Inapproximability of dominating set on power law graphs

We give logarithmic lower bounds for the approximability of the Minimum Dominating Set problem in connected (alpha,beta)-Power Law Graphs. We give also a best up to now upper approximation bound on the problem for the case of the parameters beta>2. We develop also a new functional method for proving lower approximation bounds and display a sharp phase transition between approximability and inapproximability of the underlying problem. This method could also be of independent interest.

[1]  Mahtab Seddigh,et al.  Dominating Sets and Neighbor Elimination-Based Broadcasting Algorithms in Wireless Networks , 2002, IEEE Trans. Parallel Distributed Syst..

[2]  StojmenovicIvan,et al.  Dominating Sets and Neighbor Elimination-Based Broadcasting Algorithms in Wireless Networks , 2002 .

[3]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[4]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[5]  Alice Payne Hackett.  70 years of best sellers, 1895-1965 , 1967 .

[6]  Fan Chung Graham,et al.  A Random Graph Model for Power Law Graphs , 2001, Exp. Math..

[7]  J. Montoya,et al.  Small world patterns in food webs. , 2002, Journal of theoretical biology.

[8]  Andrei Z. Broder,et al.  Graph structure in the Web , 2000, Comput. Networks.

[9]  Marek Karpinski,et al.  Improved Approximation Lower Bounds for Vertex Cover on Power Law Graphs and Some Generalizations , 2012, ArXiv.

[10]  Michalis Faloutsos,et al.  Power laws and the AS-level internet topology , 2003, TNET.

[11]  P. Bourgine,et al.  Topological and causal structure of the yeast transcriptional regulatory network , 2002, Nature Genetics.

[12]  My T. Thai,et al.  On the Hardness and Inapproximability of Optimization Problems on Power Law Graphs , 2010, COCOA.

[13]  Duncan J. Watts,et al.  The Structure and Dynamics of Networks: (Princeton Studies in Complexity) , 2006 .

[14]  A. Barabasi,et al.  Lethality and centrality in protein networks , 2001, Nature.

[15]  Deryk Osthus,et al.  Popularity based random graph models leading to a scale-free degree sequence , 2004, Discret. Math..

[16]  Ying Xuan,et al.  New techniques for approximating optimal substructure problems in power-law graphs , 2012, Theor. Comput. Sci..

[17]  Eli Upfal,et al.  Stochastic models for the Web graph , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[18]  Mark E. J. Newman,et al.  Structure and Dynamics of Networks , 2009 .

[19]  Jon Kleinberg,et al.  The Structure of the Web , 2001, Science.

[20]  Shlomo Moran,et al.  Non Deterministic Polynomial Optimization Problems and their Approximations , 1977, Theor. Comput. Sci..

[21]  Michalis Faloutsos,et al.  On power-law relationships of the Internet topology , 1999, SIGCOMM '99.

[22]  Aravind Srinivasan,et al.  Structural and algorithmic aspects of massive social networks , 2004, SODA '04.

[23]  Jon M. Kleinberg,et al.  The Web as a Graph: Measurements, Models, and Methods , 1999, COCOON.

[24]  L. Amaral,et al.  The web of human sexual contacts , 2001, Nature.

[25]  Aravind Srinivasan,et al.  Modelling disease outbreaks in realistic urban social networks , 2004, Nature.

[26]  A. Rbnyi.  ON THE EVOLUTION OF RANDOM GRAPHS , 2001 .

[27]  Kihong Park,et al.  On the Hardness of Optimization in Power Law Graphs , 2007, COCOON.

[28]  Béla Bollobás,et al.  Mathematical results on scale‐free random graphs , 2005 .

[29]  Mikael Gast,et al.  Approximability of the vertex cover problem in power-law graphs , 2012, Theor. Comput. Sci..

[30]  Ran Raz,et al.  A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP , 1997, STOC '97.

[31]  Alessandro Vespignani,et al.  Epidemic spreading in scale-free networks. , 2000, Physical review letters.

[32]  M. Jovanovi.  MODELING PEER-TO-PEER NETWORK TOPOLOGIES THROUGH “ SMALL-WORLD ” MODELS AND POWER LAWS , 2001 .

[33]  Reuven Bar-Yehuda,et al.  On approximation problems related to the independent set and vertex cover problems , 1984, Discret. Appl. Math..

[34]  Fan Chung Graham,et al.  A random graph model for massive graphs , 2000, STOC '00.