Hammer, P. L., E. L. Johnson, U. N. Peled. 1975. Facets of regular 0-1 polytopes. Mathematical Programming 8, 179-206. Home/Search Document Not in Database Summary Related Articles Check |
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The Multiple Knapsack Problem With Color Constraints - Dawande, Kalagnanam (1998) (Correct)
....i2C O i W k . The cover is minimal with respect to k if P i2Cnfcg O i W k for all c 2 C. Suppose that C N is a minimal cover with respect to some knapsack k. The inequality X i2C x ik jCj Gamma 1 (3) is called the minimal cover inequality corresponding to C and k. It was shown in [1] [5] and [9] that the minimal cover inequality corresponding to C and k defines a facet for SK(C;O;W k ) 4] shows that all non trivial facets for SK(C;O;W k ) are facets for PMKP . Suppose the minimal cover C is composed of p N distinct colors, c 1 ; c 2 ; c p . Let S(c 1 ) S(c 2 ) S(c ....
Hammer, P.L., Johnson, E.L. and Peled, U.N. (1975) Facets of regular 0-1 polytopes, Mathematical Programming, 8, 179-206.
Facets for the Multiple Knapsack Problem - Ferreira, Martin, Weismantel (1993) (Correct)
....where jM j = 1. In analogy to the definition of MK let SK(N; f; F ) convfx 2 IR N j X i2N f i x i F; x i 2 f0; 1g; i 2 Ng denote the single knapsack polytope. Although a lot of emphasis has been put on studying the facial structure of SK(N; f; F ) see, for example, B75] W75] [HJP75], P75] BZ78] P80] MK and generalizations of it have not yet been studied to the same extent. In a few papers we find investigations in this direction. Crowder, Johnson and Padberg [CJP83] consider general 0 1 linear programs with no apparent structure: Let be given a matrix A 2 I Q m Thetan ....
....a il : a i ; if l = k, i 2 V; 0; otherwise. Let (N; M; f; F ) be an instance of the multiple knapsack problem. Suppose, S is a minimal cover with respect to some k 2 M . Then, the minimal cover inequality X i2S x ik jSj Gamma 1 defines a facet for the polytope SK(S; f; F k ) B75] [HJP75], W75] By applying Lemma 2.2 we can conclude that this minimal cover inequality defines a facet for MK(S Theta M; f; F ) Similarly, let N 0 [fzg; N 0 N; jN 0 j = n 0 and z 2 N nN 0 be a (1; d) configuration with respect to some knapsack k. The (1; d) configuration inequality (n ....
P. L. Hammer, E. L. Johnson and U. N. Peled, "Facets of Regular 0-1 Polytopes", Mathematical Programming 8, 179 - 206 (1975).
A Cutting Plane Based Algorithm for the Multiple Knapsack .. - Ferreira, Martin.. (1993) (1 citation) (Correct)
....of S ao Paulo, Brazil. 1 This problem is NP hard (cf. K72] and has been extensively studied in terms of approximation algorithms (see, for instance, IK75] in terms of branch and bound methods (see, for example, MT91] and from a polyhedral point of view (see, for instance, B75] [HJP75], P75] W75] The generalized assignment problem is a generalization of the multiple knapsack problem where every item i may have a particular weight f ik for each knapsack k. The corresponding polyhedron was investigated in [GR90] Our motivation for studying the multiple knapsack problem ....
....minimal cover inequality and the (1,d) configuration inequality that we want to present now. Suppose that S N is a minimal cover with respect to some knapsack k 2 M . The inequality X i2S x ik jSj Gamma 1 6 is called minimal cover inequality corresponding to S and k. In [B75] W75] and [HJP75] it was shown that the minimal cover inequality corresponding to S and k defines a facet of SK(S; f; F k ) and, thus, of MK(S Theta M; f; F ) Another well known class of individual inequalities consists of the (1,d) configuration inequalities. Suppose that N 0 [fzg N is a ....
P. L. Hammer, E. L. Johnson and U. N. Peled, "Facets of Regular 0-1 Polytopes", Mathematical Programming 8, 179 - 206 (1975).
On the 0/1 Knapsack Polytope - Weismantel (1994) (1 citation) (Correct)
....problem; the other is the discovery of beautiful concepts and results associated with minimal covers, 1; k) configurations or the lifting and complementing of variables. Most of the polyhedral studies presented so far involve two basic and general objects: minimal covers (see for instance, B75] [HJP75], W75] and (1; k) configurations (cf. P80] Let N be a subset of items, let b denote the knapsack capacity and suppose, every item i 2 N has a weight W (i) 0. A set S N is a cover if P i2S W (i) b holds. The cover is minimal, if in addition P i2Snfsg W (i) b for all s 2 S. Let ....
.... j 00 Gamma 2r b Gamma j 0 Gamma j 00 = b Gamma r Gamma r = jT 1 j j 0 Gamma r jT 1 j j 0 Gamma . This completes the proof. The inequalities defined in Proposition 2. 2 (i) have been presented in [Le93] and can be viewed as special lifted cover inequalities if N r 1 6= B75] [HJP75], W75] If N r 1 = but N j 6= for some j r 1, these inequalities are lifted (1; k) Gammaconfiguration inequalities ( P80] For the inequalities defined in (2.2) ii) the vector P i2T 1 e i e i 0 is a root. Moreover, the coefficient of item i 0 2 N j 0 is defined as the ....
P. L. Hammer, E. L. Johnson and U. N. Peled, "Facets of Regular 0-1 Polytopes", Mathematical Programming 8, 179 - 206 (1975). 28
Some Integer Programs Arising in the Design of Main .. - Ferreira.. (1993) (Correct)
....x ik jW j Gamma 1 is valid for the polytope MK(N Theta M; f; F ) and defines a facet for MK(W Theta M; f; F ) A subset W with the above properties is called a minimal cover with respect to module k. The corresponding inequality is called minimal cover inequality and was discussed in [B75] [HJP75], W75] By applying the above theorem we can conclude that the minimal cover inequality defines a facet of the polytope C(W Theta M; f; F; E) For other classes of inequalities that are valid or facet defining for MK(N Theta M; f; F ) we refer the reader to [P80] GR90a] GR90b] and ....
P. L. Hammer, E. L. Johnson and U. N. Peled, "Facets of Regular 0-1 Polytopes", Mathematical Programming 8, 179 - 206 (1975).
Some Integer Programs Arising in the Design of Main .. - Ferreira.. (1992) (Correct)
....x ik jSj Gamma 1 is valid for the polytope MK(N Theta M; f; F ) and defines a facet for MK(S Theta M; f; F ) A subset S with the above properties is called a minimal cover with respect to module k. The corresponding inequality is called minimal cover inequality and was discussed in [B75] [HJP75], W75] By applying the above theorem we can conclude that the minimal cover inequality defines a facet of the polytope C(S Theta M; f; F; E) For other classes of inequalities that are valid or facet defining for MK(N Theta M; f; F ) we refer the reader to [P80] GR90a] GR90b] and ....
P. L. Hammer, E. L. Johnson and U. N. Peled, "Facets of Regular 0-1 Polytopes", Mathematical Programming 8, 179 - 206 (1975).
Implementing the Dantzig-Fulkerson-Johnson Algorithm.. - Applegate, Bixby.. (2003) (2 citations) Self-citation (Johnson) (Correct)
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Hammer, P. L., E. L. Johnson, U. N. Peled. 1975. Facets of regular 0-1 polytopes. Mathematical Programming 8, 179-206.
Polyhedral Techniques in Combinatorial Optimization II.. - Aardal, van Hoesel (1995) (1 citation) (Correct)
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P.L. Hammer, E.L Johnson and U.N. Peled (1975) "Facets of regular 0-1 polytopes", Mathematical Programming 8 179--206.
Polyhedral Techniques in Combinatorial Optimization II.. - Aardal, van Hoesel (1995) (1 citation) (Correct)
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P.L. Hammer, E.L Johnson and U.N. Peled (1975) "Facets of regular 0-1 polytopes", Mathematical Programming 8 179--206.
Polyhedral Techniques in Combinatorial Optimization I: Theory - Aardal, van Hoesel (1995) (1 citation) (Correct)
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P.L. Hammer, E.L Johnson and U.N. Peled (1975) "Facets of regular 0-1 polytopes", Mathematical Programming 8 179--206.
A Polyhedral Approach to Single-Machine Scheduling.. - van den Akker, van.. (1997) (1 citation) (Correct)
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P.L. Hammer, E.L. Johnson, and U.N. Peled (1975) Facets of regular 0-1 polytopes. Mathematical programming 8, 179-206.
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