On Geometric Structure of Global Roundings for Graphs and Range Spaces

  title={On Geometric Structure of Global Roundings for Graphs and Range Spaces},
  author={Tetsuo Asano and Naoki Katoh and Hisao Tamaki and Takeshi Tokuyama},
Given a hypergraph \(\mathcal{H} = (V, \mathcal{F})\) and a [0,1]-valued vector a ∈ [0,1] V , its global rounding is a binary (i.e.,{0,1}-valued) vector α ∈ {0,1} V such that |∑ v ∈ F (a(v) − α(v))|< 1 holds for each \(F \in \mathcal{F}\). We study geometric (or combinatorial) structure of the set of global roundings of a using the notion of compatible set with respect to the discrepancy distance. We conjecture that the set of global roundings forms a simplex if the hypergraph satisfies… 

Recent Progress on Combinatorics and Algorithms for Low Discrepancy Roundings

The properties of low-discrepancy roundings are surveyed, especially, on the combinatorial properties of a global rounding whose discrepancy is less than 1.

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Given a connected weighted graph G=(V,E), we consider a hypergraph H(G)=(V,F(G)) corresponding to the set of all shortest paths in G. For a given real assignment a on V satisfying 0≤a(v)≤1, a global



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The structure and number of global roundings of a graph

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List total colorings of series-parallel graphs

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It is shown that the matrix rounding using L1-discrepancy for a union of two laminar families is suitable for developing a high-quality digital-halftoning software.

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Combinatorics and algorithms for low-discrepancy roundings of a real sequence

List Edge-Colorings of Series-Parallel Graphs

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On the Discrepancy of 3 Permutations

  • G. Bohus
  • Mathematics
    Random Struct. Algorithms
  • 1990
It is shown that for any constant number of orderings the discrepancy is O(log n) and the proof also gives an efficient algorithm to determine such a coloring.