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 $i^{-\beta }$ for some constant $\beta$. 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 $\beta \ge 1.5$, which belongs to the range of most real-world networks.

This work was finished when Yilin Shen was with CISE Department, University of Florida.