Abstract
A procedure is proposed that, given a linear inequalityL having positive integral coefficients in 0–1 variables, finds all the facets of the convex hull of the solutions ofL. This is done for allL by doing once and for all the computations for a particular sequenceM 1,M 2,... of such linear inequalities, called master knapsack problems. To find the facets for a givenL, it is enough to know the facets for the master knapsack problemM b, whereb is the free term inL (no matter how many variablesL might have). ThisM b has no more than order ofb logb variables. The procedure is based on polarity. All the facets forM 1,...,M 7 are tabulated.
Zusammenfassung
Es wird ein Verfahren vorgestellt, das für eine lineare UngleichungL in binären Variablen mit positiven ganzzahligen Koeffizienten alle Facetten der konvexen Hülle der Lösungen vonL bestimmt. Um diese Facetten für irgendeine UngleichungL mit rechter Seiteb zu erhalten, braucht man nur die Facetten des sogenannten Master-Knapsack-ProblemsM b zu kennen, dasO (b log b) Variablen besitzt. Fürb=1,..., 7 werden alle Facetten vonM b angegeben.
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References
Balas, E.: Facets of the knapsack polytope. Mathematical Programming8, 1975, 146–164.
Bradley, G.H., P.L. Hammer andL. Wolsey: Coefficient reduction for inequalities in 0–1 variables. Mathematical Programming7, 1974, 263–282.
Fulkerson, D.R.: Anti-blocking polyhedra. Journal of Combinatorial Theory (B)12, 1972, 50–71.
Glover, F.: Unit-coefficient inequalities for zero-one programming. University of Colorado, Management Science Report Series 73-7, July 1973.
Hall, M., Jr.: Combinatorial Theory. Blaisdell 1967.
Hammer, P.L., E.L. Johnson andU.N. Peled: Facets of regular 0–1 polytopes. Mathematical Programming8, 1975, 179–206.
—: The role of master polytopes in the unit cube. SIAM Journal on Applied Mathematics32, 1977, 711–716.
Johnson, E.L.: A class of facets of the master 0–1 knapsack polytope, IBM Thomas J. Watson Research Center, RC 5106, 1974.
Matheiss, T.H., andD.S. Rubin: A survey and comparison of methods for finding all vertices of convex polyhedral sets. Mathematics of Operations Research5, 1980, 167–185.
Padberg, M.W.: A note on zero-one programming. Operations Research23, 1975, 833–837.
Peled, U.N.: Properties of facets of binary polytopes. Annals of Discrete Mathematics1, 1977, 435–456.
Wolsey, L.A.: Faces for linear inequalities in 0–1 variables, Mathematical Programming8, 1975, 165–178.
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Hammer, P.L., Peled, U.N. Computing low-capacity 0–1 knapsack polytopes. Zeitschrift für Operations Research 26, 243–249 (1982). https://doi.org/10.1007/BF01917116
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DOI: https://doi.org/10.1007/BF01917116