# On Geometric Structure of Global Roundings for Graphs and Range Spaces

@inproceedings{Asano2004OnGS, 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}, year={2004} }

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…

## 2 Citations

### Recent Progress on Combinatorics and Algorithms for Low Discrepancy Roundings

- MathematicsGraphs Comb.
- 2007

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

- Mathematics
- 2004

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…

## References

SHOWING 1-10 OF 19 REFERENCES

### Enumerating Global Roundings of an Outerplanar Graph

- MathematicsISAAC
- 2003

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.

### Optimal Roundings of Sequences and Matrices

- Computer Science, MathematicsNord. J. Comput.
- 2000

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.

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

- MathematicsSODA '02
- 2002

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

- MathematicsElectron. J. Comb.
- 1999

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

- MathematicsGraphs Comb.
- 2004

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

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 2005

### List Edge-Colorings of Series-Parallel Graphs

- MathematicsElectron. J. Comb.
- 1999

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

- Computer Science, MathematicsSODA '01
- 2001

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.