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},
  booktitle={Scandinavian Workshop on Algorithm Theory},
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.

On Properties of a Set of Global Roundings Associated with Clique Connection of Graphs

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



Enumerating Global Roundings of an Outerplanar Graph

This paper proves that the conjecture that there are at most |V|+1 global roundings for \(\mathcal{H}_G\) if G is an outerplanar graph, and gives a polynomial time algorithm for enumerating all the globalroundings of an outer planner graph.

The structure and number of global roundings of a graph

Optimal Roundings of Sequences and Matrices

It is proved that it is NP-hard to compute an approximate solution with approximation ratio smaller than 2 with respect to the unweighted l∞-distance associated with the family W2 of all 2 × 2 square regions.

List total colorings of series-parallel graphs

Matrix rounding under the Lp-discrepancy measure and its application to digital halftoning

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.

On the Stanley-Wilf Conjecture for the Number of Permutations Avoiding a Given Pattern

It is proved that this can be achieved with $n=k^2", and the conjecture that asymptotically $n \sim (k/e)^2$ is the best achievable is proposed, and Noga Alon's conjecture that $n(\sigma)/2 $ is the threshold for random permutations is presented.

Semi-Balanced Colorings of Graphs: Generalized 2-Colorings Based on a Relaxed Discrepancy Condition

An algorithm is designed to enumerate all semi-balanced colorings of G in O(nm2) time by relaxing a certain discrepancy condition on the shortest-paths hypergraph of the graph.

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

List Edge-Colorings of Series-Parallel Graphs

It is proved that for every integer $k\ge3$, for every (simple) series-parallel graph G, there exists a proper edge-coloring of G such that for Every edge of G, the color of e belongs to $L(e)$.

Lattice approximation and linear discrepency of totally unimodular matrices

It seems to be the first time that linear programming is successfully used for a discrepancy problem, as it is shown that the lattice approximation problem for totally unimodular matrices can be solved efficiently and optimally via a linear programming approach.