When Can Graph Hyperbolicity be Computed in Linear Time?

Till Fluschnik; Christian Komusiewicz; George B. Mertzios; André Nichterlein; Rolf Niedermeier; Nimrod Talmon

doi:10.1007/s00453-018-0522-6

When Can Graph Hyperbolicity be Computed in Linear Time?

Till Fluschnik, Christian Komusiewicz, George B. Mertzios, André Nichterlein, Rolf Niedermeier, Nimrod Talmon

Department of Industrial Engineering and Management

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

Hyperbolicity is a distance-based measure of how close a given graph is to being a tree. Due to its relevance in modeling real-world networks, hyperbolicity has seen intensive research over the last years. Unfortunately, the best known algorithms used in practice for computing the hyperbolicity number of an n-vertex graph have running time O(n⁴). Exploiting the framework of parameterized complexity analysis, we explore possibilities for “linear-time FPT” algorithms to compute hyperbolicity. For example, we show that hyperbolicity can be computed in 2 ^O⁽^k⁾+ O(n+ m) time (where m and k denote the number of edges and the size of a vertex cover in the input graph, respectively) while at the same time, unless the Strong Exponential Time Hypothesis (SETH) fails, there is no 2 ^o⁽^k⁾· n²^-^ε-time algorithm for every ε> 0.

Original language	English
Pages (from-to)	2016-2045
Number of pages	30
Journal	Algorithmica
Volume	81
Issue number	5
DOIs	https://doi.org/10.1007/s00453-018-0522-6
State	Published - 15 May 2019

Keywords

Cographs
FPT in P
Parameterized complexity
Polynomial-time algorithm
Strong Exponential Time Hypothesis
Vertex cover number

ASJC Scopus subject areas

Computer Science (all)
Computer Science Applications
Applied Mathematics

Access to Document

10.1007/s00453-018-0522-6

Cite this

@article{7691ab8478264e168d1fd5229a79526e,

title = "When Can Graph Hyperbolicity be Computed in Linear Time?",

abstract = "Hyperbolicity is a distance-based measure of how close a given graph is to being a tree. Due to its relevance in modeling real-world networks, hyperbolicity has seen intensive research over the last years. Unfortunately, the best known algorithms used in practice for computing the hyperbolicity number of an n-vertex graph have running time O(n4). Exploiting the framework of parameterized complexity analysis, we explore possibilities for “linear-time FPT” algorithms to compute hyperbolicity. For example, we show that hyperbolicity can be computed in 2 O(k)+ O(n+ m) time (where m and k denote the number of edges and the size of a vertex cover in the input graph, respectively) while at the same time, unless the Strong Exponential Time Hypothesis (SETH) fails, there is no 2 o(k)· n2-ε-time algorithm for every ε> 0.",

keywords = "Cographs, FPT in P, Parameterized complexity, Polynomial-time algorithm, Strong Exponential Time Hypothesis, Vertex cover number",

author = "Till Fluschnik and Christian Komusiewicz and Mertzios, {George B.} and Andr{\'e} Nichterlein and Rolf Niedermeier and Nimrod Talmon",

note = "Funding Information: We are grateful to the anonymous reviewers of WADS{\textquoteright}17 and Algorithmica for their comments. TF acknowledges support by the DFG, Projects DAMM (NI 369/13-2) and TORE (NI 369/18). CK acknowledges support by the DFG, Project MAGZ (KO 3669/4-1). GM acknowledges support by the EPSRC Grant EP/P020372/1. AN acknowledges support by a postdoctoral fellowship of the DAAD while at Durham University. NT acknowledges support by a postdoctoral fellowship from I-CORE ALGO. Funding Information: Acknowledgements We are grateful to the anonymous reviewers of WADS{\textquoteright}17 and Algorithmica for their comments. TF acknowledges support by the DFG, Projects DAMM (NI 369/13-2) and TORE (NI 369/18). CK acknowledges support by the DFG, Project MAGZ (KO 3669/4-1). GM acknowledges support by the EPSRC Grant EP/P020372/1. AN acknowledges support by a postdoctoral fellowship of the DAAD while at Durham University. NT acknowledges support by a postdoctoral fellowship from I-CORE ALGO. Publisher Copyright: {\textcopyright} 2018, Springer Science+Business Media, LLC, part of Springer Nature.",

year = "2019",

month = may,

day = "15",

doi = "10.1007/s00453-018-0522-6",

language = "English",

volume = "81",

pages = "2016--2045",

journal = "Algorithmica",

issn = "0178-4617",