Abstract
The quality of the solutions to a combinatorial optimization problem is usually measured using a mathematical function, named objective function. This function is also used to guide heuristic procedures through the solution space, helping to detect promising search directions (i.e., it helps to compare the quality of different solutions). However, this task becomes very hard when many solutions are evaluated with the same value by the objective function. This fact commonly occurs in either max-min/min-max optimization problems. In those situations, a key strategy relies on the introduction of an alternative objective function. This function helps to determine which solution is more promising when the compared ones achieve the same value of the original objective function. In this paper we study the Cyclic Cutwidth Minimization Problem (CCMP), which is an example of a min-max optimization problem. Particularly, we analyze the influence in the search of using alternative objective functions within local search procedures. Also, we propose two alternative objective functions for the CCMP and compare its performance against a previously introduced one. Finally, we explored the combination of more than one alternative function.
This research has been supported by the Ministerio de Ciencia, Innovación y Universidades (Grant Refs. PGC2018-095322-B-C22 and FPU19/04098) and by C. Madrid and European Regional Development Fund (Grant Ref. P2018/TCS-4566).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abbott, H.: Hamiltonian circuits and paths on the n-cube. Can. Math. Bull. 9(5), 557–562 (1966)
Aschenbrenner, R.: A proof for the cyclic cutwidth of q5. Cal State University, San Bernardino, REU Project (2001)
Castillo, C.: A proof for the cyclic cutwidth of q6. Cal State University, San Bernardino, REU Project (2003)
Cavero, S., Pardo, E.G., Laguna, M., Duarte, A.: Multistart search for the cyclic cutwidth minimization problem. Comput. Oper. Res. 126, 105–116 (2021)
Cohoon, J., Sahni, S.: Heuristics for the backplane ordering. J. VLSI Comput. Syst. 2(1–2), 37–60 (1987)
DÃaz, J., Petit, J., Serna, M.: A survey of graph layout problems. ACM Comput. Surv. (CSUR) 34(3), 313–356 (2002)
Duarte, A., Pantrigo, J.J., Pardo, E.G., Sánchez-Oro, J.: Parallel variable neighbourhood search strategies for the cutwidth minimization problem. IMA J. Manag. Math. 27(1), 55–73 (2016)
Duff, I.S., Grimes, R.G., Lewis, J.G.: Users’ guide for the harwell-boeing sparse matrix collection (release i) (1992)
Erbele, J., Chavez, J., Trapp, R.: The cyclic cutwidth of qn. California State University, San Bernardino USA, Manuscript (2003)
Gavril, F.: Some np-complete problems on graphs. Technical Report, Computer Science Department, Technion (2011)
Harper, L.H.: Optimal numberings and isoperimetric problems on graphs. J. Comb. Theory 1(3), 385–393 (1966)
Jain, P., Srivastava, K., Saran, G.: Minimizing cyclic cutwidth of graphs using a memetic algorithm. J. Heuristics 22(6), 815–848 (2016)
James, B.: The cyclical cutwidth of the three-dimensional and four dimensional cubes. Cal State University, San Bernardino McNair Scholar’s Program Summer Research Journal (1996)
Johnson, M.: The linear and cyclic cutwidth of the complete bipartite graph. Cal State University, San Bernardino, REU Project (2003)
Makedon, F., Sudborough, I.: On minimizing width in linear layouts. Discrete Appl. Math. 23(3), 243–265 (1989)
MartÃ, R., Pantrigo, J.J., Duarte, A., Pardo, E.G.: Branch and bound for the cutwidth minimization problem. Comput. Oper. Res. 40(1), 137–149 (2013)
Pantrigo, J.J., MartÃ, R., Duarte, A., Pardo, E.G.: Scatter search for the cutwidth minimization problem. Ann. Oper. Res. 199(1), 285–304 (2012)
Pardo, E.G., MartÃ, R., Duarte, A.: Linear layout problems. In: MartÃ, R., Panos, P., Resende, M.G. (eds.) Handbook of Heuristics, pp. 1–25. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-07153-4_45-1
Pardo, E.G., Mladenović, N., Pantrigo, J.J., Duarte, A.: Variable formulation search for the cutwidth minimization problem. Appl. Soft Comput. 13(5), 2242–2252 (2013)
Rios, F.R.: Complete graphs as a first step toward finding the cyclic cutwidth of the n-cube. Cal State University. San Bernardino McNair Scholar’s Program Summer Research Journal (1996)
Rodriguez-Tello, E., Lardeux, F., Duarte, A., Narvaez-Teran, V.: Alternative evaluation functions for the cyclic bandwidth sum problem. Eur. J. Oper. Res. 273(3), 904–919 (2019)
Rodriguez-Tello, E., Narvaez-Teran, V., Lardeux, F.: Dynamic multi-armed bandit algorithm for the cyclic bandwidth sum problem. IEEE Access 7, 40258–40270 (2019)
Rolim, J., Sýkora, O., Vrt’o, I.: Optimal cutwidths and bisection widths of 2- and 3-dimensional meshes. In: Nagl, M. (ed.) WG 1995. LNCS, vol. 1017, pp. 252–264. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-60618-1_80
Santos, V.G.M., de Carvalho, M.A.M.: Tailored heuristics in adaptive large neighborhood search applied to the cutwidth minimization problem. Eur. J. Oper. Res. 289(3), 1056–1066 (2021)
Schröder, H., Sýykoa, O., Vrt’o, I.: Cyclic cutwidth of the mesh. In: Pavelka, J., Tel, G., Bartošek, M. (eds.) SOFSEM 1999. LNCS, vol. 1725, pp. 449–458. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-47849-3_33
Schröder, H., Sỳkora, O., Vrt’o, I.: Cyclic cutwidths of the two-dimensional ordinary and cylindrical meshes. Discrete Appl. Math. 143(1-3), 123–129 (2004)
Sciortino, V., Chavez, J., Trapp, R.: The cyclic cutwidth of a \(p_2\times p_2\times p_n\) mesh. Cal State University, San Bernardino, REU Project (2002)
Thilikos, D.M., Serna, M., Bodlaender, H.L.: Cutwidth II: algorithms for partial w-trees of bounded degree. J. Algorithms 56(1), 25–49 (2005)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Cavero, S., Pardo, E.G., Duarte, A. (2021). Influence of the Alternative Objective Functions in the Optimization of the Cyclic Cutwidth Minimization Problem. In: Alba, E., et al. Advances in Artificial Intelligence. CAEPIA 2021. Lecture Notes in Computer Science(), vol 12882. Springer, Cham. https://doi.org/10.1007/978-3-030-85713-4_14
Download citation
DOI: https://doi.org/10.1007/978-3-030-85713-4_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-85712-7
Online ISBN: 978-3-030-85713-4
eBook Packages: Computer ScienceComputer Science (R0)