Abstract
Many real-life scheduling, routing and location problems can be formulated as combinatorial optimization problems whose goal is to find a linear layout of an input graph in such a way that the number of edge crossings is minimized. In this paper, we study a restricted version of the linear layout problem where the order of vertices on the line is fixed, the so-called fixed linear crossing number problem (FLCNP). We show that this \(\mathcal{NP}\)-hard problem can be reduced to the well-known maximum cut problem. The latter problem was intensively studied in the literature; efficient exact algorithms based on the branch-and-cut technique have been developed. By an experimental evaluation on a variety of graphs, we show that using this reduction for solving FLCNP compares favorably to earlier branch-and-bound algorithms.
Partially supported by the Marie Curie RTN ADONET 504438 funded by the EU.
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References
Cimikowski, R.: Algorithms for the fixed linear crossing number problem. Disc. Appl. Math. 122, 93–115 (2002)
Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Algorithms for drawing graphs: an annotated bibliography. Computational Geometry: Theory and Applications 4, 235–282 (1994)
Galil, Z., Kannan, R., Szemerédi, E.: On nontrivial separators for k-page graphs and simulations by nondeterministic one-tape Turing machines. J. Comput. System Sci. 38(1), 134–149 (1989)
Garey, M.R., Johnson, D.S.: Crossing number is NP-complete. SIAM J. Alg. Disc. Meth. 4, 312–316 (1983)
Gilbert, R.S., Kleinöder, W.K.: CNMgraf – graphic presentation services for network management. In: Proc. 9th Symposium on Data Communication, pp. 199–206 (1985)
Harju, T., Ilie, L.: Forbidden subsequences and permutations sortable on two parallel stacks. In: Where mathematics, computer science, linguistics and biology meet, pp. 267–275. Kluwer, Dordrecht (2001)
Laurent, M.: The max-cut problem. In: Dell’Amico, M., Maffioli, F., Martello, S. (eds.) Annotated Bibliography in Combinatorial Optimization. Wiley, Chichester (1997)
Laurent, M., Rendl, F.: Semidefinite programming and integer programming. In: Discrete Optimization, pp. 393–514. Elsevier, Amsterdam (2005)
Liers, F., Jünger, M., Reinelt, G., Rinaldi, G.: Computing exact ground states of hard Ising spin glass problems by branch-and-cut. In: New Optimization Algorithms in Physics, pp. 47–69. Wiley-VCH, Chichester (2004)
Masuda, S., Nakajima, K., Kashiwabara, T., Fujisawa, T.: Crossing minimization in linear embeddings of graphs. IEEE Trans. Comput. 39(1), 124–127 (1990)
Munro, J.I., Raman, V.: Succinct representation of balanced parentheses and static trees. SIAM J. Comput. 31(3), 762–776 (2001)
Nicholson, T.A.J.: Permutation procedure for minimizing the number of crossings in a network. Proc. IEEE 115, 21–26 (1968)
Rosenberg, A.L.: DIOGENES, circa 1986. In: Makedon, F., Mehlhorn, K., Papatheodorou, T.S., Spirakis, P.G. (eds.) AWOC 1986. LNCS, vol. 227, pp. 96–107. Springer, Heidelberg (1986)
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Buchheim, C., Zheng, L. (2006). Fixed Linear Crossing Minimization by Reduction to the Maximum Cut Problem. In: Chen, D.Z., Lee, D.T. (eds) Computing and Combinatorics. COCOON 2006. Lecture Notes in Computer Science, vol 4112. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11809678_53
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DOI: https://doi.org/10.1007/11809678_53
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