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 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, 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.
Work partially supported by funds for research from MIUR ex 60% 2005.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aiello, W., Chung, F.R.K., Lu, L.: A Random Graph Model for Massive Graphs. In: Proceedings of STOC 2000, pp. 171–180. ACM Press, New York (2000)
Aiello, W., Chung, F.R.K., Lu, L.: A random graph model for power law graphs. In Experimental Mathematics 10, 53–66 (2000)
Barabasi, A.: Emergence of Scaling in Complex Networks. In: Bornholdt, S., Schuster, H. (eds.) Handbook of Graphs and Networks, Wiley, Chichester (2003)
Bollobas, B., Riordan, O.: Mathematical Results on Scale-free Random Graphs. In: Bornholdt, S., Schuster, H. (eds.) Handbook of Graphs and Networks (2003)
Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. North-Holland, Amsterdam (1976)
Eubank, S., Kumar, V.S.A., Marathe, M.V., Srinivasan, A., Wang, N.: Structural and Algorithmic Aspects of Massive Social Networks. In: SODA 2004. Proceedings of 15th ACM-SIAM Symposium on Discrete Algorithms, pp. 711–720. ACM Press, New York (2004)
Ferrante, A., Pandurangan, G., Park, K.: On the Hardness of Optimization in Power-Law Graphs, http://www.cs.purdue.edu/homes/gopal/papers-by-date.html
Gkantsidis, C., Mihail, M., Saberi, A.: Throughput and Congestion in Power-Law Graphs. In: Proceedings of SIGMETRICS 2003, pp. 148–159. ACM Press, New York (2003)
Koyuturk, M., Grama, A., Szpankowski, W.: Assessing significance of connectivity and conservation in protein interaction networks. In: Apostolico, A., Guerra, C., Istrail, S., Pevzner, P., Waterman, M. (eds.) RECOMB 2006. LNCS (LNBI), vol. 3909, pp. 45–49. Springer, Heidelberg (2006)
Mihail, M., Papadimitriou, C., Saberi, A.: On Certain Connectivity Properties of the Internet Topology. In: Proc. of FOCS 2003, pp. 28–35. IEEE Computer Society Press, Los Alamitos (2003)
Park, K., Lee, H.: On the effectiveness of route-based packet filtering for distributed DoS attack prevention in power-law internets. In: Proceedings of SIGCOMM 2001, pp. 15–26. ACM Press, New York (2001)
Park, K.: The Internet as a complex system. In: Park, K., Willinger, W. (eds.) The Internet as a Large-Scale Complex System. Santa Fe Institute Studies on the Sciences of Complexity, Oxford University Press, Oxford (2005)
Yannakakis, M.: Node- and Edge-Deletion NP-Complete Problems. In: Proceedings of STOC 1978. SIAM 1978, San Diego, California, pp. 253–264 (1978)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ferrante, A., Pandurangan, G., Park, K. (2007). On the Hardness of Optimization in Power Law Graphs. In: Lin, G. (eds) Computing and Combinatorics. COCOON 2007. Lecture Notes in Computer Science, vol 4598. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73545-8_41
Download citation
DOI: https://doi.org/10.1007/978-3-540-73545-8_41
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73544-1
Online ISBN: 978-3-540-73545-8
eBook Packages: Computer ScienceComputer Science (R0)