Abstract
This paper deals with the problem of designing a least-cost digital data service (DDS) network that connects a given set of locations through digital switching offices with bridging capabilities. We present several alternative mixed 0–1 integer programming formulations and evaluate analytically their relative strengths by comparing their respective linear programming relaxations. By exploiting the structures inherent in a particularly strong formulation, we develop several classes of valid inequalities and cutting planes in order to tighten the initial formulation. For several problems of real-world data, computational results show that the strong formulation with valid inequalities and cutting planes generates a very tight lower bound (over 98% of the optimality) and so finds an optimal solution well within an acceptable time bound.
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References
Y.P. Aneja, An integer linear programming approach to the Steiner problem in graphs, Networks 10(1980)167–178.
J.E. Beasley, An algorithm for the Steiner problem in graphs, Networks 14(1984)147–159.
J.E. Beasley, An SST-based algorithm for the Steiner problem on graphs, Networks 19(1989)1–16.
S. Chopra and E. Gorres, On the node weighted Steiner tree problem, Working Paper, Northwestern University (1990).
S. Chopra and M.R. Rao, On the Steiner tree problem, I and II, Working Paper, New York University (1989).
S. Chopra, E. Gorres and M.R. Rao, Solving the Steiner tree problem on a graph using branch and cut, ORSA J. Comput. 4(1992)320–336.
CPLEX Optimization, Inc., Using the CPLEX callable library and CPLEX mixed integer library (1992).
H. Crowder, E.L. Johnson and M.W. Padberg, Solving large-scale zero-one linear programming problems, Oper. Res. 31(1983)803–834.
M. Fischetti, Facets of two Steiner arborescence polyhedra, Technical Report No. OR-89-4, University of Bologna (1989).
S.L. Hakimi, Steiner's problem in graphs and its implications, Networks 1(1971)113–133.
K.L. Hoffman and M.W. Padberg, Improving LP-representations of zero-one linear programs for branch- and-cut, ORSA J. Comput. 3(1991)121–134.
F.K. Hwang and D.S. Richards, Steiner tree problems, Networks 22(1992)55–89.
F. Glover, Least cost circuit layout for telecommunication networks, University of Colorado, Boulder, CO (1992).
N. Maculan, The Steiner problem in graphs, Ann. Discr. Math. 31(1987)185–222.
K.R. Martin, Generating alternative mixed-integer programming models using variable redefinition, Oper. Res. 35(1987)820–831.
S. McCrady, Mathematical model for least cost networks with switch hierarchies, US West Advanced Technologies, Boulder, CO (1992).
M.W. Padberg and G. Rinaldi, A branch- and cut algorithm for the resolution of large-scale symmetric traveling salesman problems, SIAM Rev. 33(1991)60–100.
T.J. Van Roy and L.A. Wolsey, Solving mixed integer programs by automatic reformulation, Oper. Res. 35(1987)45–57.
A. Segev, The node-weighted Steiner tree problem, Networks 17(1987)1–17.
H.D. Sherali, Tight linear programming representations: A key to solving nonconvex programming problems, Research Report, Department of Industrial and Systems Engineering, Virginia Polytechic Institute and State University, Blacksburg, VA (1992).
M.L. Shore, L.R. Foulds and P.B. Gibbons, An algorithm for the Steiner problem in graphs, Networks 12(1982)323–333.
P. Winter, Steiner problem in networks: A survey, Networks 17(1987)129–167.
L.A. Wolsey, Strong formulations for mixed integer programming: A survey, Math. Progr. 45(1989)173–191.
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Lee, Y., Lu, L., Qiu, Y. et al. Strong formulations and cutting planes for designing digital data service networks. Telecommunication Systems 2, 261–274 (1993). https://doi.org/10.1007/BF02109861
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DOI: https://doi.org/10.1007/BF02109861