Theory of linear and integer programming
Theory of linear and integer programming
Digital halftones by dot diffusion
ACM Transactions on Graphics (TOG)
Convex separable optimization is not much harder than linear optimization
Journal of the ACM (JACM)
Network flows: theory, algorithms, and applications
Network flows: theory, algorithms, and applications
Handbook of combinatorics (vol. 2)
Polynomial Methods for Separable Convex Optimization in Unimodular Linear Spaces with Applications
SIAM Journal on Computing
Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems
Journal of the ACM (JACM)
Lattice approximation and linear discrepency of totally unimodular matrices
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Optimal roundings of sequences and matrices
Nordic Journal of Computing
Non-independent randomized rounding
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Non-independent Randomized Rounding and an Application to Digital Halftoning
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
The structure and number of global roundings of a graph
Theoretical Computer Science - Special papers from: COCOON 2003
Combinatorics and algorithms for low-discrepancy, roundings of a real sequence
Theoretical Computer Science - Automata, languages and programming
Non-independent randomized rounding and coloring
Discrete Applied Mathematics - Special issue: Efficient algorithms
Non-independent randomized rounding and coloring
Discrete Applied Mathematics - Special issue: Efficient algorithms
The structure and number of global roundings of a graph
COCOON'03 Proceedings of the 9th annual international conference on Computing and combinatorics
Combinatorial and geometric problems related to digital halftoning
Proceedings of the 11th international conference on Theoretical foundations of computer vision
Discrepancy-based digital halftoning: automatic evaluation and optimization
Proceedings of the 11th international conference on Theoretical foundations of computer vision
A generalization of magic squares with applications to digital halftoning
ISAAC'04 Proceedings of the 15th international conference on Algorithms and Computation
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In this paper we study the problem of rounding a real-valued matrix into an integer-valued matrix to minimize an Lp-discrepancy measure between them. To define the Lp-discrepancy measure, we introduce a family F of regions (rigid submatrices) of the matrix, and consider a hypergraph defined by the family. The difficulty of the problem depends on the choice of the region family F. We first investigate the rounding problem by using integer programming problems with convex piecewise-linear objective functions, and give some nontrivial upper bounds for the Lp-discrepancy. Then, we propose "laminar family" for constructing a practical and well-solvable class of F. Indeed, we show that the problem is solvable in polynomial time if F is a union of two laminar families. Finally, we show that the matrix rounding using L1-discrepancy for a union of two laminar families is suitable for developing a high-quality digital-halftoning software.