Approximation algorithms for optimization problems in random power-law graphs

Abstract

Many large-scale real-world networks are well-known to have the power law distribution in their degree sequences: the number of ver­tices with degree i is proportional to i-βfor some constant β. It is a common belief that solving optimization problems in power-law graphs is easier. Unfortunately, many problems have been proven NP-hard, along with their inapproximability factors in power-law graphs. Therefore, it is of great importance to develop an algorithm framework such that these optimization problems can be approximated in power-law graphs, with provable theoretical approximation ratios. In this paper, we propose an algorithmic framework, called Low-Degree Percolation (LDP) Framework, for solving Minimum Dominating Set, Minimum Vertex Cover and Maximum Independent Set problems in power-law graphs. Using this framework, we further show a theoreti­cal framework to derive the approximation ratios for these optimization problems in two well-known random power-law graphs. Our numerical analysis shows that, these optimization problems can be approximated into near 1 factor with high probability, using our proposed LDP algo­rithms, in power-law graphs with exponential factor β ≥ 1.5, which belongs to the range of most real-world networks.