Bertrand Estellon

Diameters, centers, and approximating trees of delta-hyperbolicgeodesic spaces and graphs

δ-Hyperbolic metric spaces have been defined by M. Gromov via a simple 4-point condition: for any four points <i>u,v,w,x</i>, the two larger of the sums <i>d</i>(<i>u,v</i>)+<i>d</i>(<i>w,x</i>), <i>d</i>(<i>u,w</i>)+<i>d</i>(<i>v,x</i>), <i>d</i>(<i>u,x</i>)+<i>d</i>(<i>v,w</i>) differ by at most 2δ. Given a finite set <i>S</i> of points of a δ-hyperbolic space, we present simple and fast methods for approximating the diameter of <i>S</i> with an additive error 2δ and computing an approximate radius and center of a smallest enclosing ball for <i>S</i> with an additive error 3δ. These algorithms run in linear time for classical hyperbolic spaces and for δ-hyperbolic graphs and networks. Furthermore, we show that for δ-hyperbolic graphs <i>G</i>=(<i>V,E</i>) with uniformly bounded degrees of vertices, the exact center of <i>S</i> can be computed in linear time <i>O</i>(|E|). We also provide a simple construction of distance approximating trees of δ-hyperbolic graphs <i>G</i> on <i>n</i> vertices with an additive error <i>O</i>(δlog₂ <i>n</i>). This construction has an additive error comparable with that given by Gromov for <i>n</i>-point δ-hyperbolic spaces, but can be implemented in <i>O</i>(|E|) time (instead of <i>O</i>(<i>n</i>²)). Finally, we establish that several geometrical classes of graphs have bounded hyperbolicity.

Notes on diameters, centers, and approximating trees of δ-hyperbolic geodesic spaces and graphs

We present simple methods for approximating the diameters, radii, and centers of finite sets in δ-hyperbolic geodesic spaces and graphs. We also provide a simple construction of distance approximating trees of δ-hyperbolic graphs G on n vertices with an additive error O(δ log 2 n) comparable with that given by M. Gromov.

Packing and Covering δ-Hyperbolic Spaces by Balls

We consider the problem of covering and packing subsets ofΔ-hyperbolic metric spaces and graphs by balls.These spaces, defined via a combinatorial Gromov condition, haverecently become of interest in several domains of computer science.Specifically, given a subset Sof aΔ-hyperbolic graph Gand a positive numberR, let Δ(S,R) be theminimum number of balls of radius Rcovering S.It is known that computing Δ(S,R)or approximating this number within a constant factor is hard evenfor 2-hyperbolic graphs. In this paper, using a primal-dualapproach, we show how to construct in polynomial time a covering ofSwith at most Δ(S,R)balls of (slightly larger) radius R+ Δ.This result is established in the general framework ofΔ-hyperbolic geodesic metric spaces and is extendedto some other set families derived from balls. The coveringalgorithm is used to design better approximation algorithms for theaugmentation problem with diameter constraints and for thek-center problem in Δ-hyperbolicgraphs.

Mixed Covering of Trees and the Augmentation Problem with Odd Diameter Constraints

In this paper we present a polynomial time algorithm for solving the problem of partial covering of trees with n1 balls of radius R1 and n2 balls of radius R2 (R1 < R2) to maximize the total number of covered vertices. The solutions provided by this algorithm in the particular case R1 = R – 1, R2 = R can be used to obtain for any integer δ > 0 a factor (2+1/δ) approximation algorithm for solving the following augmentation problem with odd diameter constraints D = 2R + 1: Given a tree T, add a minimum number of new edges such that the augmented graph has diameter ≤ D. The previous approximation algorithm of Ishii, Yamamoto, and Nagamochi (2003) has factor 8.

Covering Planar Graphs with a Fixed Number of Balls

In this note we prove that there exists a constant ρ such that any planar graph G of diameter ≤ 2R can be covered with at most ρ balls of radius R, a result conjectured by Gavoille, Peleg, Raspaud, and Sopena in 2001.

Mixed covering of trees and the augmentation problem with odd diameter constraints

Abstract In this talk, we will outline a polynomial time algorithm for solving the problem of partial covering of trees with n1 balls of radius R1 and n2 balls of radius R 2 ( R 1 R 2 ) so as to maximize the total number of covered vertices. We will then show that the solutions provided by this algorithm in the particular case R 1 = R − 1 , R 2 = R can be used to obtain for any integer δ > 0 a factor ( 2 + 1 δ ) approximation algorithm for solving the following augmentation problem with odd diameter constraints D = 2 R + 1 : given a tree T, add a minimum number of new edges such that the augmented graph has diameter ≤D. The previous approximation algorithm of Ishii, Yamamoto, and Nagamochi (2003) has factor 8.