2002, pp. 896--904]. We present a randomized algorithm computing roundings with expected error at most 0.5463 per box, improving the 0.75 nonconstructive bound of Asano et al. Our algorithm is the first to solve this problem fast enough for practical application, namely, in linear time. Of broader interest might be our rounding scheme, which is a modification of randomized rounding. Instead of independently rounding the variables, we impose a number of suitable dependencies. Thus, by equipping the rounding process with some of the problem information, we reduce the rounding error significantly compared to independent randomized rounding, which leads to an expected error of 0.82944 per box. Finally, we give a characterization of realizable dependencies.