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Journal of Intelligent Manufacturing

, Volume 28, Issue 3, pp 625–632 | Cite as

Uncertain programming model for uncertain minimum weight vertex covering problem

  • Lin Chen
  • Jin Peng
  • Bo Zhang
  • Shengguo Li
Article

Abstract

In this paper, the minimum weight vertex covering problem with uncertain vertex weights is investigated. By virtue of the uncertainty distribution operation of independent uncertain variables, the uncertainty distribution of the minimum weight of vertex cover is derived, and the concept of the \(\alpha \)-minimum cover among uncertain weight vertex covers is proposed within the framework of uncertain programming. Then an \(\alpha \)-minimum model for uncertain weight vertex covering problem is established and discussed. Taking advantage of some properties of uncertainty theory, the model can be transformed into the corresponding deterministic form. At last, a numerical example is presented to show the performance of the model.

Keywords

Vertex covering problem \(\alpha \)-minimum cover \(\alpha \)-minimum model Uncertainty theory Uncertain programming 

Notes

Acknowledgments

This work is supported by the Projects of the Humanity and Social Science Foundation of Ministry of Education of China (No.13YJA630065), the Key Project of Hubei Provincial Natural Science Foundation (No.2012FFA065), the Scientific and Technological Innovation Team Project (No.T201110) of Hubei Provincial Department of Education, China, and the Fundamental Research Funds for the Central Universities (No. 31541411209).

References

  1. Bar-Yehuda, R., & Even, S. (1981). A linear-time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms, 2, 198–203.CrossRefGoogle Scholar
  2. Bouamama, S., Blum, C., & Boukerram, A. (2012). A population-based iterated greedy algorithm for the minimum weight vertex cover problem. Applied Soft Computing, 12, 1632–1639.CrossRefGoogle Scholar
  3. Feige, U., Hajiaghayi, M. T., & Lee, J. R. (2008). Improved approximation algorithms for minimum weight vertex separators. SIAM Journal on Computing, 38, 629–657.CrossRefGoogle Scholar
  4. Gao, X., Gao, Y., & Ralescu, D. A. (2010). On Liu’s inference rule for uncertain systems. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 18, 1–11.CrossRefGoogle Scholar
  5. Gao, Y. (2011). Shortest path problem with uncertain arc lengths. Computers & Mathematics with Applications, 62, 2591–2600.CrossRefGoogle Scholar
  6. Han, Q., Punnen, A. P., & Ye, Y. (2009). An edge-reduction algorithm for the vertex cover problem. Operations Research Letters, 37, 181–186.CrossRefGoogle Scholar
  7. Han, S., Peng, Z., & Wang, S. (2014). The maximum flow problem of uncertain network. Information Sciences, 265, 167–175.CrossRefGoogle Scholar
  8. Hartmann, A. K., & Weigt, M. (2001). Statistical mechanics perspective on the phase transition in vertex covering of finite-connectivity random graphs. Theoretical Computer Science, 265, 199–225.CrossRefGoogle Scholar
  9. Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47, 263–292.CrossRefGoogle Scholar
  10. Karp, R. M. (1972). Reducibility among combinatorial problems. In R. E. Miller, J. W. Thatcher, & J. D. Bohlinger (Eds.), Complexity of Computer Computations, 85–103. New York: Plenum Press.Google Scholar
  11. Kovac, P., Rodic, D., Pucovsky, V., Savkovic, B., & Gostimirovic, M. (2013). Application of fuzzy logic and regression analysis for modeling surface roughness in face milliing. Journal of Intelligent Manufacturing, 24, 755–762.CrossRefGoogle Scholar
  12. Kratsch, S., & Neumann, F. (2013). Fixed-parameter evolutionary algorithms and the vertex cover problem. Algorithmica, 65, 754–771.CrossRefGoogle Scholar
  13. Li, X., & Liu, B. (2009). Hybrid logic and uncertain logic. Journal of Uncertain Systems, 3, 83–94.Google Scholar
  14. Liu, B. (2007). Uncertainty theory (2nd ed.). Berlin: Springer-Verlag.Google Scholar
  15. Liu, B. (2008). Fuzzy process, hybrid process and uncertain process. Journal of Uncertain Systems, 2, 3–16.Google Scholar
  16. Liu, B. (2009a). Theory and practice of uncertain programming (2nd ed.). Berlin: Springer-Verlag.Google Scholar
  17. Liu, B. (2009b). Some research problems in uncertainty theory. Journal of Uncertain Systems, 3, 3–10.Google Scholar
  18. Liu, B. (2010a). Uncertainty theory: A branch of mathematics for modeling human uncertainty. Berlin: Springer-Verlag.Google Scholar
  19. Liu, B. (2010b). Uncertain risk analysis and uncertain reliability analysis. Journal of Uncertain Systemsm, 4, 163–170.Google Scholar
  20. Liu, B. (2012). Why is there a need for uncertainty theory? Journal of Uncertain Systems, 6, 3–10.Google Scholar
  21. Liu B.,(2013) Toward uncertain finance theory, Journal of Uncertainty Analysis and Applications, 1, Article 1.Google Scholar
  22. Liu, B. (2015). Uncertainty theory (4th ed.). Berlin: Springer-Verlag.Google Scholar
  23. Murat, C., & Paschos, V. T. (2002). The probabilistic minimum vertex-covering problem. International Transactions in Operational Research, 9, 19–32.CrossRefGoogle Scholar
  24. Ni, Y., Xie, L., & Liu, Z. (2010). Minimizing the expected complete influence time of a social network. Information Sciences, 180, 2514–2527.CrossRefGoogle Scholar
  25. Ni, Y. (2012). Minimum weight covering problems in stochastic environments. Information Sciences, 214, 91–104.CrossRefGoogle Scholar
  26. Norman, R. Z., & Rabin, M. O. (1959). An algorithm for a minimum cover of a graph. Proceedings of the American Mathematical Society, 10, 315–319.CrossRefGoogle Scholar
  27. Oliveto, P. S., He, J., & Yao, X. (2009). Analysis of the (1+1)-EA for finding approximate solutions to vertex cover problems. IEEE Transactions on Evolutionary Computation, 13, 1006–1029.Google Scholar
  28. Peng, Z., & Iwamura, K. (2010). A sufficient and necessary condition of uncertainty distribution. Journal of Interdisciplinary Mathematics, 13, 277–285.CrossRefGoogle Scholar
  29. Peng, J., Zhang, B., & Li, S. (2014). Towards uncertain network optimization, Journal of Uncertainty Analysis and Applications, to be published.Google Scholar
  30. Shiue, W. T. (2005). Novel state minimization and state assignment in finite state machine design for low-power portable devices. Integration, the VLSI Journal, 38, 549–570.CrossRefGoogle Scholar
  31. Shyu, S. J., Yin, P. Y., & Lin, B. M. T. (2004). An ant colony optimization algorithm for the minimum weight vertex cover problem. Annals of Operations Research, 131, 283–304.CrossRefGoogle Scholar
  32. Sun, G., Liu, Y. K., & Lan, Y. (2011). Fuzzy two-stage material procurement planning problem. Journal of Intelligent Manufacturing, 22, 319–331.CrossRefGoogle Scholar
  33. Weigt, M., & Hartmann, A. K. (2000). Number of guards needed by a museum: A phase transition in vertex covering of random graphs. Physical Review Letters, 84, 6118–6121.CrossRefGoogle Scholar
  34. Yang, X., & Gao, J. (2013). Uncertain differential games with application to capitalism, Journal of Uncertainty Analysis and Applications, 1, Article 17.Google Scholar
  35. Yao, K. (2013a). Extreme values and integral of solution of uncertain differential equation, Journal of Uncertainty Analysis and Applications, 1, Article 2.Google Scholar
  36. Yao, K. (2013b). A type of nonlinear uncertain differential equations with analytic solution, Journal of Uncertainty Analysis and Applications, 1, Article 8.Google Scholar
  37. Zhang, B., & Peng, J. (2012). Uncertain programming model for Chinese postman problem with uncertain weights. Industrial Engineering & Management Systems, 11, 18–25.CrossRefGoogle Scholar
  38. Zhu, Y. (2010). Uncertain optimal control with application to a portfolio selection model. Cybernetics and Systems, 41, 535–547.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.College of Mathematics and ScienceShanghai Normal UniversityShanghaiChina
  2. 2.Institute of Uncertain SystemsHuanggang Normal UniversityHubeiChina
  3. 3.School of Statistics and MathematicsZhongnan University of Economics and LawHubeiChina

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