On the Hardness and Inapproximability of Optimization Problems on Power Law Graphs

The discovery of power law distribution in degree sequence (i.e. the number of vertices with degree i is proportional to i−β for some constant β) of many large-scale real networks creates a belief that it may be easier to solve many optimization problems in such networks. Our works focus on the hardness and inapproximability of optimization problems on power law graphs (PLG). In this paper, we show that the Minimum Dominating Set, Minimum Vertex Cover and Maximum Independent Set are still APX-hard on power law graphs. We further show the inapproximability factors of these optimization problems and a more general problem (ρ-Minimum Dominating Set), which proved that a belief of (1 + o(1))-approximation algorithm for these problems on power law graphs is not always true. In order to show the above theoretical results, we propose a general cycle-based embedding technique to embed any d-bounded graphs into a power law graph. In addition, we present a brief description of the relationship between the exponential factor β and constant greedy approximation algorithms. keyword: Theory, Complexity, Inapproximability, Power Law Graphs