Journal of Combinatorial Optimization

, Volume 27, Issue 3, pp 487–503 | Cite as

On the approximability of positive influence dominating set in social networks

  • Thang N. DinhEmail author
  • Yilin Shen
  • Dung T. Nguyen
  • My T. Thai


In social networks, there is a tendency for connected users to match each other’s behaviors. Moreover, a user likely adopts a behavior, if a certain fraction of his family and friends follows that behavior. Identifying people who have the most influential effect to the others is of great advantages, especially in politics, marketing, behavior correction, and so on. Under a graph-theoretical framework, we study the positive influence dominating set (PIDS) problem that seeks for a minimal set of nodes \(\mathcal{P}\) such that all other nodes in the network have at least a fraction ρ>0 of their neighbors in \(\mathcal{P}\). We also study a different formulation, called total positive influence dominating set (TPIDS), in which even nodes in \(\mathcal{P}\) are required to have a fraction ρ of neighbors inside \(\mathcal{P}\). We show that neither of these problems can be approximated within a factor of (1−ϵ)lnmax{Δ,|V|1/2}, where Δ is the maximum degree. Moreover, we provide a simple proof that both problems can be approximated within a factor lnΔ+O(1). In power-law networks, where the degree sequence follows a power-law distribution, both problems admit constant factor approximation algorithms. Finally, we present a linear-time exact algorithms for trees.


Hardness of approximation Approximation algorithm Social networks Information diffusion 


  1. Aiello W, Chung F, Lu L (2000) A random graph model for massive graphs. In: STOC ’00. ACM, New York, pp 171–180. CrossRefGoogle Scholar
  2. Chlebík M, Chlebíková J (2004) Approximation hardness of dominating set problems. In: ESA’04, pp. 192–203 Google Scholar
  3. Cicalese F, Milanič M, Vaccaro U (2011) Hardness, approximability, and exact algorithms for vector domination and total vector domination in graphs. In: Proceedings of the 18th international conference on fundamentals of computation theory, FCT’11. Springer, Berlin, pp 288–297. CrossRefGoogle Scholar
  4. Dinh TN, Dung NT, Thai MT (2012) Cheap, easy, and massively effective viral marketing in social networks: truth or fiction? In: Proceedings of the 23rd ACM conference on hypertext and social media, HT ’12. ACM, Milwaukee Google Scholar
  5. Domingos P, Richardson M (2001) Mining the network value of customers. In: KDD ’01. ACM, New York, pp 57–66. CrossRefGoogle Scholar
  6. Feige U (1998) A threshold of ln n for approximating set cover. J ACM 45(4):634–652. CrossRefzbMATHMathSciNetGoogle Scholar
  7. Ferrante A, Pandurangan G, Park K (2008) On the hardness of optimization in power-law graphs. Theor Comput Sci 393(1–3):220–230. CrossRefzbMATHMathSciNetGoogle Scholar
  8. Girvan M, Newman ME (2002) Community structure in social and biological networks. Proc Natl Acad Sci USA 99(12):7821–7826. doi: 10.1073/pnas.122653799. CrossRefzbMATHMathSciNetGoogle Scholar
  9. Gkantsidis C, Mihail M, Saberi A (2003) Conductance and congestion in power law graphs. In: SIGMETRICS ’03. ACM, New York, pp 148–159. CrossRefGoogle Scholar
  10. Hill KG, Hawkins JD, Catalano RF, Abbott RD, Guo J (2005) Family influences on the risk of daily smoking initiation. J Adolescent Health 37(3):202–210. CrossRefGoogle Scholar
  11. Kempe D, Kleinberg J, Tardos É. (2003) Maximizing the spread of influence through a social network. In: KDD’03. ACM, New York, pp 137–146 Google Scholar
  12. Kempe D, Kleinberg J, Tardos É (2005) Influential nodes in a diffusion model for social networks. In: ICALP’05. pp 1127–1138 Google Scholar
  13. Leskovec J, Krause A, Guestrin C, Faloutsos C, VanBriesen J, Glance N (2007) Cost-effective outbreak detection in networks. In: KDD ’07. ACM, New York, pp 420–429. CrossRefGoogle Scholar
  14. Rajagopalan S, Vazirani VV (1993) Primal-dual rnc approximation algorithms for (multi)-set (multi)-cover and covering integer programs. In: STOC ’93. IEEE Comput Soc, Washington, pp 322–331. Google Scholar
  15. Shakarian P, Paulo D (2012) Large social networks can be targeted for viral marketing with small seed sets. In: IEEE/ACM international conference on advances in social networks analysis and mining (ASONAM) Google Scholar
  16. Standridge JB, Zylstra RG, Adams SM (2004) Alcohol consumption: an overview of benefits and risks. Southern Med J.
  17. Trevisan L (2001) Non-approximability results for optimization problems on bounded degree instances. In: ACM symposium on theory of computing ’01. ACM, New York, pp 453–461. Google Scholar
  18. Wang F, Camacho E, Xu K (2009) Positive influence dominating set in online social networks. In: COCOA ’09. Springer, Berlin, pp 313–321. Google Scholar
  19. Zou F, Zhang Z, Wu W (2009) Latency-bounded minimum influential node selection in social networks. In: Liu B, Bestavros A, Du DZ, Wang J (eds) WASA. Lecture notes in computer science, pp 519–526. Google Scholar
  20. Zhang W, Zhang Z, Wang W, Zou F, Lee W (2010) Polynomial time approximation scheme for t-latency bounded information propagation problem in wireless networks. Journal of Combinatorial Optimization, 1–11.
  21. Zhu X, Yu J, Lee W, Kim D, Shan S, Du DZ (2010) New dominating sets in social networks. J Glob Optim 48:633–642. CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Thang N. Dinh
    • 1
    Email author
  • Yilin Shen
    • 1
  • Dung T. Nguyen
    • 1
  • My T. Thai
    • 1
  1. 1.Department of Computer & Information Science & EngineeringUniversity of FloridaGainesvilleUSA

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