TY - GEN

T1 - On the complexities of the optimal rounding problems of sequences and matrices

AU - Asano, Tetsuo

AU - Matsui, Tomomi

AU - Tokuyama, Takeshi

PY - 2000/1/1

Y1 - 2000/1/1

N2 - In this paper, we discuss the problem of computing an opti-mal 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 DistF,w∞ (A;B) between an input sequence (resp. matrix) A and the output B. The distance is dependent on a family F of inter-vals (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 l1 distance, and the other for any weight function w, for any family F of intervals. We give an algorithm that computes a ma-trix 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.

AB - In this paper, we discuss the problem of computing an opti-mal 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 DistF,w∞ (A;B) between an input sequence (resp. matrix) A and the output B. The distance is dependent on a family F of inter-vals (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 l1 distance, and the other for any weight function w, for any family F of intervals. We give an algorithm that computes a ma-trix 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.

UR - http://www.scopus.com/inward/record.url?scp=84956862020&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84956862020&partnerID=8YFLogxK

U2 - 10.1007/3-540-44985-X

DO - 10.1007/3-540-44985-X

M3 - Conference contribution

AN - SCOPUS:84956862020

SN - 3540676902

SN - 9783540676904

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 476

EP - 489

BT - Algorithm Theory - SWAT 2000 - 7th Scandinavian Workshop on Algorithm Theory, 2000, Proceedings

A2 - Halldórsson, Magnús M.

PB - Springer-Verlag

T2 - 7th Scandinavian Workshop on Algorithm Theory, SWAT 2000

Y2 - 5 July 2000 through 7 July 2000

ER -