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ISAAC '01 Proceedings of the 12th International Symposium on Algorithms and Computation
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In this paper, we discuss the problem of computing an optimal rounding of a real sequence (resp. matrix) into an integral sequence (resp. matrix). Our criterion of the optimality is to minimize the weighted l∞ distance Dist∞F, w (A, B) between an input sequence (resp. matrix) A and the output B. The distance is dependent on a family F of intervals (resp. rectangular regions) for the sequence rounding (resp. matrix rounding) and positive valued weight function w on the family. We give efficient polynomial time algorithms for the sequence-rounding problem, one for the weighted l∞ distance, and the other for any weight function w, for any family F of intervals. We give an algorithm that computes a matrix rounding with an error at most 1:75 with respect to the unweighted l∞ distance associated with the family W2 of all 2 × 2 square regions, whereas we prove that it is NP-hard to compute an approximate solution to the matrix-rounding problem with an approximate ratio smaller than 2 for the same distance.