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

Bläsius, Thomas ; Friedrich, Tobias; Katzmann, Maximilian

## 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 $\sqrt{2}$. 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 $\mathcal{O}(m \log(n))$.
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 Zugehörige Institution(en) am KIT Institut für Theoretische Informatik (ITI) Publikationstyp Forschungsbericht/Preprint Publikationsjahr 2021 Sprache Englisch Identifikator KITopen-ID: 1000138405 Nachgewiesen in arXiv Relationen in KITopen Verweist aufEfficiently approximating vertex cover on scale-free networks with underlying hyperbolic geometry. Bläsius, Thomas; Friedrich, Tobias; Katzmann, Maximilian (2021) Proceedingsbeitrag (1000138386)
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