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

TL;DR

This paper presents an efficient algorithm for approximating the minimum vertex cover in hyperbolic random graphs, closely modeling real-world networks, achieving near-optimal solutions with tunable tradeoffs between accuracy and speed.

Contribution

It introduces a novel algorithm that combines greedy strategies with improvements tailored for hyperbolic graphs, bridging the gap between theoretical hardness and practical efficiency.

Findings

Achieves a (1 + o(1))-approximation asymptotically almost surely.

Runs in O(m log n) time, scalable to large networks.

Empirical results outperform standard greedy algorithms on real-world data.

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 $\mathcal{O}(m \log(n))$. The proposed algorithm is an adaptation 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.