New techniques for approximating optimal substructure problems in power-law graphs

The remarkable discovery of many large-scale real networks is the power-law distribution in degree sequence: the number of vertices with degree i is proportional to i−β for some constant β > 1. A lot of researchers believe that it may be easier to solve some optimization problems in powerlaw graphs. Unfortunately, many problems have been proved NP-hard even in power-law graphs. Intuitively, a theoretical question is raised: Are these problems on power-law graphs still as hard as on general graphs? In this paper, we show that many optimal substructure problems, such as Minimum Dominating Set, Minimum Vertex Cover and Maximum Independent Set, are easier to solve in power-law graphs by illustrating better inapproximability factors. An optimization problem has the property of optimal substructure if its optimal solution on some given graph is essentially the union of the optimal sub-solutions on all maximal connected components. In particular, we prove the above problems and a more general problem (ρ-Minimum Dominating Set) remain APX-hard and their constant inapproximability factors on general power-law graphs by using the cycle-based embedding technique to embed any d-bounded graphs into a power-law graph. In addition, in simple power-law graphs, we further prove the corresponding inapproximability factors of these problems based on the graphic embedding technique as well as that of Maximum Clique and Minimum Coloring using the embedding technique in [1]. As a result of these inapproximability factors, the belief that there exists some (1 + o(1))-approximation algorithm for these problems on power-law graphs is proven not always true. At last, we do the in-depth investigations in the relationship between the exponential factor β and constant greedy approximation algorithms.