Skip to main content
SpringerLink
Log in

Degree-constrained minimum spanning tree problem of uncertain random network

  • Original Research
  • Published:
Journal of Ambient Intelligence and Humanized Computing Aims and scope Submit manuscript

Abstract

A degree-constrained minimum spanning tree (DCMST) problem involving any network aims to find the least weighted spanning tree of that network, subject to constraints on node degrees. In this paper, we first define a DCMST problem in an uncertain random network, where some weights are uncertain variables and others are random variables. We also introduce the concept of an ideal chance distribution for DCMST problem. In order to seek out the degree-constrained spanning tree (DCST) closest to the ideal chance distribution, an uncertain random programming model is formulated. An algorithm is presented to solve the DCMST problem. The effectiveness of our method and algorithm are exhibited by solving a numerical example.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • Borüvka O (1926) O jistém problému minimálním. Práce Mor. Přírodovéd. Spol. v Brně 3:37–58

    Google Scholar 

  • Bau Y, Ho C, Eve H (2008) Ant colony optimization approaches to the degree-constrained minimum spanning tree problem. J Assoc Inf Sci Tech 24(4):1081–1094

    Google Scholar 

  • Chen X, Liu B (2010) Existence and uniqueness theorem for uncertain differential equations. Fuzzy Optim Decis Mak 9(1):69–81

    Article  MathSciNet  MATH  Google Scholar 

  • Chen X, Gao J (2013) Uncertain term structure model of interest rate. Soft Comput 17(4):597–604

    Article  MATH  Google Scholar 

  • Ding S (2014) Uncertain minimum cost flow problem. Soft Comput 18(11):2201–2207

    Article  MATH  Google Scholar 

  • Frank H, Hakimi S (1965) Probabilistic flows through a communication network. IEEE Trans Circuit Theory 12:413–414

    Article  Google Scholar 

  • Gao J, Yang X, Liu D (2017) Uncertain Shapley value of coalitional game with application to supply chain alliance. Appl Soft Comput. doi:10.1016/j.asoc.2016.06.018

    Google Scholar 

  • Gao J (2013) Uncertain bimatrix game with applications. Fuzzy Optim Decis Mak 12(1):65–78

    Article  MathSciNet  Google Scholar 

  • Gao J, Yao K (2015) Some concepts and theorems of uncertain random process. Int J Intell Syst 30(1):52–65

    Article  Google Scholar 

  • Gao X, Jia L (2017) Degree-constrained minimum spanning tree problem with uncertain edge weights. Appl Soft Comput. doi:10.1016/j.asoc.2016.07.054

    Google Scholar 

  • Gao X, Gao Y, Ralescu D (2010) On Liu’s inference rule for uncertain systems. Int J Uncertain Fuzziness Knowl Syst 18(1):1–11

    Article  MathSciNet  MATH  Google Scholar 

  • Gao Y (2011) Shortest path problem with uncertain arc lengths. Comput Math Appl 62(6):2591–2600

    Article  MathSciNet  MATH  Google Scholar 

  • Gao Y, Yang L, Li S, Kar S (2015) On distribution function of the diameter in uncertain graph. Inf Sci 296:61–74

    Article  MathSciNet  MATH  Google Scholar 

  • Gao Y, Qin Z (2016) On computing the edge-connectivity of an uncertain graph. IEEE Trans Fuzzy Syst 24(4):981–991

    Article  Google Scholar 

  • Gao Y, Yang L, Li S (2016) Uncertain models on railway transportation planning problem. Appl Math Model 40:4921–4934

    Article  MathSciNet  Google Scholar 

  • Guo C, Gao J (2017) Optimal dealer pricing under transaction uncertainty. J Intell Manuf 28(3):657–665

    Article  Google Scholar 

  • Han S, Peng Z, Wang S (2014) The maximum flow problem of uncertain network. Inf Sci 265:167–175

    Article  MathSciNet  MATH  Google Scholar 

  • Hou Y (2014) Subadditivity of chance measure. J Uncertainty Anal Appl 2(Article 14)

  • Knowles J, Corne D (2000) A new evolutionary approach to the degree-constrained minimum spanning tree problem. IEEE T Evol Comput 4:125–134

    Article  Google Scholar 

  • Liu B (2007) Uncertainty theory, 2nd edn. Springer-Verlag, Berlin

    MATH  Google Scholar 

  • Liu B (2008) Fuzzy process, hybrid process and uncertain process. J Uncertain Syst 2(1):3–16

    MathSciNet  Google Scholar 

  • Liu B (2009a) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3–10

    MathSciNet  Google Scholar 

  • Liu B (2009b) Theory and practice of uncertain programming, 2nd edn. Springer-Verlag, Berlin

    Book  MATH  Google Scholar 

  • Liu B (2010a) Uncertainty theory: a branch of mathematics for modeling human uncertainty. Springer-Verlag, Berlin

    Book  Google Scholar 

  • Liu B (2010b) Uncertain set theory and uncertain inference rule with application to uncertain control. J Uncertain Syst 4(2):83–98

    Google Scholar 

  • Liu B (2014) Uncertain random graph and uncertain random network. J Uncertain Syst 8(1):3–12

    Google Scholar 

  • Liu Y (2013a) Uncertain random variables: A mixture of uncertainty and randomness. Soft comput 17(4):625–634

    Article  MATH  Google Scholar 

  • Liu Y (2013b) Uncertain random programming with applications. Fuzzy Optim Decis Mak 12(2):153–169

    Article  MathSciNet  Google Scholar 

  • Liu Y, Ralescu D (2014) Risk index in uncertain random risk analysis. Int J Uncertain Fuzziness Knowl Syst 22(4):491–504

    Article  MathSciNet  MATH  Google Scholar 

  • Mu Y, Zhou G (2008) Genetic algorithm based on prufer number for solving degree-constrained minimum spanning tree problem. Com Eng App 44(12):53–56

    Google Scholar 

  • Narula S, Ho C (1980) Degree-constrained minimum spanning tree. Comput Oper Res 7:239–249

    Article  Google Scholar 

  • Raidl G (2000) An efficient evolutionary algorithm for the degree-constrained minimum spanning tree problem. IEEE Trans Evol Comput 1(104–111):2000

    Google Scholar 

  • Sheng Y, Qin Z, Shi G (2017) Minimum spanning tree problem of uncertain random network. J Intell Manuf 28(3):565–574

    Article  Google Scholar 

  • Sheng Y, Gao J (2014) Chance distribution of the maximum flow of uncertain random network. J Uncertain Anal Appl 2(Article 14)

  • Sheng Y, Gao Y (2016) Shortest path problem of uncertain random network. Comput Ind Eng 99:97–105

    Article  Google Scholar 

  • Torkestani J (2012) Degree-constrained minimum spanning tree problem in stochastic graph. Cybernet Syst 43(1):1–21

    Article  MATH  Google Scholar 

  • Torkestani J (2013) A learning automata-based algorithm to the stochastic min-degree constrained minimum spanning tree problem. Int J Found Comput 24(3):329–348

    Article  MathSciNet  MATH  Google Scholar 

  • Volgenant A (1989) A lagrangean approach to the degree-constrained minimum spanning tree problem. Eur J Oper Res 39(3):325–331

    Article  MathSciNet  MATH  Google Scholar 

  • Yang X, Gao J (2017) Bayesian equilibria for uncertain bimatrix game with asymmetric information. J Intell Manuf 28(3):515–525

    Article  Google Scholar 

  • Yang X, Gao J (2013) Uncertain differential games with applications to capitalism. J Uncertainty Anal Appl 1(Article 17)

  • Yang X, Gao J (2016) Linear-quadratic uncertain differential games with applicatians to resource extraction problem. IEEE T Fuzzy Syst 24(4):819–826

    Article  MathSciNet  Google Scholar 

  • Yang X, Yao K (2016) Uncertain partial differential equation with application to heat conduction. Fuzzy Optim Decis Mak. doi:10.1007/s10700-016-9253-9

  • Yao K, Gao J (2016) Law of large numbers for uncertain random variables. IEEE T Fuzzy Syst 24(3):615–621

    Article  Google Scholar 

  • Zhang X, Wang Q, Zhou J (2013) Two uncertain programming models for inverse minimum spanning tree problem. Ind Eng Manag Syst 12(1):9–15

    Google Scholar 

  • Zhou G, Gen M (1997) Approach to the degree-constrained minimum spanning tree problem using genetic algorithm. Eng Des Autom 3:157–165

    Google Scholar 

  • Zhou J, He X, Wang K (2014) Uncertain quadratic minimum spanning tree problem. J Commum 9(5):385–390

    Article  Google Scholar 

  • Zhou J, Chen L, Wang K (2015) Path optimality conditions for minimum spanning tree problem with uncertain edge weights. Int J Unc Fuzz Knowl Based Syst 23(1):49–71

    Article  MathSciNet  MATH  Google Scholar 

  • Zhu Y (2010) Uncertain optimal control with application to a portfolio selection model. Cybernet Syst 41(7):535–547

    Article  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the Fundamental Research Funds for the Central Universities No.2016MS65 and National Natural Science Foundation of China Grant No.71671064.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences and Physics, North China Electric Power University, Beijing, 102206, China

    Xin Gao

  2. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Lifen Jia

  3. Department of Mathematics, National Institute of Technology, Durgapur, 713209, India

    Samarjit Kar

Authors
  1. Xin Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lifen Jia
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Samarjit Kar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lifen Jia.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gao, X., Jia, L. & Kar, S. Degree-constrained minimum spanning tree problem of uncertain random network. J Ambient Intell Human Comput 8, 747–757 (2017). https://doi.org/10.1007/s12652-017-0493-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12652-017-0493-5

Keywords

Search

Navigation