Skip to main content
SpringerLink
Log in

On solving bi-objective constrained minimum spanning tree problems

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

This paper investigates two approaches for solving bi-objective constrained minimum spanning tree problems. The first seeks to minimize the tree weight, keeping the problem’s additional objective as a constraint, and the second aims at minimizing the other objective while constraining the tree weight. As case studies, we propose and solve bi-objective generalizations of the Hop-Constrained Minimum Spanning Tree Problem (HCMST) and the Delay-Constrained Minimum Spanning Tree Problem (DCMST). First, we present an Integer Linear Programming (ILP) formulation for the HCMST. Then, we propose a new compact mathematical model for the DCMST based on the well-known Miller–Tucker–Zemlin subtour elimination constraints. Next, we extend these formulations as bi-objective models and solve them using an Augmented \(\epsilon \)-constraints method. Computational experiments performed on classical instances from the literature evaluated two different implementations of the Augmented \(\epsilon \)-constraints method for each problem. Results indicate that the algorithm performs better when minimizing the tree weight while constraining the other objective since this implementation finds shorter running times than the one that minimizes the additional objective and constrains the tree weight.

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

Similar content being viewed by others

Data Availability

Data and materials can be obtained upon request to the authors.

Notes

  1. https://www.ibm.com/products/ilog-cplex-optimization-studio.

References

  1. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Minimum spanning trees. In: Introduction to Algorithms, pp. 624–632. MIT Press, Cambridge (2009)

  2. Aschner, R., Katz, M.J.: Bounded-angle spanning tree: modeling networks with angular constraints. Algorithmica 77(2), 349–373 (2017). https://doi.org/10.1007/s00453-015-0076-9

    Article  MathSciNet  MATH  Google Scholar 

  3. Salama, H.F., Reeves, D.S., Viniotis, Y.: The delay-constrained minimum spanning tree problem, in: Proceedings Second IEEE Symposium on Computer and Communications, pp. 699–703. IEEE (1997). https://doi.org/10.1109/ISCC.1997.616089

  4. de Almeida, A.M., Martins, P., de Souza, M.C.: Min-degree constrained minimum spanning tree problem: complexity, properties, and formulations. Int. Trans. Oper. Res. 19(3), 323–352 (2012). https://doi.org/10.1111/j.1475-3995.2011.00830.x

    Article  MathSciNet  MATH  Google Scholar 

  5. Gouveia, L.: Using the Miller–Tucker–Zemlin constraints to formulate a minimal spanning tree problem with hop constraints. Comput. Oper. Res. 22(9), 959–970 (1995). https://doi.org/10.1016/0305-0548(94)00074-I

    Article  MATH  Google Scholar 

  6. Deo, N., Micikevicius, P.: A heuristic for a leaf constrained minimum spanning tree problem, Congressus Numerantium, pp. 61–72 (1999)

  7. Akgün, I., Tansel, B.C.: New formulations of the hop-constrained minimum spanning tree problem via Miller–Tucker–Zemlin constraints. Eur. J. Oper. Res. 212(2), 263–276 (2011). https://doi.org/10.1016/j.ejor.2011.01.051

    Article  MathSciNet  MATH  Google Scholar 

  8. Noronha, T.F., Ribeiro, C.C., Santos, A.C.: Solving diameter-constrained minimum spanning tree problems by constraint programming. Int. Trans. Oper. Res. 17(5), 653–665 (2010). https://doi.org/10.1111/j.1475-3995.2010.00780.x

    Article  MathSciNet  MATH  Google Scholar 

  9. Martinez, L.C., Da Cunha, A.S.: The min-degree constrained minimum spanning tree problem: formulations and branch-and-cut algorithm. Discret. Appl. Math. 164, 210–224 (2014). https://doi.org/10.1016/j.dam.2011.08.008

    Article  MathSciNet  MATH  Google Scholar 

  10. Gouveia, L., Paias, A., Sharma, D.: Modeling and solving the rooted distance-constrained minimum spanning tree problem. Comput. Oper. Res. 35(2), 600–613 (2008). https://doi.org/10.1016/j.cor.2006.03.022

    Article  MathSciNet  MATH  Google Scholar 

  11. Mavrotas, G.: Effective implementation of the \(\epsilon \)-constraint method in multi-objective mathematical programming problems. Appl. Math. Comput. 213(2), 455–465 (2009). https://doi.org/10.1016/j.amc.2009.03.037

    Article  MathSciNet  MATH  Google Scholar 

  12. Carvalho, I.A., Ribeiro, M.A.: An exact approach for the minimum-cost bounded-error calibration tree problem. Ann. Oper. Res. 287(1), 109–126 (2020). https://doi.org/10.1007/s10479-019-03443-4

    Article  MathSciNet  MATH  Google Scholar 

  13. Akgün, I.: New formulations for the hop-constrained minimum spanning tree problem via Sherali and Driscoll’s tightened miller-tucker-zemlin constraints. Comput. Oper. Res. 38(1), 277–286 (2011). https://doi.org/10.1016/j.cor.2010.05.003

    Article  MathSciNet  MATH  Google Scholar 

  14. Gouveia, L.: Multicommodity flow models for spanning trees with hop constraints. Eur. J. Oper. Res. 95(1), 178–190 (1996). https://doi.org/10.1016/0377-2217(95)00090-9

    Article  MathSciNet  MATH  Google Scholar 

  15. Gouveia, L., Simonetti, L., Uchoa, E.: Modeling hop-constrained and diameter-constrained minimum spanning tree problems as steiner tree problems over layered graphs. Math. Program. 128(1–2), 123–148 (2011). https://doi.org/10.1007/s10107-009-0297-2

    Article  MathSciNet  MATH  Google Scholar 

  16. Fernandes, M., Gouveia, L., Voß, S.: Determining hop-constrained spanning trees with repetitive heuristics. J. Telecommun. Inf. Technol. pp 16–22 (2007)

  17. Gouveia, L., Paias, A., Sharma, D.: Restricted dynamic programming based neighborhoods for the hop-constrained minimum spanning tree problem. J. Heurist. 17(1), 23–37 (2011). https://doi.org/10.1007/s10732-009-9123-5

    Article  MATH  Google Scholar 

  18. Gouveia, L.: Using variable redefinition for computing lower bounds for minimum spanning and Steiner trees with hop constraints. INFORMS J. Comput. 10(2), 180–188 (1998). https://doi.org/10.1287/ijoc.10.2.180

    Article  MathSciNet  MATH  Google Scholar 

  19. Gouveia, L., Requejo, C.: A new lagrangean relaxation approach for the hop-constrained minimum spanning tree problem. Eur. J. Oper. Res. 132(3), 539–552 (2001). https://doi.org/10.1016/S0377-2217(00)00143-0

    Article  MathSciNet  MATH  Google Scholar 

  20. Berlakovich, M., Ruthmair, M., Raidl, G.R.: A multilevel heuristic for the rooted delay-constrained minimum spanning tree problem. In: International Conference on Computer Aided Systems Theory, pp. 256–263. Springer (2011). https://doi.org/10.1007/978-3-642-27549-4_33

  21. Salama, H.F., Reeves, D.S., Viniotis, Y.: An efficient delay-constrained minimum spanning tree heuristic. In: Proceedings of the 5th International Conference on Computer Communications and Networks, pp. 1–16. IEEE (1996)

  22. Ruthmair, M., Raidl, G.R.: A kruskal-based heuristic for the rooted delay-constrained minimum spanning tree problem. In: International Conference on Computer Aided Systems Theory, pp. 713–720. Springer (2009). https://doi.org/10.1007/978-3-642-04772-5_92

  23. Xiong, X., Chen, X., Ning, A.: Competitive decision algorithm for the rooted delay-constrained minimum spanning tree. In: The International Conference on Artificial Intelligence and Software Engineering (ICAISE 2013), pp. 82–86. Atlantis Press (2013). https://doi.org/10.2991/icaise.2013.19

  24. Ruthmair, M., Raidl, G.R.: Variable neighborhood search and ant colony optimization for the rooted delay-constrained minimum spanning tree problem. In: International Conference on Parallel Problem Solving from Nature, Springer, pp. 391–400 (2010). https://doi.org/10.1007/978-3-642-15871-1_40

  25. Ruthmair, M., Raidl, G.R.: A memetic algorithm and a solution archive for the rooted delay-constrained minimum spanning tree problem, in: International Conference on Computer Aided Systems Theory, pp. 351–358. Springer (2011). https://doi.org/10.1007/978-3-642-27549-4_45

  26. Miller, C.E., Tucker, A.W., Zemlin, R.A.: Integer programming formulation of traveling salesman problems. J. ACM 7(4), 326–329 (1960). https://doi.org/10.1145/321043.321046

    Article  MathSciNet  MATH  Google Scholar 

  27. Deb, K.: Multi-objective optimization. In: Search Methodologies, pp. 403–449. Springer (2014)

  28. Corley, H.: Efficient spanning trees. J. Optim. Theory Appl. 45(3), 481–485 (1985). https://doi.org/10.1007/BF00938448

    Article  MathSciNet  MATH  Google Scholar 

  29. Emelichev, V., Perepelitsa, V.: Multiobjective problems on the spanning trees of a graph. Soviet Math. Doklady 37(1), 114–117 (1988)

    MathSciNet  MATH  Google Scholar 

  30. Sourd, F., Spanjaard, O.: A multiobjective branch-and-bound framework: Application to the biobjective spanning tree problem. INFORMS J. Comput. 20(3), 472–484 (2008). https://doi.org/10.1287/ijoc.1070.0260

    Article  MathSciNet  MATH  Google Scholar 

  31. Steiner, S., Radzik, T.: Computing all efficient solutions of the biobjective minimum spanning tree problem. Comput. Oper. Res. 35(1), 198–211 (2008). https://doi.org/10.1016/j.cor.2006.02.023

    Article  MathSciNet  MATH  Google Scholar 

  32. Pugliese, L.D.P., Guerriero, F., Santos, J.L.: Dynamic programming for spanning tree problems: application to the multi-objective case. Optim. Lett. 9(3), 437–450 (2015). https://doi.org/10.1007/s11590-014-0759-1

    Article  MathSciNet  MATH  Google Scholar 

  33. Santos, J.L., Pugliese, L.D.P., Guerriero, F.: A new approach for the multiobjective minimum spanning tree. Comput. Oper. Res. 98, 69–83 (2018). https://doi.org/10.1016/j.cor.2018.05.007

    Article  MathSciNet  MATH  Google Scholar 

  34. Arroyo, J.E.C., Vieira, P.S., Vianna, D.S.: A grasp algorithm for the multi-criteria minimum spanning tree problem. Ann. Oper. Res. 159(1), 125–133 (2008). https://doi.org/10.1007/s10479-007-0263-4

    Article  MathSciNet  MATH  Google Scholar 

  35. Fernandes, I.F., Maia, S.M., Goldbarg, E.F., Goldbarg, M.C.: A multi-agent transgenetic algorithm for the bi-objective spanning tree problem. Electron. Notes Theoret. Comput. Sci. 346, 449–460 (2019). https://doi.org/10.1016/j.entcs.2019.08.040

    Article  MathSciNet  MATH  Google Scholar 

  36. Fernandes, I., Goldbarg, E., Maia, S., Goldbarg, M.: Empirical study of exact algorithms for the multi-objective spanning tree. Comput. Optim. Appl. 75(2), 561–605 (2020). https://doi.org/10.1007/s10589-019-00154-1

    Article  MathSciNet  MATH  Google Scholar 

  37. Maia, S.M., Goldbarg, E.F., Goldbarg, M.C.: On the biobjective adjacent only quadratic spanning tree problem. Electron. Notes Discrete Math. 41, 535–542 (2013). https://doi.org/10.1016/j.endm.2013.05.135

    Article  Google Scholar 

  38. Maia, S.M., Goldbarg, E.F., Pinheiro, L.D. d.S., Goldbarg, M.C.: Tabu search with ejection chain for the biobjective adjacent-only quadratic spanning tree. In: 2016 5th Brazilian Conference on Intelligent Systems (BRACIS), pp. 313–318. IEEE (2016). https://doi.org/10.1109/bracis.2016.064

  39. Ho, J.-M., Lee, D., Chang, C.-H., Wong, C.: Minimum diameter spanning trees and related problems. SIAM J. Comput. 20(5), 987–997 (1991). https://doi.org/10.1137/0220060

    Article  MathSciNet  MATH  Google Scholar 

  40. Santos, A.C., Lima, D.R., Aloise, D.J.: Modeling and solving the bi-objective minimum diameter-cost spanning tree problem. J. Global Optim. 60(2), 195–216 (2014). https://doi.org/10.1007/s10898-013-0124-4

    Article  MathSciNet  MATH  Google Scholar 

  41. De Sousa, E.G., Santos, A.C., Aloise, D.J.: An exact method for solving the bi-objective minimum diameter-cost spanning tree problem. RAIRO-Oper. Res. 49(1), 143–160 (2015). https://doi.org/10.1051/ro/2014029

    Article  MathSciNet  MATH  Google Scholar 

  42. Marathe, M.V., Ravi, R., Sundaram, R., Ravi, S., Rosenkrantz, D.J., Hunt, H.B., III.: Bicriteria network design problems. J. Algorithms 28(1), 142–171 (1998). https://doi.org/10.1006/jagm.1998.0930

    Article  MathSciNet  MATH  Google Scholar 

  43. Prakash, V.P., Patvardhan, C., Srivastav, A.: A novel hybrid multi-objective evolutionary algorithm for the bi-objective minimum diameter-cost spanning tree (bi-mdcst) problem. Eng. Appl. Artif. Intell. 87, 103237 (2020). https://doi.org/10.1016/j.engappai.2019.103237

    Article  Google Scholar 

  44. Knowles, J.D., Corne, D.W.: Benchmark problem generators and results for the multiobjective degree-constrained minimum spanning tree problem. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2001), pp. 424–431 (2001)

  45. Garey, M.R., Johnson, D.S.: Computers and Intractability, Vol. 174. Freeman, San Francisco (1979)

  46. Goldbarg, E.F.G., de Souza, G.R., Goldbarg, M.C.: Particle swarm optimization for the bi-objective degree constrained minimum spanning tree. In: 2006 IEEE International Conference on Evolutionary Computation, pp. 420–427. IEEE (2006). https://doi.org/10.1109/CEC.2006.1688339

  47. Ghosh, A., Incel, Ö.D., Kumar, V.A., Krishnamachari, B.: Multichannel scheduling and spanning trees: throughput-delay tradeoff for fast data collection in sensor networks. IEEE/ACM Trans. Netw. 19(6), 1731–1744 (2011). https://doi.org/10.1109/TNET.2011.2146273

    Article  Google Scholar 

  48. An, M.K., Lam, N.X., Huynh, D.T., Nguyen, T.N.: Bounded-degree minimum-radius spanning trees in wireless sensor networks. Theoret. Comput. Sci. 498, 46–57 (2013). https://doi.org/10.1016/j.tcs.2013.05.033

    Article  MathSciNet  MATH  Google Scholar 

  49. Akcan, H.: On the complexity of energy efficient pairwise calibration in embedded sensors. Appl. Soft Comput. 13(4), 1766–1773 (2013). https://doi.org/10.1016/j.asoc.2013.01.013

    Article  Google Scholar 

  50. Carvalho, I.A., Ribeiro, M.A.: A node-depth phylogenetic-based artificial immune system for multi-objective network design problems. Swarm Evol. Comput. 50, 100491 (2019). https://doi.org/10.1016/j.swevo.2019.01.007

    Article  Google Scholar 

  51. Haimes, Y., Ladson, L., Wismer, D.: On a bicriterion formulation of the problems of integrated system identification and system optimization. IEEE Trans. Syst. Man Cybern. 1, 296–297 (1971). https://doi.org/10.1109/TSMC.1971.4308298

    Article  MathSciNet  MATH  Google Scholar 

  52. Marler, R., Arora, J.: Survey of multi-objective optimization methods for engineering. Struct. Multidiscip. Optim. 26(6), 369–395 (2004). https://doi.org/10.1007/s00158-003-0368-6

    Article  MathSciNet  MATH  Google Scholar 

  53. Chankong, V., Haimes, Y.Y.: Methods for Generating Noninferior Solutions, vol. 6, pp. 221–290. Courier Dover Publications. New York (2008)

  54. Wendell, R.E., Lee, D.N.: Efficiency in multiple objective optimization problems. Math. Program. 12(1), 406–414 (1977). https://doi.org/10.1007/BF01593807

    Article  MathSciNet  MATH  Google Scholar 

  55. Carvalho, I.A.: On the statistical evaluation of algorithmic’s computational experimentation with infeasible solutions. Inf. Process. Lett. 143, 24–27 (2019). https://doi.org/10.1016/j.ipl.2018.11.003

    Article  MathSciNet  MATH  Google Scholar 

  56. Shapiro, S.S., Wilk, M.B.: An analysis of variance test for normality (complete samples). Biometrika 52(3/4), 591–611 (1965). https://doi.org/10.2307/2333709

    Article  MathSciNet  MATH  Google Scholar 

  57. Wilcoxon, F.: Individual comparisons by ranking methods. Biom. Bull. 1(6), 80–83 (1945). https://doi.org/10.2307/3001968

    Article  Google Scholar 

  58. Gorski, J., Klamroth, K., Ruzika, S.: Generalized multiple objective bottleneck problems. Oper. Res. Lett. 40(4), 276–281 (2012). https://doi.org/10.1016/j.orl.2012.03.007

    Article  MathSciNet  MATH  Google Scholar 

  59. Leitner, M., Ruthmair, M., Raidl, G.R.: Stabilizing branch-and-price for constrained tree problems. Networks 61(2), 150–170 (2013). https://doi.org/10.1002/net.21484

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Science Department, Universidade Federal de Alfenas, Av. Jovino Fernandes de Sales 2600, Alfenas, MG, 37133-840, Brazil

    Iago A. Carvalho

  2. ORKAD, CRIStAL, UMR 9189, CNRS, Centrale Lille, Université de Lille, Lille, France

    Amadeu A. Coco

Authors
  1. Iago A. Carvalho
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Amadeu A. Coco
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Iago A. Carvalho.

Ethics declarations

Conflict of interest

The authors declare that they have no conflicts/competing interests.

Code availability

Code can be obtained upon request to the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This study was financed in part by the Coordenação de Aperfeiç oamento de Pessoal de Nível Superior–Brasil (CAPES)–Finance Code 001, the Conselho Nacional de Desenvolvimento Científico e Tecnológico–Brasil (CNPq), and the Fundação de Amparo à Pesquisa do Estado de Minas Gerais–Brasil (FAPEMIG).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carvalho, I.A., Coco, A.A. On solving bi-objective constrained minimum spanning tree problems. J Glob Optim 87, 301–323 (2023). https://doi.org/10.1007/s10898-023-01295-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-023-01295-8

Keywords

Search

Navigation