On the hardness of optimization in power law graphs

  • Authors:
  • Alessandro Ferrante;Gopal Pandurangan;Kihong Park

  • Affiliations:
  • Dipartimento di Informatica ed Applicazioni "R.M. Capocelli", University of Salerno, Via Ponte don Melillo - Fisciano (SA), Italy;Department of Computer Science, Purdue University, West Lafayette, IN;Department of Computer Science, Purdue University, West Lafayette, IN

  • Venue:
  • COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
  • Year:
  • 2007

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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 exhibit a power-law distribution: the number of nodes yi of a given degree i is proportional to iβ 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, remains NP-hard for all β ≥ 1. Moreover, we show that all the problems that satisfy the so-called "optimal substructure property" remains NP-hard for all β 0. This includes classical problems such as MIN VERTEX-COVER, MAX INDEPENDENT-SET, and MIN DOMINATING-SET. Our proofs involve designing efficient algorithms for constructing graphs with prescribed degree sequences that are tractable with respect to various optimization problems.