Abstract
For a given inequality with 0–1 variables, there are many other “equivalent” inequalities with exactly the same 0–1 feasible solutions. The set of all equivalent inequalities is characterized, and methods to construct the equivalent inequality with smallest coefficients are described.
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References
V.J. Bowman, “Constraint classification of the unit hypercube”, Management Science Res. Rept. No. 287, Carnegie-Mellon University, Pittsburgh, Pa. (March 1972; revised July 1972).
V.J. Bowman and G. Nemhauser, “Deep cuts in integer programming”,Opsearch 8 (1971) 89–111.
G.H. Bradley, “Equivalent integer programs and canonical problems”,Management Science 17 (1971) 354–366.
G.H. Bradley, “Transformation of integer programs to knapsack problems”,Discrete Mathematics 1 (1971) 29–45.
G.H. Bradley and P.N. Wahi, “An algorithm for integer linear programming: a combined algebraic and enumeration approach”,Operations Research 21 (1973) 45–60.
A.V. Cabot, “An enumeration algorithm for knapsack problems”,Operations Research 18 (1970) 306–307.
B.H. Faaland and F.S. Hillier, “The accelerated bound-and-scan algorithm for integer programming”, Tech. Rept. No. 34, Department of Operations Research, Stanford University Stanford, Calif. (May 1972).
F. Glover and R.E. Woolsey, “Aggregating diophantine equations”,Unternehmensforschung 16 (1972) 1–10.
R.E. Gomory, “On the relation between integer and non-integer solutions to linear programs”,Proceedings of the National Academy of Science 53 (1965) 260–265.
F. Granot and P.L. Hammer, “On the use of Boolean functions in 0–1 programming”, Technion Mimeograph Series on Operations Research, Statistics and Economics, No. 70 (August 1970).
P.L. Hammer and L.A. Wolsey, “Equivalent forms of integer inequalities in 0–1 variables”, CORE Discussion Paper No. 7220, Université de Louvain, Belgium (August 1972).
S.T. Hu,Threshold logic (University of California Press, Berkeley, Calif., 1965).
F. Kianfar, “Stronger inequalities for 0–1 integer programming using knapsack functions”,Operations Research 19 (1971) 1374–1392.
A.H. Land and S. Powell, “Fortran programs for linear and nonlinear programming”, London School of Economics (1971).
S. Muroga,Threshold logic and its applications (Wiley, New York, 1971).
M. Padberg, “Equivalent knapsack-type formulations of bounded integer linear programs: an alternative approach”,Naval Research Logistics Quarterly 19 (1972) 699–708.
C.C. Petersen, “Computational experience with variants of the Balas algorithm applied to the selection of R&D projects”,Management Science 13 (1967) 736–750.
C.L. Sheng,Threshold logic (Academic Press, New York, 1969).
H.P. Williams, “Experiments in the formulation of integer programming problems”, University of Sussex (March 1973).
D.G. Willis, “Minimum weights for threshold switches”, in:Symposium on the applicacations of switching theory in space technology, Eds. H. Aiken and W.F. Main (Stanford University Press, Stanford, Calif. 1963) pp. 91–108.
L.A. Wolsey, “Extensions of the group theoretic approach in integer programming”,Management Science 18 (1971) 74–83.
L.A. Wolsey, “An algorithm to determine optimal equivalent inequalities in 0–1 variables”, presented at the IBM Symposium on Discrete Optimization, Wildbad, W. Germany, October 1972.
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This paper is an extended version of [11].
Supported by National Science Foundation Grant 32158X. This work was done while the author was at Yale University.
Supported by National Research Council of Canada Grant A8552.
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Bradley, G.H., Hammer, P.L. & Wolsey, L. Coefficient reduction for inequalities in 0–1 variables. Mathematical Programming 7, 263–282 (1974). https://doi.org/10.1007/BF01585527
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DOI: https://doi.org/10.1007/BF01585527