Abstract
We prove the NP-completeness of a weighted version of the jump number problem on two-dimensional orders, by reducing the Maximum Independent Set on cubic planar graphs, using a geometrical construction.
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Arnim, A. and de la Higuera, C.: Computing the jump number on semi-orders is polynomial, Discrete Appl. Math. 51 (1994), 219–232.
Asano, T.: Difficulty of the maximum independent set problem on intersection graphs of geometric objects, In: Y. Alavi et al. (eds), Graph Theory, Combinatorics and Application, 1991, pp. 9–18.
Bouchitté, V. and Habib, M.: The calculation of invariants for ordered sets, In: I. Rival (ed.), Algorithms and Order, NATO ASI, Ser. C 255, Kluwer Acad. Publ., Dordrecht, 1989, pp. 231–279.
Ceroi, S.: Jump number of 2-dimensional orders and intersection graphs of rectangles, To appear in DMTCS, special issue on ORDAL'99, 2000.
Chaty, G. and Chein, M.: Ordered matchings and matchings without alternating cycles in bipartite graphs, Utilitas Math. 16 (1979), 183–187.
Cogis, O. and Habib, M.: Nombre de sauts et graphes série-parallèles, RAIRO Inform. Theo. 13 (1979), 3–18.
Colbourn, C. and Pulleyblank, W.: Minimizing setups in ordered sets of fixed width, Order 1 (1985), 225–228.
Duffus, D., Rival, I. and Winkler, P.: Minimizing setups for cycle-free ordered sets, Proc. Amer. Math. Soc. 85 (1982), 509–513.
El-Zahar, M. and Rival, I.: Greedy linear extensions to minimize jumps, Discrete Appl. Math. 11 (1985), 143–156.
Faigle, U., Gierz, G. and Schrader, R.: Algorithmic approach to setup minimization, SIAM J. Comput. 14 (1985), 954–965.
Faigle, U. and Schrader, R.: A setup heuristic for interval orders, Oper. Res. Lett. 4 (1985), 185–188.
Felsner, S.: A 3/2-approximation algorithm for the jump number of interval orders, Order 6(4) (1990), 325–334.
Felsner, S.: Interval orders: combinatorial structure and algorithms, Technische Universität Berlin, Fachbereich Mathematik, 1992.
Franzblau, D. S. and Kleitman, D. J.: An algorithm for covering polygons with rectangles, Information and Control 63 (1984), 164–189.
Garey, M. R., Johnson, D. S. and Stockmeyer, L.: Some simplified NP-complete graph problems, Theoret. Comput. Sci. 1 (1976), 237–267.
Liu, Y., Morgana, A. and Simeone, B.: A linear algorithm for 2-bend embeddings of planar graphs in the two-dimensional grid, Discrete Appl. Math. 81 (1998), 69–91.
Lubiw, A.: The boolean basis problem and how to cover some polygons by rectangles, SIAM J. Discrete Math. 3(1) (1990), 98–115.
Möhring, R. H.: Computationally tractable classes of ordered sets, In: I. Rival (ed.), Algorithms and Order, NATO ASI, Ser. C 255, Kluwer Acad. Publ., Dordrecht, 1989, pp. 105–193.
Mitas, J.: Tackling the jump number of interval orders, Order 8 (1991), 115–132.
Müller, H.: Alternating cycle-free matchings, Order 7 (1990), 11–21.
Pulleyblank, W.: Alternating cycle-free matchings, Technical Report 82-18, Dept. of Combinatorics and Optimization, University of Waterloo, 1982.
Rim, C. S. and Nakajima, K.: On rectangle intersection and overlap graphs, IEEE Trans. Circuits Systems I 42(9) (1995), 549–553.
Sharary, A.: The jump number of Z-free ordered sets, Order 8 (1991), 267–273.
Steiner, G. and Stewart, L. K.: A linear time algorithm to find the jump number of 2-dimensional bipartite partial orders, Order 3 (1987), 359–367.
Sysło, M. M.: The jump number problem on interval orders: A 3/2 approximation algorithm, Discrete Math. 1–3 (1995), 119–130.
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Ceroi, S. A Weighted Version of the Jump Number Problem on Two-Dimensional Orders is NP-Complete. Order 20, 1–11 (2003). https://doi.org/10.1023/A:1024417802690
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DOI: https://doi.org/10.1023/A:1024417802690