Abstract
Degree-constrained minimum spanning tree problem is an NP-hard bicriteria combinatorial optimization problem seeking for the minimum weight spanning tree subject to an additional degree constraint on graph vertices. Due to the NP-hardness of the problem, heuristics are more promising approaches to find a near optimal solution in a reasonable time. This paper proposes a decentralized learning automata-based heuristic called LACT for approximating the DCMST problem. LACT is an iterative algorithm, and at each iteration a degree-constrained spanning tree is randomly constructed. Each vertex selects one of its incident edges and rewards it if its weight is not greater than the minimum weight seen so far and penalizes it otherwise. Therefore, the vertices learn how to locally connect them to the degree-constrained spanning tree through the minimum weight edge subject to the degree constraint. Based on the martingale theorem, the convergence of the proposed algorithm to the optimal solution is proved. Several simulation experiments are performed to examine the performance of the proposed algorithm on well-known Euclidean and non-Euclidean hard-to-solve problem instances. The obtained results are compared with those of best-known algorithms in terms of the solution quality and running time. From the results, it is observed that the proposed algorithm significantly outperforms the existing method.
Similar content being viewed by others
References
Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. Freeman, San Francisco
Papadimitriou CH, Vazirani UV (1984) On two geometric problems related to the travelling salesman problem. J Algorithms 5(2):231–246
Ausiello G, Crescenzi P, Gambosi G, Kann V, Marchetti Spaccamela A, Protasi M (1999) Approximate solution of NP-hard optimization problems. Springer, Berlin
Monma C, Suri S (1992) Transitions in geometric minimum spanning trees. Discrete Comput Geom 8:265–293
Bau Y-T, Ho C-K, Ewe H-T (2008) Ant colony optimization approaches to the degree-constrained minimum spanning tree problem. J Inf Sci Eng 24(4):1081–1094
de Souza MC, Martins P (2008) Skewed VNS enclosing second order algorithm for the degree constrained minimum spanning tree problem. Eur J Oper Res 191:677–690
Krishnamoorthy M, Ernst AT (2001) Comparison of algorithms for the degree constrained minimum spanning tree. J Heuristics 7:587–611
Bau Y-T, Ho C-K, Ewe H-T (2005) An ant colony optimization approach to the degree-constrained minimum spanning tree problem. In: Computational intelligence and security. Lecture notes in computer science, vol 3801. Springer, Berlin, pp 657–662
Soak S-M, Corne D, Ahn B-H (2004) A new encoding for the degree constrained minimum spanning tree problem. In: Knowledge-based intelligent information and engineering systems. Lecture notes in computer science, vol 3213, pp 952–958
Binh HTT, Nguyen TB (2008) New particle swarm optimization algorithm for solving degree constrained minimum spanning tree problem. In: Trends in artificial intelligence. Lecture notes in computer science, vol 5351, pp 1077–1085
Raidl GR (2000) An efficient evolutionary algorithm for the degree-constrained minimum spanning tree problem. In: Proceedings of the 2000 congress on evolutionary computation, pp 104–111
Andrade R, Lucena A, Maculan N (2006) Using Lagrangian dual information to generate degree constrained spanning trees. Discrete Appl Math 154:703–717
Hanr L, Wang Y (2006) A novel genetic algorithm for degree-constrained minimum spanning tree problem. Int J Comput Sci Netw Secur 6(7A):50–57
Raidl GR, Julstrom BA (2000) A weighted coding in a genetic algorithm for the degree-constrained minimum spanning tree problem. In: Proceedings of the 2000 ACM symposium on applied computing, pp 440–445
Bui TN, Zrncic CM (2006) An ant-based algorithm for finding degree-constrained minimum spanning tree. In: Proceedings of the 8th annual conference on genetic and evolutionary computation, pp 11–18
Martins P, de Souza MC (2009) VNS and second order heuristics for the min-degree constrained minimum spanning tree problem. Comput Oper Res 36:2969–2982
Ning A, Ma L, Xiong X (2008) A new algorithm for degree-constrained minimum spanning tree based on the reduction technique. Prog Nat Sci 18:495–499
Kawatra R, Bricker D (2004) Design of a degree-constrained minimal spanning tree with unreliable links and node outage costs. Eur J Oper Res 156:73–82
Ribeiro CC, Souza MC (2002) Variable neighborhood search for the degree-constrained minimum spanning tree problem. Discrete Appl Math 118:43–54
Volgenant A (1989) A Lagrangean approach to the degree-constrained minimum spanning tree problem. Eur J Oper Res 39:325–331
Pereira TL, Carrano EG, Takahashi RHC, Wanner EF, Neto OM (2009) Continuous-space embedding genetic algorithm applied to the degree constrained minimum spanning tree problem. In: Proceedings of IEEE congress on evolutionary computation, pp 1391–1398
Doan MN (2007) An effective ant-based algorithm for the degree-constrained minimum spanning tree problem. In: Proceedings of IEEE congress on evolutionary computation, pp 485–491
Zhou G, Gen M, Wu T (1996) A new approach to the degree-constrained minimum spanning tree problem using genetic algorithm. In: Proceedings of IEEE international conference on systems, man, and cybernetics, pp 2683–2688
Ernst AT (2010) A hybrid Lagrangian particle swarm optimization algorithm for the degree-constrained minimum spanning tree problem. In: Proceedings of IEEE congress on evolutionary computation, pp 1–8
Goldbarg EFG, de Souza GR, Goldbarg MC (2006) Particle swarm optimization for the bi-objective degree constrained minimum spanning tree. In: Proceedings of IEEE congress on evolutionary computation, pp 420–427
Zeng Y, Wang Y-P (2003) A new genetic algorithm with local search method for degree-constrained minimum spanning tree problem. In: Proceedings of fifth international conference on computational intelligence and multimedia applications, pp 218–222
Bui TN, Deng X, Zrncic CM (2012) An improved ant-based algorithm for the degree-constrained minimum spanning tree problem. IEEE Trans Evol Comput 16(2):266–278
Han L-X, Wang Y, Guo F-Y (2005) A new genetic algorithm for the degree-constrained minimum spanning tree problem. In: Proceedings of IEEE international workshop on VLSI design and video technology, pp 125–128
Knowles J, Corne D (2000) A new evolutionary approach to the degree-constrained minimum spanning tree problem. IEEE Trans Evol Comput 4(2):125–134
Guo W-Z, Gao H-L, Chen G-L, Yu L (2009) Particle swarm optimization for the degree-constrained MST problem in WSN topology control. In: Proceedings of international conference on machine learning and cybernetics, pp 1793–1798
Mladenović N, Hansen P (1997) Variable neighborhood search. Comput Oper Res 24:1097–1100
Gruber M, Hemert J, Raidl GR (2006) Neighborhood searches for the bounded diameter minimum spanning tree problem embedded in a VNS, EA and ACO. In: Proceedings of genetic and evolutionary computational conference (GECCO’2006)
Oncan T (2007) Design of capacitated minimum spanning tree with uncertain cost and demand parameters. Inf Sci 177:4354–4367
Oncan T, Cordeau JF, Laporte G (2008) A tabu search heuristic for the generalized minimum spanning tree problem. Eur J Oper Res 191(2):306–319
Parsa M, Zhu Q, Garcia-Luna-Aceves JJ (1998) An iterative algorithm for delay-constrained minimum-cost multicasting. IEEE/ACM Trans Netw 6(4):461–474
Salama HF, Reeves DS, Viniotis Y (1997) The delay-constrained minimum spanning tree problem. In: Proceedings of the second IEEE symposium on computers and communications, pp 699–703
Gouveia L, Simonetti L, Uchoa E (2009) Modeling hop-constrained and diameter-constrained minimum spanning tree problems as Steiner tree problems over layered graphs. J Math Program
Kruskal JB (1956) On the shortest spanning sub tree of a graph and the traveling salesman problem. In: Proceedings of American Mathematical Society, pp 748–750
Prim RC (1957) Shortest connection networks and some generalizations. Bell Syst Tech J 36:1389–1401
Karp RM (1972) Reducibility among combinatorial problems. In: Complexity of computer computations. Plenum, New York, pp 85–103
Narendra KS, Thathachar KS (1989) Learning automata: an introduction. Prentice-Hall, New York
Thathachar MAL, Harita BR (1987) Learning automata with changing number of actions. IEEE Trans Syst Man Cybern 17:1095–1100
Narula SC, Ho CA (1980) Degree constrained minimum spanning tree. Comput Oper Res 7:239–249
Savelsbergh M, Volgenant T (1985) Edge exchanges in the degree-constrained minimum spanning tree problem. Comput Oper Res 12:341–348
Delbem A, de Carvalho A, Policastro C, Pinto A, Honda K, Garcia A (2004) Node-depth encoding for evolutionary algorithms applied to network design. In: Genetic evolutionary computation. Lecture notes in computer science, vol 3102. Springer, Berlin, pp 678–687
Boldon B, Deo N, Kumar N (1996) Minimum-weight degree constrained spanning tree problem: heuristics and implementation on an SIMD parallel machine. Parallel Comput 22:369–382
Zhou G, Gen M (1998) A note on genetic algorithms for degree constrained spanning tree problems. Networks 30(2):91–95
Akbari Torkestani J (2012) A new distributed job scheduling algorithm for grid systems. Cybern Syst (to appear)
Akbari Torkestani J (2012) An adaptive learning to rank algorithm: learning automata approach. Decis Support Syst 54(1):574–583
Akbari Torkestani J (2012) A distributed resource discovery algorithm for P2P grids. J Netw Comput Appl 35(6):2028–2036
Akbari Torkestani J (2012) Backbone formation in wireless sensor networks. Sens Actuators A, Phys 185:117–126
Akbari Torkestani J (2012) An adaptive heuristic to the bounded diameter minimum spanning tree problem. Soft Comput 16(11):1977–1988
Akbari Torkestani J (2012) An adaptive focused web crawling algorithm based on learning automata. Appl Intell 37(4):586–601
J Akbari Torkestani (2012) LAAP: a learning automata-based adaptive polling scheme for clustered wireless ad-hoc networks. Wirel Pers Commun (to appear)
Akbari Torkestani J (2012) Mobility prediction in mobile wireless networks. J Netw Comput Appl 35(5):1633–1645
Akbari Torkestani J, Meybodi MR (2012) Finding minimum weight connected dominating set in stochastic graph based on learning automata. Inf Sci 200:57–77
Akbari Torkestani J (2012) A learning automata-based solution to the bounded diameter minimum spanning tree problem. J Chin Inst Eng (to appear)
Akbari Torkestani J (2012) An adaptive learning automata-based ranking function discovery algorithm. J Intell Inf Syst 39(2):441–459
Akbari Torkestani J (2012) A new approach to the job scheduling problem in computational grids. Clust Comput 15(3):201–210
Akbari Torkestani J (2012) Mobility-based backbone formation in wireless mobile ad-hoc networks. Wirel Pers Commun (to appear)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Akbari Torkestani, J. Degree constrained minimum spanning tree problem: a learning automata approach. J Supercomput 64, 226–249 (2013). https://doi.org/10.1007/s11227-012-0851-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11227-012-0851-1