On the hardness of optimization in power-law graphs

Abstract

Our motivation for this work is the remarkable discovery that many large-scale real-world graphs ranging from Internet and World Wide Web to social and biological networks appear to exhibit a power-law distribution: the number of nodes y i of a given degree i is proportional to i −β where β > 0 is a constant that depends on the application domain. There is practical evidence that combinatorial optimization in power-law graphs is easier than in general graphs, prompting the basic theoretical question: Is combinatorial optimization in power-law graphs easy? Does the answer depend on the power-law exponent β? Our main result is the proof that many classical NP-hard graph-theoretic optimization problems remain NP-hard on power-law graphs for certain values of β. In particular, we show that some classical problems, such as CLIQUE and COLORING, remain NP-hard for all β ≥ 1. Moreover, we show that all the problems that satisfy the so-called "optimal substructure property" remain NP-hard for all β > 0. This includes classical problems such as MINIMUM VERTEX COVER, MAXIMUM INDEPENDENT SET, and MINIMUM DOMINATING SET. Our proofs involve designing efficient algorithms for constructing graphs with prescribed degree sequences that are tractable with respect to various optimization problems.

On positive influence dominating sets in social networks

“…Furthermore, we propose a greedy approximation algorithm and give theoretical analysis about its approximation ratio. [6] studied the hardness of optimization in power-law graph and found that dominating set is theoretically an easier problem in a powerlaw graph than in a general graph. We are interested in studying PIDS in a power-law graph since most social networks follow the power-law.…”

Positive Influence Dominating Set in Online Social Networks

“…In the case of college drinking where participants of an intervention program are selected accord with the greedy PIDS, by moderately increasing the participation related cost, the probability of positive influencing the entire community is significantly higher. We also found that power law graphs have larger dominating set size than random graphs even though dominating set problem is theoretically an easier problem in a power law graph than in a random graph [6].…”

Decompositions of triangle-dense graphs

“…For example, the property of having small diameter indicates little about the structure of a network -every network can be rendered small-diameter by adding one extra vertex connected to all other vertices. Similarly, merely assuming a power-law degree distribution does not seem to impose significant restrictions on a graph [FPP06]. For example, the Chung-Lu model [CL02b] generates power-law graphs with no natural decompositions.…”