Abstract
In recent years there has been a growing interest for the problem of the minimal partitioning of graphs with supply and demand, due to its close connection to electrical distribution systems, especially in the context of smartgrids. In this paper we present a new version of the problem which is more suitable for practical applications in modeling such systems. More precisely, the constraint of having a unique supply node in a subgraph (partition) is substituted with a limit on the number of subgraphs and the capacity for each of them. The problem is initially solved by a two stage greedy method. With the goal of further improving the quality of found solutions, a corresponding GRASP and an ant colony optimization algorithm are developed. Due to the novelty of the problem, we include a description of a method for generating test instances with known optimal solutions. In our computational experiments we evaluate the performance of the proposed algorithms on both trees and general graphs. The tests show that the proposed ant colony approach manages to frequently find optimal solutions. It has an average relative error of less than 2 % when compared to known optimal solutions. Moreover, it outperform the GRASP.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andreev K, Räcke H (2004) Balanced graph partitioning. In: Proceedings of the sixteenth annual ACM symposium on parallelism in algorithms and architectures. SPAA ’04. ACM, New York, pp 120–124
Arefifar S, Mohamed Y, EL-Fouly THM (2012) Supply-adequacy-based optimal construction of microgrids in smart distribution systems. IEEE Trans Smart Grid 3(3):1491–1502
Arefifar S, Mohamed YR, EL-Fouly T (2013a) Comprehensive operational planning framework for self-healing control actions in smart distribution grids. IEEE Trans Power Syst 28(4):4192–4200
Arefifar S, Mohamed YR, EL-Fouly T (2013b) Optimum microgrid design for enhancing reliability and supply-security. IEEE Trans Smart Grid 4(3):1567–1575
Barnes E, Vannelli A, Walker J (1988) A new heuristic for partitioning the nodes of a graph. SIAM J Discrete Math 1(3):299–305
Comellas F, Sapena E (2006) A multiagent algorithm for graph partitioning. In: Rothlauf F, Branke J, Cagnoni S, Costa E, Cotta C, Drechsler R, Lutton E, Machado P, Moore J, Romero J, Smith G, Squillero G, Takagi H (eds) Applications of evolutionary computing. Lecture notes in computer science, vol 3907. Springer, Berlin, pp 279–285
Dorigo M, Blum C (2005) Ant colony optimization theory: a survey. Theor Comput Sci 344(2):243–278
Dorigo M, Gambardella LM (1997) Ant colony system: a cooperative learning approach to the traveling salesman problem. IEEE Trans Evol Comput 1(1):53–66
Feo TA, Resende MG (1995) Greedy randomized adaptive search procedures. J Global Optim 6(2):109–133
Hart JP, Shogan AW (1987) Semi-greedy heuristics: an empirical study. Oper Res Lett 6:107–114
Hatziargyriou N, Asano H, Iravani R, Marnay C (2007) Microgrids. IEEE Power Energ Mag 5(4):78–94
Hinne M, Marchiori E (2011) Cutting graphs using competing ant colonies and an edge clustering heuristic. In: Merz P, Hao JK (eds) Evolutionary computation in combinatorial optimization. Lecture notes in computer science, vol 6622. Springer, Berlin, pp 60–71
Ito T, Zhou X, Nishizeki T (2005) Partitioning trees of supply and demand. Int J Found Comput Sci 16(4):803–827
Ito T, Demaine ED, Zhou X, Nishizeki T (2008) Approximability of partitioning graphs with supply and demand. J Discrete Algorithms 6(4):627–650
Ito T, Hara T, Zhou X, Nishizeki T (2012) Minimum cost partitions of trees with supply and demand. Algorithmica 64(3):400–415
Jovanovic R (2015) Benchmark data sets for the problem of partitioning graphs with supply and demand. http://mail.ipb.ac.rs/~rakaj/home/graphsdlc.htm
Jovanovic R, Bousselham A (2014) A greedy method for optimizing the self-adequacy of microgrids presented as partitioning of graphs with supply and demand. In: The 2nd international renewable and sustainable energy conference Ouarzazate. IEEE conference, pp 154–159, October 17–19, 2014
Jovanovic R, Tuba M (2011) An ant colony optimization algorithm with improved pheromone correction strategy for the minimum weight vertex cover problem. Appl Soft Comput 11(8):5360–5366
Jovanovic R, Tuba M (2013) Ant colony optimization algorithm with pheromone correction strategy for the minimum connected dominating set problem. Comput Sci Inf Syst 10(3):133–149
Jovanovic R, Tuba M, Voß S (2014) An ant colony optimization algorithm for partitioning graphs with supply and demand. arXiv:1503.0089
Jovanovic R, Bousselham A, Voß S (2015) A heuristic method for solving the problem of partitioning graphs with supply and demand. Ann Oper Res. doi:10.1007/s10479-015-1930-5
Kawabata M, Nishizeki T (2013) Partitioning trees with supply, demand and edge-capacity. IEICE Trans 96–A(6):1036–1043
Morishita S, Nishizeki T (2013) Parametric power supply networks. In: Du DZ, Zhang G (eds) Computing and combinatorics. Lecture notes in computer science, vol 7936. Springer, Berlin, pp 245–256
Narayanaswamy NS, Ramakrishna G (2012) Linear time algorithm for tree t-spanner in outerplanar graphs via supply-demand partition in trees arXiv:1210.7919, accepted in Discrete Applied Mathematics (2014)
Popa A (2013) Modelling the power supply network - hardness and approximation. In: Chan TH, Lau L, Trevisan L (eds) Theory and applications of models of computation. Lecture notes in computer science, vol 7876. Springer, Berlin, pp 62–71
Reinelt G, Wenger KM (2010) Generating partitions of a graph into a fixed number of minimum weight cuts. Discrete Optim 7(12):1–12
Reinelt G, Theis DO, Wenger KM (2008) Computing finest mincut partitions of a graph and application to routing problems. Discrete Appl Math 156(3):385–396
Tashkova K, Korosec P, Silc J (2011) A distributed multilevel ant-colony algorithm for the multi-way graph partitioning. Int J Bio-Inspired Comput 3(5):286–296
Voß S, Fink A, Duin C (2005) Looking ahead with the pilot method. Ann Oper Res 136(1):285–302
Acknowledgments
The comments of two anonymous referees are greatly appreciated.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jovanovic, R., Bousselham, A. & Voß, S. Partitioning of supply/demand graphs with capacity limitations: an ant colony approach. J Comb Optim 35, 224–249 (2018). https://doi.org/10.1007/s10878-015-9945-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-015-9945-z