Cop and Robber Game and Hyperbolicity

Chalopin, Jérémie; Chepoi, Victor; Papasoglu, Panos; Pecatte, Timothée

doi:10.1137/130941328

Cited by 16 publications

(12 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [19], approximation algorithms are given to compute a (1 + )approximation in O( −1 n 3.38 ) time and a (2 + )-approximation in O( −1 n 2.38 ) time. As a direct application of the characterization of hyperbolicity of graphs via a cop and robber game and dismantlability, [9] presents a simple constant factor approximation algorithm for hyperbolicity of G running in optimal O(n 2 ) time. Its approximation ratio is huge (1569), however it is believed that its theoretical performance is much better and the factor of 1569 is mainly due to the use in the proof of the definition of hyperbolicity via linear isoperimetric inequality.…”

Section: Introductionmentioning

confidence: 99%

Fast Approximation and Exact Computation of Negative Curvature Parameters of Graphs

Chalopin

Chepoi

Dragan

et al. 2019

Discrete Comput Geom

Self Cite

View full text Add to dashboard Cite

In this paper, we study Gromov hyperbolicity and related parameters, that represent how close (locally) a metric space is to a tree from a metric point of view. The study of Gromov hyperbolicity for geodesic metric spaces can be reduced to the study of graph hyperbolicity. The main contribution of this paper is a new characterization of the hyperbolicity of graphs, via a new parameter which we call rooted insize. This characterization has algorithmic implications in the field of large-scale network analysis. A sharp estimate of graph hyperbolicity is useful, e.g., in embedding an undirected graph into hyperbolic space with minimum distortion [Verbeek and Suri, SoCG'14]. The hyperbolicity of a graph can be computed in polynomialtime, however it is unlikely that it can be done in subcubic time. This makes this parameter difficult to compute or to approximate on large graphs. Using our new characterization of graph hyperbolicity, we provide a simple factor 8 approximation algorithm (with an additive constant 1) for computing the hyperbolicity of an n-vertex graph G = (V, E) in optimal time O(n 2 ) (assuming that the input is the distance matrix of the graph). This algorithm leads to constant factor approximations of other graph-parameters related to hyperbolicity (thinness, slimness, and insize). We also present the first efficient algorithms for exact computation of these parameters. All of our algorithms can be used to approximate the hyperbolicity of a geodesic metric space.We also show that a similar characterization of hyperbolicity holds for all geodesic metric spaces endowed with a geodesic spanning tree. Along the way, we prove that any complete geodesic metric space (X, d) has such a geodesic spanning tree. 1 The O(·) notation hides polyloglog factors. 2 Informally, (y|z)w can be viewed as half the detour you make, when going over w to get from y to z.

show abstract

Section: Introductionmentioning

confidence: 99%

Fast Approximation and Exact Computation of Negative Curvature Parameters of Graphs

Chalopin

Chepoi

Dragan

et al. 2019

Discrete Comput Geom

Self Cite

View full text Add to dashboard Cite

show abstract

“…This statement plays a fundamental role in the cubulation process in proving the cocompactness of the cube complex associated with a finite set of quasiconvex codimension-1 subgroups [22,33,35]. The Helly property for balls proved in [15] is also important in the dismantlability and cop-and-robber game characterizations of hyperbolic graphs established in [12].…”

Section: Introductionmentioning

confidence: 95%

Core congestion is inherent in hyperbolic networks

Chepoi¹,

Dragan²,

Vaxès³

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

Self Cite

View full text Add to dashboard Cite

Abstract. We investigate the impact the negative curvature has on the traffic congestion in large-scale networks. We prove that every Gromov hyperbolic network G admits a core, thus answering in the positive a conjecture by Jonckheere, Lou, Bonahon, and Baryshnikov, Internet Mathematics, 7 (2011) which is based on the experimental observation by Narayan and Saniee, Physical Review E, 84 (2011) that real-world networks with small hyperbolicity have a core congestion. Namely, we prove that for every subset X of vertices of a graph with δ-thin geodesic triangles (in particular, of a δ-hyperbolic graph) G there exists a vertex m of G such that the ball B(m, 4δ) of radius 4δ centered at m intercepts at least one half of the total flow between all pairs of vertices of X, where the flow between two vertices x, y ∈ X is carried by geodesic (or quasi-geodesic) (x, y)-paths. Moreover, we prove a primaldual result showing that, for any commodity graph R on X and any r ≥ 8δ, the size σr(R) of the least r-multi-core (i.e., the number of balls of radius r) intercepting all pairs of R is upper bounded by the maximum number of pairwise (2r − 5δ)-apart pairs of R and that an r-multi-core of size σ r−5δ (R) can be computed in polynomial time for every finite set X.Our result about total r-multi-cores is based on a Hellytype theorem for quasiconvex sets in δ-hyperbolic graphs (this is our second main result). Namely, we show that for any finite collection Q of pairwise intersecting -quasiconvex sets of a δ-hyperbolic graph G there exists a single ball B(c, 2 + 5δ) intersecting all sets of Q. More generally, we prove that if Q is a collection of 2r-close (i.e., any two sets of Q are at distance ≤ 2r) -quasiconvex sets of a δ-hyperbolic graph G, then there exists a ball B(c, r * ) of radius r * := max{2 + 5δ, r + + 3δ} intersecting all sets of Q. These kind of Helly-type results are also useful in geometric group theory.Using the Helly theorem for quasiconvex sets and a primal-dual approach, we show algorithmically that the minimum number of balls of radius 2 + 5δ intersecting all sets of a family Q of -quasiconvex sets does not exceed the packing number of Q (maximum number of pairwise disjoint sets of Q). We extend the covering and packing result to set-families κ Q in which each set is a union of at most κ -quasiconvex sets of a δ-hyperbolic graph G. Namely, we show that if r ≥ + 2δ and πr( κ Q) is the maximum number of mutually 2r-apart members of κ Q, then the minimum number of balls of radius r + 2 + 6δ intersecting all members of κ Q is at most 2κ 2 πr( κ Q) and such a hitting set and a packing can be constructed in polynomial time for every finite κ Q (this is our third main result). For setfamilies consisting of unions of κ balls in δ-hyperbolic graphs a similar result was obtained by Chepoi and Estellon (2007 In case of δ = 0 (trees) and = r = 0, (subtrees of a tree)we recover the result of Alon (2002) about the transversal and packing numbers of a set-family in which each set is a union of at most κ subtrees of a tree.

show abstract

“…In [12] the authors give a polynomial algorithm which computes for such graphs an augmented graph of at most a given diameter, and whose number of added edges is within a constant factor of the minimum number of added edges that are needed such that the augmented graph has at most such a diameter. Finally, in [11] it is shown that all cop-win graphs in which the cop and the robber move at different speeds have small hyperbolicity, and also a constant-factor approximation of δ H in time O(n 2 log δ) is given. Moreover, the concept of hyperbolicity turns out to be useful for many applied problems such as visualization of the Internet, the Web graph, and other complex networks [20,21], routing, navigation, and decentralized search in these networks [7,19].…”

Section: Introductionmentioning

confidence: 99%

On the Hyperbolicity of Random Graphs

Mitsche¹,

Prałat²

2014

Electron. J. Combin.

View full text Add to dashboard Cite

Let G = (V, E) be a connected graph with the usual (graph) distance metric d : V ×V → N∪{0}. Introduced by Gromov, G is δ-hyperbolic if for every four vertices u, v, x, y ∈ V , the two largest values of the three sums d(u, v) + d(x, y), d(u, x) + d(v, y), d(u, y) + d(v, x) differ by at most 2δ. In this paper, we determine precisely the value of this hyperbolicity for most binomial random graphs.1991 Mathematics Subject Classification. 05C12, 05C35, 05C80.

show abstract

Cop and Robber Game and Hyperbolicity

Cited by 16 publications

References 22 publications

Fast Approximation and Exact Computation of Negative Curvature Parameters of Graphs

Fast Approximation and Exact Computation of Negative Curvature Parameters of Graphs

Core congestion is inherent in hyperbolic networks

On the Hyperbolicity of Random Graphs

Contact Info

Product

Resources

About