Rounding of Sequences and Matrices, with Applications

We show that any real matrix can be rounded to an integer matrix in such a way that the rounding errors of all row sums are less than one, and the rounding errors of all column sums as well as all sums of consecutive row entries are less than two.Such roundings can be computed in linear time. This extends and improves previous results on rounding sequences and matrices in several directions. It has particular applications in just-in-time scheduling, where balanced schedules on machines with negligible switch over costs are sought after. Here we extend existing results to multiple machines and non-constant production rates.

[1]  N. Braunera,et al.  The maximum deviation just-intime scheduling problem , 2003 .

[2]  Jon Louis Bentley.  Algorithm Design Techniques , 1984, Commun. ACM.

[3]  Benjamin Doerr,et al.  Lattice approximation and linear discrepency of totally unimodular matrices , 2001, SODA '01.

[4]  Y. Monden.  Toyota Production System: Practical Approach to Production Management , 1983 .

[5]  Jon Bentley,et al.  Programming pearls: algorithm design techniques , 1984, CACM.

[6]  Joel Spencer.  Ten Lectures on the Probabilistic Method: Second Edition , 1994 .

[7]  Yves Crama,et al.  The maximum deviation just-in-time scheduling problem , 2004, Discret. Appl. Math..

[8]  B. Causey,et al.  Applications of Transportation Theory to Statistical Problems , 1985 .

[9]  J. Beck,et al.  Discrepancy Theory , 1996 .

[10]  Kunihiko Sadakane,et al.  Combinatorics and algorithms for low-discrepancy roundings of a real sequence , 2005, Theor. Comput. Sci..

[11]  John B. Kidd,et al.  Toyota Production System , 1993 .

[12]  Donald E. Knuth.  Two-Way Rounding , 1995, SIAM J. Discret. Math..

[13]  Benjamin Doerr,et al.  Global roundings of sequences , 2004, Inf. Process. Lett..

[14]  Takeshi Tokuyama,et al.  Discrepancy-Based Digital Halftoning: Automatic Evaluation and Optimization , 2002, Theoretical Foundations of Computer Vision.

[15]  Anand Srivastav,et al.  Multicolour Discrepancies , 2003, Comb. Probab. Comput..

[16]  J. Spencer.  Ten lectures on the probabilistic method , 1987 .

[17]  G. Steiner,et al.  Level Schedules for Mixed-Model, Just-in-Time Processes , 1993 .