Abstract
We solve for the first time to proven optimality the small instances in the classical literature benchmark of Minimum Linear Arrangement. This is achieved by formulating the problem as an ILP in a somehow unintuitive way, using variables expressing the fact that a vertex is between two other adjacent vertices in the arrangement. Using (only) these variables appears to be the key idea of the approach. Indeed, with these variables already the use of very simple constraints leads to good results, which can however be improved with a more detailed study of the underlying polytope.
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References
Amaral A.R.S.: On the exact solution of a facility layout problem. Eur. J. Oper. Res. 173, 508–518 (2006)
Amaral A.R.S.: An exact approach to the one-dimensional facility layout problem. Oper. Res. 56, 1026–1033 (2008)
Amaral A.R.S.: A mixed 0-1 linear programming formulation for the exact solution of the minimum linear arrangement problem. Optim. Lett. 3, 513–520 (2009)
Amaral A.R.S.: A new lower bound for the single row facility layout problem. Discr. Appl. Math. 157, 183–190 (2009)
Amaral A.R.S., Letchford A.N.: A polyhedral approach to the single row facility layout problem. http://www.optimization-online.org/DB_FILE/2008/03/1931.pdf (in preparation) (2011)
Barahona F., Mahjoub A.R.: On the cut polytope. Math. Program. 36, 157–173 (1986)
Booth K.S., Lueker G.S.: Testing for the consecutive ones property, interval graphs, and planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13, 335–379 (1976)
Caprara A., Letchford A.N., Salazar J.J.: Decorous lower bounds for minimum linear arrangement. INFORMS J. Comput. 23, 26–40 (2011)
Caprara A., Salazar J.J.: Laying out sparse graphs with provably minimum bandwidth. INFORMS J. Comput. 17, 356–373 (2005)
Christof T., Oswald M., Reinelt G.: Consecutive ones and a betweenness problem in computational biology. In: Bixby, R.E., Boyd, E.A., Rios-Mercado, R.Z. (eds) Proceedings of 6th Conference on Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science, vol. 1412., pp. 213–228. Springer, Berlin (1998)
Díaz J., Petit J., Serna M.: A survey of graph layout problems. ACM Comput. Surv. 34, 313–356 (2002)
Koren Y., Harel D.: A multi-scale algorithm for the linear arrangement problem. In: Kucera, L. (eds) Proceedings of 28th International Workshop on Graph-Theoretic Concepts in Computer Science. Lecture Notes in Computer Science, vol. 2573, pp. 293–306. Springer, Berlin (2002)
Liu W., Vannelli A.: Generating lower bounds for the linear arrangement problem. Discr. Appl. Math. 59, 137–151 (1995)
Petit J.: Layout Problems. PhD thesis, Universitat Politècnica de Catalunya, Barcelona (2001)
Petit J.: Experiments on the linear arrangement problem. J. Exp. Algorithmics, 8 (2003). article 2.3
Schwarz R.: A Branch-and-Cut Algorithm with Betweennes Variables for the Linear Arrangement Problem. Master Thesis, Universität Heidelberg (2010)
Seitz H.: Contributions to the Minimum Linear Arrangement Problem. PhD Thesis, Universität Heidelberg (2010)
Traversi E.: Orientation and Layout Problems on Graphs, with Applications. PhD Thesis, Università di Bologna (2010)
Tucker A.: A structure theorem for the consecutive ones property. J. Comb. Theory Ser. B 12, 153–162 (1972)
ZIB Optimization Suite (Version 1.2), Zuse Institut Berlin. http://zibopt.zib.de (2009)
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Caprara, A., Oswald, M., Reinelt, G. et al. Optimal linear arrangements using betweenness variables. Math. Prog. Comp. 3, 261–280 (2011). https://doi.org/10.1007/s12532-011-0027-7
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DOI: https://doi.org/10.1007/s12532-011-0027-7