Hardness Complexity of Optimal Substructure Problems on Power-Law Graphs

Abstract

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 power-law graphs. Unfortunately, many problems have been proved NP-hard even in power-law graphs as Ferrante proposed in Ferrante et al. (Theoretical Computer Science 393(1–3):220–230, 2008). Intuitively, a theoretical question is raised: Are these problems on power-law graphs still as hard as on general graphs? The chapter shows 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 subsolutions on all maximal connected components. In particular, the above problems and a more general problem (ρ-minimum dominating set) are proven to 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, the corresponding inapproximability factors of these problems are further proven in simple power-law graphs based on the graphic embedding technique as well as that of maximum clique and minimum coloring using the embedding technique in Ferrante et al. (Theoretical Computer Science 393 (1–3):220–230, 2008). 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. In addition, this chapter contains the in-depth investigations in the relationship between the exponential factor β and constant greedy approximation algorithms. The last part includes some minor NP-hardness results on simple power-law graphs for small β<1.