Abstract
A labeling of a graph with n vertices and m edges is a one-to-one mapping from the union of the set of vertices and edges onto the set {1,2,…,m + n} . Such a labeling is defined as magic, if one or both of the following two conditions is fulfilled: the sum of an edge label and the labels of its endpoint vertices is constant for all edges; the sum of a vertex label and the labels of its incident edges is constant for all vertices. We present effective IP and Sat based algorithms for the problem of finding a magic labeling for a given graph. We experimentally compare the resulted algorithms by applying it to random graphs. Finally, we demonstrate its performance by solving five open problems within the theory of magic graphs, posed in the book of Wallis.
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References
Arnold, F.: Totally Magic Graphs – A Complete Search On Small Graphs. Master Thesis, Clausthal University of Technology, Germany (2013)
Baker, A., Sawada, J.: Magic Labelings on Cycles and Wheels. In: Yang, B., Du, D.-Z., Wang, C.A. (eds.) COCOA 2008. LNCS, vol. 5165, pp. 361–373. Springer, Heidelberg (2008)
Bertault, F., Feria-Purón, R., Miller, M., Pérez-Rosés, H., Vaezpour, E.: A Heuristic for Magic and Antimagic Graph Labellings. In: Proc. VII Spanish Congress on Metaheuristics and Evolutive and Bioinspired Algorithms, Valencia, Spain (2010)
Bloom, G.S., Golomb, S.W.: Applications of Numbered Undirected Graphs. Proc. IEEE 65(4), 562–570 (1977)
Bloom, G.S., Golomb, S.W.: Numbered Complete Graphs, Unusual Rulers, and Assorted Applications. In: Theory and Applications of Graphs. Lecture Notes in Mathematics, vol. 642, pp. 53–65 (1978)
Eén, N., Sörensson, N.: An Extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004)
Exoo, G., Ling, A.C.H., McSorley, J.P., Philipps, N.C., Wallis, W.D.: Totally Magic Graphs. Discrete Math. 254(1-3), 103–113 (2002)
Gallian, J.A.: A Dynamic Survey of Graph Labelings. Electron. J. Combin. 15, DS6 (2008)
Gent, I.P., Lynce, I.: A Sat Encoding for the Social Golfer Problem. In: 19th International Joint Conference on Artificial Intelligence (IJCAI), Workshop on Modelling and Solving Problems with Constraints (2005)
Gomes, C.P., Kautz, H., Sabharwal, A., Selman, B.: Satisfiability Solvers. In: van Harmelen, F., Lifschitz, V., Porter, B. (eds.) Handbook of Knowledge Representation. Foundations of Artificial Intelligence, vol. 3, pp. 89–134. Elsevier (2008)
Hoos, H.H.: Sat-Encodings, Search Space Structure, and Local Search Performance. In: Proc. 16th International Joint Conference on Artificial Intelligence (IJCAI), pp. 296–303. Morgan Kaufmann (1999)
Jäger, G., Zhang, W.: An Effective Algorithm for and Phase Transitions of the Directed Hamiltonian Cycle Problem. J. Artificial Intelligence Res. 39, 663–687 (2010)
Kalantari, B., Khosrovshahi, G.B.: Magic Labeling in Graphs: Bounds, Complexity, and an Application to a Variant of TSP. Networks 28(4), 211–219 (1996)
Lynce, I., Marques-Silva, J., Prestwich, S.D.: Boosting Haplotype Inference with Local Search. Constraints 13(1-2), 155–179 (2008)
Lynce, I., Ouaknine, J.: Sudoku as a Sat Problem. In: Proc. 9th International Symposium on Artificial Intelligence and Mathematics, AIMATH (2006)
MacDougall, J.A., Miller, M., Slamin, Wallis, W.D.: Vertex-magic Total Labelings of Graphs. Util. Math. 61, 3–21 (2002)
MacDougall, J.A., Miller, M., Wallis, W.D.: Vertex-magic Total Labelings of Wheels and Related Graphs. Util. Math. 62, 175–183 (2002)
Prestwich, S.D.: Sat Problems with Chains of Dependent Variables. Discrete Appl. Math. 130(2), 329–350 (2003)
Sun, G.C., Guan, J., Lee, S.-M.: A Labeling Algorithm for Magic Graph. Congr. Numer. 102, 129–137 (1994)
Vanderbei, R.J.: Linear Programming: Foundations and Extensions, 3rd edn. International Series in Operations Research & Management Science, vol. 114. Springer (2008)
Velev, M.N., Gao, P.: Efficient Sat Techniques for Absolute Encoding of Permutation Problems: Application to Hamiltonian Cycles. In: Proc. 8th Symposium on Abstraction, Reformulation and Approximation (SARA), pp. 159–166 (2009)
Wallis, W.D.: Magic Graphs. Birkhäuser, Boston (2001)
Source Code of [6] (MiniSat), http://minisat.se/
Homepage of IP solver Cplex, http://www.ilog.com/products/optimization/archive.cfm
International Sat Competition, http://www.satcompetition.org/
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Jäger, G. (2013). SAT and IP Based Algorithms for Magic Labeling with Applications. In: Lecroq, T., Mouchard, L. (eds) Combinatorial Algorithms. IWOCA 2013. Lecture Notes in Computer Science, vol 8288. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45278-9_22
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DOI: https://doi.org/10.1007/978-3-642-45278-9_22
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