Abstract
Given a linear inequality in 0–1 variables we attempt to obtain the faces of the integer hull of 0–1 feasible solutions. For the given inequality we specify how faces of a variety of lower-dimensional inequalities can be raised to give full-dimensional faces. In terms of a set, called a “strong cover”, we obtain necessary and sufficient conditions for any inequality with 0–1 coefficients to be a face, and characterize different forms that the integer hull must take.
In general the suggested procedures fail to produce the complete integer hull. Special subclasses of inequalities for which all faces can be generated are demonstrated. These include the “matroidal” and “graphic” inequalities, where a count on the number of such inequalities is obtained, and inequalities where all faces can be derived from lower dimensional faces.
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References
E. Balas and R. Jeroslow, “Canonical cuts on the unit hypercube”,SIAM Journal on Applied Mathematics 23 (1) (1972) 61–69.
E. Balas, “Facets of the knapsack polytope”,Mathematical Programming 8 (1975) 146–164 (this issue).
G.H. Bradley, P.L. Hammer and L.A. Wolsey, “Coefficient reduction for inequalities in 0–1 variables”,Mathematical Programming 7 (1974) 263–282.
V. Chvatal, “On certain polytopes associated with graphs”, Tech. Rept. CRM-238, Université de Montréal (1970).
V. Chvatal and P.L. Hammer, “Set packing problems and threshold graphs”, Centre de Recherches Mathématiques, Université de Montréal (August 1973).
J. Edmonds, “Matroids and the Greedy Algorithm”,Mathematical Programming 1 (1971) 127–136.
J. Edmonds, “Maximum matching and a polyhedron with 0–1 vertices”,Journal of Research of the National Bureau of Standards 6913 (1965) 125–130.
J. Edmonds, “Submodular functions, matroids and certain polyhedra”, in: H. Guy, ed.,Combinatorial structures and their applications (Gordon and Breach, New York, 1969).
F. Glover, “Unit coefficient inequalities for zero–one programming”, Management Science Report Series No. 73-7, University of Colorado (July 1973).
B. Grünbaum,Convex polytopes (Wiley, New York, 1967).
P.L. Hammer, E.L. Johnson and U.N. Peled, “Regular 0–1 programs”, Research Rept. CORR No. 73-19, University of Waterloo (September 1973).
P.L. Hammer, E.L. Johnson and U.N. Peled, “Facets of regular 0–1 polytopes”,Mathematical Programming 8 (1975) 179–206 (this issue).
G.L. Nemhauser and L.E. Trotter, Jr., “Properties of vertex packing and independence systems polyhedra”,Mathematical Programming 6 (1974) 48–61.
M.W. Padberg, “On the facial structure of set covering problems”, Reprint Series of the International Institute of Management, Berlin (April 1972).
M.W. Padberg, “A note on 0–1 programming”, Reprint 173/24, International Institute of Management, Berlin (March 1973).
L.E. Trotter, Jr., “Solution characteristics and algorithms for the vertex packing problem”, Tech. Rept. No. 168, Dept. of Operations Research, Cornell University (January 1973).
L.A. Wolsey, “An algorithm to determine optimal equivalent inequalities in 0–1 variables”, IBM Symposium on Discrete Optimization, Wildbad, Germany, October 1972.
L.A. Wolsey, “Faces for linear inequalities in 0–1 variables”, Mimeo, Louvain (July 1973).
L.A. Wolsey, “Faces for linear inequalities in 0–1 variables”, CORE Discussion Paper No. 7338, Louvain (November 1973).
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Wolsey, L.A. Faces for a linear inequality in 0–1 variables. Mathematical Programming 8, 165–178 (1975). https://doi.org/10.1007/BF01580441
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DOI: https://doi.org/10.1007/BF01580441