On Computing the Hyperbolicity of Real-World Graphs

Borassi, Michele; Coudert, David; Crescenzi, Pierluigi; Marino, Andrea

doi:10.1007/978-3-662-48350-3_19

Cited by 23 publications

(87 citation statements)

References 25 publications

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“…The hyperbolicity of a given 4-tuple is upper-bounded by the minimum distance between two vertices of the 4-tuple [16,22,81]. So, let us consider the distances in H.…”

Section: Hardness Results For Clique-widthmentioning

confidence: 99%

Fully polynomial FPT algorithms for some classes of bounded clique-width graphs

Coudert¹,

Ducoffe²,

Popa³

2018

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

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Recently, hardness results for problems in P were achieved using reasonable complexity theoretic assumptions such as the Strong Exponential Time Hypothesis. According to these assumptions, many graph theoretic problems do not admit truly subquadratic algorithms. A central technique used to tackle the difficulty of the above mentioned problems is fixed-parameter algorithms with polynomial dependency in the fixed parameter (P-FPT). Applying this technique to clique-width, an important graph parameter, remained to be done. In this paper we study several graph theoretic problems for which hardness results exist such as cycle problems, distance problems and maximum matching. We give hardness results and P-FPT algorithms, using cliquewidth and some of its upper-bounds as parameters. We believe that our most important result is an O(k 4 · n + m)-time algorithm for computing a maximum matching where k is either the modular-width or the P 4 -sparseness. The latter generalizes many algorithms that have been introduced so far for specific subclasses such as cographs. Our algorithms are based on preprocessing methods using modular decomposition and split decomposition. Thus they can also be generalized to some graph classes with unbounded clique-width.

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“…The hyperbolicity of a given 4-tuple is upper-bounded by the minimum distance between two vertices of the 4-tuple [16,22,81]. So, let us consider the distances in H.…”

Section: Hardness Results For Clique-widthmentioning

confidence: 99%

Fully polynomial FPT algorithms for some classes of bounded clique-width graphs

Coudert¹,

Ducoffe²,

Popa³

2018

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

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“…Furthermore, it turns out that not all 4-tuples in the graph need to be considered for the computation of hyperbolicity. This crucial point is the cornerstone of the most efficient algorithms so far to compute this parameter [7,13]. Here we will use this observation to gain more insights on 4-tuples with maximum hyperbolicity in our proofs.…”

Section: Definitions and Notationsmentioning

confidence: 92%

On the hyperbolicity of bipartite graphs and intersection graphs

Coudert¹,

Ducoffe²

2016

Discrete Applied Mathematics

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International audienceHyperbolicity is a measure of the tree-likeness of a graph from a metric perspective. Recently , it has been used to classify complex networks depending on their underlying geometry. Motivated by a better understanding of the structure of graphs with bounded hyperbolicity, we here investigate on the hyperbolicity of bipartite graphs. More precisely, given a bipartite graph B = (V0 ∪ V1 , E) we prove it is enough to consider any one side Vi of the bipartition of B to obtain a close approximate of its hyperbolicity δ(B) — up to an additive constant 2. We obtain from this result the sharp bounds δ(G) − 1 ≤ δ(L(G)) ≤ δ(G) + 1 and δ(G) − 1 ≤ δ(K(G)) ≤ δ(G) + 1 for every graph G, with L(G) and K(G) being respectively the line graph and the clique graph of G. Finally, promising extensions of our techniques to a broader class of intersection graphs are discussed and illustrated with the case of the biclique graph BK(G), for which we prove (δ(G) − 3)/2 ≤ δ(BK(G)) ≤ (δ(G) + 3)/2

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“…Moreover, roughly quadratic lower bounds are known [5,15,22]. In practice, however, the best known algorithm still has an O(n 4 )-time worst-case bound but uses several clever tricks when compared to the brute-force algorithm [4]. Based on empirical studies, an O(mn) running time is claimed, where m is the number of edges in the graph.…”

Section: Introductionmentioning

confidence: 99%

Fast Approximation and Exact Computation of Negative Curvature Parameters of Graphs

Chalopin

Chepoi

Dragan

et al. 2019

Discrete Comput Geom

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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.

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On Computing the Hyperbolicity of Real-World Graphs

Cited by 23 publications

References 25 publications

Fully polynomial FPT algorithms for some classes of bounded clique-width graphs

Fully polynomial FPT algorithms for some classes of bounded clique-width graphs

On the hyperbolicity of bipartite graphs and intersection graphs

Fast Approximation and Exact Computation of Negative Curvature Parameters of Graphs

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