Abstract
Computing the Pathwidth of a graph is the problem of finding a linear ordering of the vertices such that the width of its corresponding path decomposition is minimized. This problem has been proven to be NP-hard. Currently, some of the best exact methods for generic graphs can be found in the mathematical software project called SageMath. This project provides an integer linear programming model (IPSAGE) and an enumerative algorithm (EASAGE), which is exponential in time and space. The algorithm EASAGE uses an array whose size grows exponentially with respect to the size of the problem. The purpose of this array is to improve the performance of the algorithm. In this chapter we propose two exact methods for computing pathwidth. More precisely, we propose a new integer linear programming formulation (IPPW) and a new enumerative algorithm (BBPW). The formulation IPPW generates a smaller number of variables and constraints than IPSAGE. The algorithm BBPW overcomes the exponential space requirement by using a last-in-first-out stack. The experimental results showed that, in average, IPPW reduced the number of variables by 33.3 % and the number of constraints by 64.3 % with respect to IPSAGE. This reduction of variables and constraints allowed IPPW to save approximately 14.9 % of the computing time of IPSAGE. The results also revealed that BBPW achieved a remarkable use of memory with respect to EASAGE. In average, BBPW required 2073 times less amount of memory than EASAGE for solving the same set of instances.
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References
Díaz, J., Petit, J., & Serna, M. (2002). A survey of graph layout problems. ACM Computing Surveys (CSUR), 34(3), 313–356.
Lengauer, T. (1981). Black-white pebbles and graph separation. Acta Informatica, 16(4), 465–475.
Leiserson, C. E. (1980, October). Area-efficient graph layouts. In Foundations of Computer Science, 1980, 21st Annual Symposium on (pp. 270–281). IEEE.
Linhares, A., & Yanasse, H. H. (2002). Connections between cutting-pattern sequencing, VLSI design, and flexible machines. Computers & Operations Research, 29(12), 1759–1772.
De Oliveira, A., & Lorena, L. A. (2002). A constructive genetic algorithm for gate matrix layout problems. Computer-Aided Design of Integrated Circuits and Systems, IEEE Transactions on, 21(8), 969–974.
Bodlaender, H., Gustedt, J., & Telle, J. A. (1998, January). Linear-time register allocation for a fixed number of registers. In SODA (Vol. 98, pp. 574–583).
Kornai, A., & Tuza, Z. (1992). Narrowness, pathwidth, and their application in natural language processing. Discrete Applied Mathematics, 36(1), 87–92.
Lopes, I. C., & Carvalho, J. M. (2010). Minimization of open orders using interval graphs. International Journal of Applied Mathematics, 40.
Dinneen, M. J. (1996). VLSI Layouts and DNA physical mappings. Technical Report, Los Alamos National Laboratory.
Bollobás, B., & Leader, I. (1991). Edge-isoperimetric inequalities in the grid. Combinatorica, 11(4), 299–314.
Castillo-García, N., Huacuja, H. J. F., Rangel, R. A. P., Flores, J. A. M., Barbosa, J. J. G., & Valadez, J. M. C. (2014). On the Exact Solution of VSP for General and Structured Graphs: Models and Algorithms. In Recent Advances on Hybrid Approaches for Designing Intelligent Systems (pp. 519–532). Springer International Publishing.
Ellis, J. A., Sudborough, I. H., & Turner, J. S. (1994). The vertex separation and search number of a graph. Information and Computation, 113(1), 50–79.
Skodinis, K. (2000). Computing optimal linear layouts of trees in linear time (pp. 403–414). Springer Berlin Heidelberg.
Bodlaender, H. L., & Möhring, R. H. (1993). The pathwidth and treewidth of cographs. SIAM Journal on Discrete Mathematics, 6(2), 181–188.
Bodlaender, H. L., Kloks, T., & Kratsch, D. (1995). Treewidth and pathwidth of permutation graphs. SIAM Journal on Discrete Mathematics, 8(4), 606–616.
Duarte, A., Escudero, L. F., Martí, R., Mladenovic, N., Pantrigo, J. J., & Sánchez-Oro, J. (2012). Variable neighborhood search for the vertex separation problem. Computers & Operations Research, 39(12), 3247–3255.
Sánchez-Oro, J., Pantrigo, J. J., & Duarte, A. (2014). Combining intensification and diversification strategies in VNS. An application to the Vertex Separation problem. Computers & Operations Research, 52, 209–219.
Kinnersley, N. G. (1992). The vertex separation number of a graph equals its path-width. Information Processing Letters, 42(6), 345–350.
Cohen, N., & Coudert, D. (2010). Integer linear programming formulation and enumerative algorithm for computing the vertex separation number. http://sagemanifolds.obspm.fr/preview/reference/graphs/sage/graphs/graph_decompositions/vertex_separation.html.
Suchan, K., & Villanger, Y. (2009). Computing pathwidth faster than 2n. In Parameterized and Exact Computation (pp. 324–335). Springer Berlin Heidelberg.
Huacuja, H. F., Castillo-García, N., Rangel, R. A. P., Flores, J. A. M., Barbosa, J. J. G., & Valadez, J. M. C. (2015). Two New Exact Methods for the Vertex Separation Problem. International Journal of Combinatorial Optimization Problems and Informatics, 6(1), 31–41.
Acknowledgments
This research was partially supported by the Mexican Council of Science and Technology (CONACYT). The second author would like to thank CONACYT for his Ph.D. scholarship. We also wish to thank the IBM Academic Initiative for allowing us to use the optimization software CPLEX.
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Fraire-Huacuja, H.J. et al. (2017). Integer Linear Programming Formulation and Exact Algorithm for Computing Pathwidth. In: Melin, P., Castillo, O., Kacprzyk, J. (eds) Nature-Inspired Design of Hybrid Intelligent Systems. Studies in Computational Intelligence, vol 667. Springer, Cham. https://doi.org/10.1007/978-3-319-47054-2_44
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DOI: https://doi.org/10.1007/978-3-319-47054-2_44
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