Approximating Optimization Problems using EAs on Scale-Free Networks

It has been experimentally observed that real-world networks follow certain topological properties, such as small-world, power-law etc. To study these networks, many random graph models, such as Preferential Attachment, have been proposed. In this paper, we consider the deterministic properties which capture power-law degree distribution and degeneracy. Networks with these properties are known as scale-free networks in the literature. Many interesting problems remain NP-hard on scale-free networks. We study the relationship between scale-free properties and the approximation ratio of some commonly used evolutionary algorithms. For the Vertex Cover, we observe experimentally that the (1+1) EA always gives the better result than a greedy local search, even when it runs for only $O(n \log n)$ steps. We give the construction of a scale-free network in which a multi-objective algorithm and a greedy algorithm obtain optimal solutions, while the (1+1) EA obtains the worst possible solution with constant probability. We prove that for the Dominating Set, Vertex Cover, Connected Dominating Set and Independent Set, the (1+1) EA obtains constant-factor approximation in expected run time $O(n \log n)$ and $O(n^4)$ respectively. Whereas, GSEMO gives even better approximation than (1+1) EA in expected run time $O(n^3)$ for Dominating Set, Vertex Cover and Connected Dominating Set on such networks.