Abstract
The role of 0–1 programming problems having monotone or regular feasible sets was pointed out in [6]. The solution sets of covering and of knapsack problems are examples of monotone and of regular sets respectively. Some connections are established between prime implicants of a monotone or a regular Boolean functionβ on the one hand, and facets of the convex hullH of the zeros ofβ on the other. In particular (Corollary 2) a necessary and sufficient condition is given for a constraint of a covering problem to be a facet of the corresponding integer polyhedron. For any prime implicantP ofβ, a nonempty familyF(P) of facets ofH is constructed. Proposition 17 gives easy-to-determine sharp upper bounds for the coefficients of these facets whenβ is regular. A special class of prime implicants is described for regular functions and it is shown that for anyP in this class,F(P) consists of one facet ofH, and this facet has 0–1 coefficients. Every nontrivial facet ofH with 0–1 coefficients is obtained from this class.
Similar content being viewed by others
References
E. Balas and R. Jeroslow, “Cannonical cuts on the unit hypercube”,Management Sciences Research Report Series, No. 198(R), Carnegie-Mellon University, Pittsburgh, Pa. August–December 1969, revised February 1971).
V. Chvátal, “Edmonds polyhedra and a hierarchy of combinatorial problems”,Discrete Mathematics 4 (1973) 305–337.
F. Glover, “Unit-coefficient inequalities for zero–one programming”, Management ScienceReport Series, No. 73-7University of Colorado, Boulder, Colo. (July 1973).
F. Glover, “Polyhedra Annexations in mixed integer programming”,Management Science Report Series, No. 73-9University of Colorado, Boulder, Colo. (August 1973).
R.E. Gomory, “Some polyhedra related to combinatorial problems”,Linear Algebra and its Applications 2 (1969) 451–558.
P.L. Hammer, E.L. Johnson and U.N. Peled: “Regular 0–1 programs”, Combinatorics and Optimization Research Rept., University of Waterloo, Waterloo, Ont., CORR No. 73-18 (September 1973).
S.T. Hu,Threshold logic (University of California Press, Berkely, Calif., 1965).
S. Muroga,Threshold logic and its applications (Wiley—Interscience, New York, 1971).
G.L. Nemhauser and L.E. Trotter, Jr., “Properties of vertex packing and independence system polyhedra”,Mathematical Programming 6 (1974) 48–61.
M.W. Padberg, “A note on zero–one programming”, Combinatorics and Optimization Research Rept., CORR No. 73-5, University of Waterloo, Waterloo, Ont. (March 1973).
U.N. Peled: “An Elementary Calculation of the Circulant Determinant”, University of Waterloo, Ont. Combinatorics and Optimization Research Rept. CORR No. 73-27 (October 1973).
M.A. Pollatschek, “Algorithms on finite weighted graphs”, Ph.D. Thesis, Technion-Israel Institute of Technology, Faculty of Industrial and Management Engineering (1970) (in Hebrew, with English Synopsis).
K. Spielberg, “Minimal preferred variable reduction for zero–one programming, Tech. Rept. No. 320-3013 IBM Philadelphia Scientific Center (July 1972).
R.O. Winder, “Threshold logic”, Ph. D. Thesis, Princeton University, Princeton, N.J.Mathematics Department (1962).
L.A. Wolsey, “Faces for linear inequalities in 0–1 variables”, Centre for Operations Research and Econometrics, Louvain (July 1973).
Additional references
E. Balas, “Facets of the knapsack polytope”,Mathematical Programming 8 (1975) 146–164 (this issue).
L.A. Wolsey, “Faces for linear inequalities in 0–1 variables”, CORE Discussion Paper No. 7338 (November 1973).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hammer, P.L., Johnson, E.L. & Peled, U.N. Facet of regular 0–1 polytopes. Mathematical Programming 8, 179–206 (1975). https://doi.org/10.1007/BF01580442
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01580442