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Efficiently Approximating Vertex Cover on Scale-Free Networks with Underlying Hyperbolic Geometry

Authors Thomas Bläsius, Tobias Friedrich , Maximilian Katzmann

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Author Details

Thomas Bläsius
  • Karlsruhe Institute of Technology, Germany
Tobias Friedrich
  • Hasso Plattner Institute, University of Potsdam, Germany
Maximilian Katzmann
  • Hasso Plattner Institute, University of Potsdam, Germany

Funding

This research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Grant No. 390859508.

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Thomas Bläsius, Tobias Friedrich, and Maximilian Katzmann. Efficiently Approximating Vertex Cover on Scale-Free Networks with Underlying Hyperbolic Geometry. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ESA.2021.20

Abstract

Finding a minimum vertex cover in a network is a fundamental NP-complete graph problem. One way to deal with its computational hardness, is to trade the qualitative performance of an algorithm (allowing non-optimal outputs) for an improved running time. For the vertex cover problem, there is a gap between theory and practice when it comes to understanding this tradeoff. On the one hand, it is known that it is NP-hard to approximate a minimum vertex cover within a factor of √2. On the other hand, a simple greedy algorithm yields close to optimal approximations in practice. A promising approach towards understanding this discrepancy is to recognize the differences between theoretical worst-case instances and real-world networks. Following this direction, we close the gap between theory and practice by providing an algorithm that efficiently computes nearly optimal vertex cover approximations on hyperbolic random graphs; a network model that closely resembles real-world networks in terms of degree distribution, clustering, and the small-world property. More precisely, our algorithm computes a (1 + o(1))-approximation, asymptotically almost surely, and has a running time of 𝒪(m log(n)). The proposed algorithm is an adaption of the successful greedy approach, enhanced with a procedure that improves on parts of the graph where greedy is not optimal. This makes it possible to introduce a parameter that can be used to tune the tradeoff between approximation performance and running time. Our empirical evaluation on real-world networks shows that this allows for improving over the near-optimal results of the greedy approach.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Random network models
  • Mathematics of computing → Approximation algorithms
  • Mathematics of computing → Random graphs
Keywords
  • vertex cover
  • approximation
  • random graphs
  • hyperbolic geometry
  • efficient algorithm

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