Abstract
The jump number of a partially ordered set (poset) P isthe minimum number of incomparable adjacent pairs (jumps) in some linearextension of P. The problem of finding a linear extension of Pwith minimum number of jumps (jump number problem) is known to beNP-hard in general and, at the best of our knowledge, no exactalgorithm for general posets has been developed. In this paper, wegive examples of applications of this problem and propose for thegeneral case a new heuristic algorithm and an exactalgorithm. Performances of both algorithms are experimentallyevaluated on a set of randomly generated test problems.
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L. Bianco, P. Dell'Olmo, and S. Giordani, "Exact and heuristic algorithms for the jump number problem," I.A.S.I., C.N.R. Tech. Rep., I.A.S.I., C.N.R., Rome, Italy, vol. 376, 1994.
L.D. Bodin, B.L. Golden, A.A. Assad, and M.O. Ball, "Routing and scheduling of vehicles and crews," Comput. & Opns. Res., vol. 10, pp. 63–211, 1983.
V. Bouchitté and M. Habib, "NP-completeness properties about linear extensions," Order, vol. 4, pp. 143–154, 1987.
O. Cogis and M. Habib, "Nombre de Sauts et Graphes Série-parallèles," RAURO Inform. Théor., vol. 13, pp. 3–18, 1979.
C.J. Colbourn and W.R. Pulleyblank, "Minimizing setups in ordered sets of fixed width," Order, vol. 1, pp. 225–229, 1985.
C.F. Daganzo and R.W. Hall, "A routing model for pickups and deliveries: No capacity restrictions on the secondary items," Trans. Sci., vol. 27, pp. 315–329, 1993.
D. Duffus, I. Rival, and P. Winkler "Minimizing setups for cycle-free ordered sets," Proc. Am. Math. Soc., vol. 85, pp. 509–513, 1982.
U. Faugle and R. Schrader, "Minimizing completion time for a class of scheduling problems," Inform. Process. Lett., vol. 19, pp. 27–29, 1984.
U. Faugle and R. Schrader, "A setup heuristic for interval orders," Opns. Res. Lett., vol. 4, pp. 185–188, 1985.
S. Felsner, "A 3/2-approximation algorithm for the jump number of interval orders," Orders, vol. 6, pp. 325–334, 1990.
G. Gierz and W. Poguntke, "Minimizing setups for ordered sets: A linear algebrauc approach," SIAM J. Alg. Disc. Meth., vol. 4, pp. 132–144, 1983.
J. Mitas, "Tackling the jump number problem for interval orders," Order, vol. 8, pp. 115–132, 1991.
R.H. Möhring, "Computationally tractable classes of ordered sets," in Algorithms and Order, I. Rival (Ed.), Kluwer Academic Publisher, pp. 105–193, 1989.
M.G. Norman and P. Thanisch, "Models of machines and computation for mapping in multicomputers," ACM Comput. Surv., vol. 25, pp. 263–302, 1993.
I. Rival, "Optimal linear extensions by interchanging chauns," Proc. Amer. Math. Soc., vol. 89, pp. 387–394, 1983.
M.W.P. Savelsbergh and M. Sol, "The general pickup and delivery problem," Trans. Sci., vol. 29, pp. 17–29, 1995.
G. Steiner, "On the complexity of dynamic programming for sequencing problems with precedence constraunts," Anns. Opns. Res., vol. 26, pp. 103–123, 1990.
M.M. Syslo, "Minimizing the jump-number for partially ordered sets: A graph-theoretic approach," Order, vol. 1, pp. 7–19, 1984.
M.M. Syslo, "A graph-theoretic approach to the jump-number problem," in Graphs and Orders, I. Rival (Ed.), D. Reidel Publishing Company, pp. 185–215, 1985.
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Bianco, L., Dell‘Olmo, P. & Giordani, S. An Optimal Algorithm to Find the Jump Number of Partially Ordered Sets. Computational Optimization and Applications 8, 197–210 (1997). https://doi.org/10.1023/A:1008625405476
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DOI: https://doi.org/10.1023/A:1008625405476