H. Crowder, E. Johnson, and M. Padberg. Solving large scale zero-one linear programming problem. Operations Research, 31:803--834, 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....planes for MIP. There are many important polyhedral studies on special cases of the mixed integer knapsack set K. The most studied is probably the 0 1 knapsack set (u = 1 and C = for which seminal works [5, 7, 19, 33] date back to 70 s; see also [16, 28, 32, 37] Crowder, Johnson, and Padberg [13] demonstrate the e#ectiveness of cutting planes from individual 0 1 knapsack constraints in solving 0 1 programming problems. Date: March 2002, December 2002. Alper Atamturk: Department of Industrial Engineering and Operations Research, University of California at Berkeley, Berkeley, CA ....

H. Crowder, E. L. Johnson, and M. W. Padberg. Solving large--scale zero--one linear programming problems. Operations Research, 31:803--834, 1983.

....which we refer to as packs here. Weismantel [35] studies a special case of the knapsack set with two distinct coe#cients. See also Padberg [28] for a survey on covering, packing, and knapsack problems. In an influential paper in computational integer programming, Crowder, Johnson, and Padberg [12] show how to e#ectively incorporate strong valid inequalities for the 0 1 knapsack set in cut generation procedures for solving 0 1 programming problems. Using strong valid inequalities for 0 1 knapsacks, they were able to solve much bigger problems than that were possible until then. Even ....

H. Crowder, E. L. Johnson, and M. W. Padberg. Solving large--scale zero--one linear programming problems. Operations Research, 31:803--834, 1983.

....[BCC93] An alternative approach to obtain cutting planes for an integer program follows essentially the scheme to derive relaxations associated with certain substructures of the underlying constraint matrix, and tries to find valid inequalites for these relaxations. Crowder, Johnson and Padberg [CJP83] applied this methodology by interpreting each single row of the constraint matrix as a knapsack relaxation and strengthened the integer program by adding violated knapsack inequalities. An analysis of other important relaxations of an integer program allows to incorporate odd hole and clique ....

H. Crowder, E. L. Johnson and M. W. Padberg, Solving Large-Scale Zero-One Linear Programming Problems, Operations Research Vol 31, No. 5, 803 -- 834 (1983).

....same side of the line kl (see constraints (13) 16) Thus, ij and kl do not cross. 3 Comparative Sizing for Graph Families Research in integer and nonlinear programming provides some hope that some useful instances of QCF can be solved. One breakthrough was made by Crowder, Johnson and Padberg [11] whose paper won the Lanchester Prize for solving binary integer programming (BIP) problems with up to 2,756 variables and no special structure. Roy and Wolsey [33] succeeded in solving mixed BIPs with nearly 1,000 binary variables and an even larger number of real variables. Their paper won the ....

H. Crowder, E. I. Johnson and M. Padberg, Solving large-scale zero-one linear programming problems, Operations Research, 31 (1983) 803-834.

....(i.e. an objective function value within 10 of the exact solution) A detailed description of two specific examples is included at the end of our report. In the course of the last few decades, many efficient 0 1 mixed linear algorithms have been developed and implemented to solve larger problems [10, 7, 18]. More recently, interior point approaches have been successfully used to solve very large 0 1 mixed integer problems optimally or approximately. The success of these approaches is due to very efficient implementations of interior point methods for solving very large scale linear programs [13] We ....

E.L. Johnson H. Crowder and M.W. Padberg. Solving large-scale zero-one linear programming problems. Operations Research, 31:803 834, 1982.

....or presolving techniques aimed at improving a given integer programming formulation to facilitate the solution process. A detailed description of the main techniques proposed in literature for the general case of integer problems can be found in the survey papers of Crowder, Johnson and Padberg [10], in the textbook Nemhauser and Wolsey [17] and the references therein. Special problems offer additional potential for designing specific techniques that can take further advantage of the model s structure. 4.1 The combined relaxation The choice of the relaxation constitutes a crucial issue ....

H. P. Crowder, E. L. Johnson, M. W. Padberg, Solving large-scale zero-one linear programming problems, Operations Research 31 (1983) 803-834.

....value y 23 for the sensitive cell can be computed in a perfectly analogous way by solving the linear program of maximizing y 23 subject to the same constraints as before. In the example, y 23 = 5 and y 23 = 30, i.e. the sensitive information is protected within the protection interval [5,30]. If this interval is considered suciently wide by the statistical oce, the sensitive cell is called protected; otherwise new suppressions are needed. Notice that the extreme values of interval [5; 30] are only attained if the cell corresponding to Activity II and Region A takes the quite ....

Crowder, H. P., Johnson, E. L., and Padberg, M. W. (1983), \Solving Large-Scale Zero-One Linear Programming Problems," Operations Research, 31, 803-834.

....have been derived without using the fact that x variables must be 0 1 valued, a property that can be exploited to derive strong families of additional constraints which are useful to tighten the LP relaxation of our model. In particular, following the seminal work of Crowder, Johnson and Padberg [5] we can observe that each single inequality in (7) 8) can be viewed as a knapsack constraint, hence it implies a number of more combinatorial restrictions. To be speci c, let P (i;j)2A ij x ij 0 12 represent any inequality in (7) and (8) in the strengthened form discussed at the end ....

H.P. Crowder, E.L. Johnson, and M.W. Padberg, \Solving Large-Scale Zero-One Linear Programming Problems", Operations Research, 31 (1983) 803-834.

....heuristic, we rst attempt to identify a set S yielding a violation of the weaker cover inequalities (13) If this is successful then constraint (14) associated to S is also violated. Otherwise, we still check whether a violation of (14) has been identi ed. As in Crowder, Johnson and Padberg [6], constraints (13) can easily be separated by solving the 0 1 Knapsack Problem (KP) max f X v i 2M k y i u i : X v i 2V k q ik u i d k ; u i 2 f0; 1g for all v i 2 M k g; where is a small positive value (if all q ik and d k are integer numbers, then : 1) Indeed, S is ....

H. Crowder, E.L. Johnson, M.W. Padberg, \Solving large-scale zero-one linear programming problems", Operations Research 31 (1983) 803-834.

....have been derived without using the fact that x variables must be 0 1 valued, a property that can be exploited to derive strong families of additional constraints which are useful to tighten the LP relaxation of our model. In particular, following the seminal work of Crowder, Johnson and Padberg [3] we can observe that each single capacity inequality implies a number of more combinatorial restrictions. To be speci c, let P i s i x i s 0 represent any inequality in (5) in the strengthened form discussed at the end of Section 2. By means of the variable transformation x i = 1 i we ....

H.P. Crowder, E.L. Johnson, and M.W. Padberg, \Solving Large-Scale Zero-One Linear Programming Problems", Operations Research, 31 (1983) 803-834.

....23 and y 23 for the sensitive cell, respectively, can be computed by solving two linear programs in which the values y ij for the missing cells (i; j) are treated as unknowns. In the example, y 23 = 5 and y 23 = 30, i.e. the sensitive information is protected within the protection interval [5,30]. If this interval is considered suciently wide by the statistical oce, the sensitive cell is called protected; otherwise new suppressions are needed. The Cell Suppression Problem (CSP) consists in nding a set of cells whose suppression guarantees the protection of all the sensitive cells ....

Crowder, H. P., Johnson, E. L., and Padberg, M. W. (1983), \Solving Large-Scale Zero-One Linear Programming Problems," Operations Research, 31, 803-834.

....have been derived without using the fact that x variables must be 0 1 valued, a property that can be exploited to derive strong families of additional constraints which are useful to tighten the LP relaxation of our model. In particular, following the seminal work of Crowder, Johnson and Padberg [3] we can observe that each single capacity inequality can be viewed as a knapsack constraint, hence it implies a number of more combinatorial restrictions. To be speci c, let P i s i x i s 0 represent any inequality in (5) in the strengthened form discussed at the end of Section 2. By means ....

H.P. Crowder, E.L. Johnson, and M.W. Padberg, \Solving Large-Scale Zero-One Linear Programming Problems", Operations Research, 31 (1983) 803-834.

....differences among stations. Recently, cutting plane methods have performed successfully in a variety of applications. Even though valid inequalities used as cutting planes are problem specific, the polyhedral characteristics of embedded structures can be applied in a more complex application [4]. This relationship motivated this research, which describes some characteristics of the polyhedral structure of the ALBP and applies that knowledge to resolve the WSP. Specifically, branch and cut has resulted in successful applications including those by Padberg and Rinaldi [17] for symmetric ....

....for Minimal Covers. The standard separation problem is a 0 1 knapsack problem with GUB constraints [16] and we solve it as described in Section 3. 2 using a pseudo polynomial time dynamic programming algorithm for small problems and switching to Dantzig s method to reduce runtime in large problems [4]. 4. PREPROCESSING METHODS Our preprocessing methods simplify the precedence graph by decomposing it into smaller subgraphs and or removing nodes and arcs, resulting in a reduced graph. They also compute bounds on parameters E i , L i , SL , SU , ij , a j (i, s) and j (i, s) as defined in ....

H. Crowder, E. L. Johnson and M. Padberg, 1983. Solving Large-Scale Zero-One Linear Programming Problems. Opns. Res. 31, 803-834.

....have been derived without using the fact that x variables must be 0 1 valued, a property that can be exploited to derive strong families of additional constraints which are useful to tighten the LP relaxation of our model. In particular, following the seminal work of Crowder, Johnson and Padberg [5] we can observe that each single inequality in (7) 8) can be viewed as a knapsack constraint, hence it implies a number of more combinatorial restrictions. To be speci c, let P (i;j)2A ij x ij 0 represent any inequality in (7) and (8) in the strengthened form discussed at the end of ....

H.P. Crowder, E.L. Johnson, and M.W. Padberg, \Solving Large-Scale Zero-One Linear Programming Problems", Operations Research, 31 (1983) 803-834. 25

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Crowder, H., E. L. Johnson, M. Padberg. 1983. Solving large-scale zero-one linear programming problems. Operations Research 31,803-834.

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H. Crowder, E. Johnson, and M. Padberg. Solving large scale zero-one linear programming problem. Operations Research, 31:803--834, 1983.

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H. Crowder, E. L. Johnson, and M. W. Padberg. Solving large-scale zero-one linear programming problems. Operations Research, 31:803--834, 1983.

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H. Crowder, E.L. Johnson, and M.W. Padberg. Solving large-scale zero-one linear programming problems. Operations Research, 31:803--834, 1983.

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H. Crowder, E.L. Johnson, and M.W. Padberg. Solving large-scale zero-one linear programming problems. Operations Research, 31:803--834, 1983.

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H. Crowder, E. Johnson, and M. W. Padberg, Solving large-scale zero-one linear programming problems, Operations Research 31 (1983), 803--834.

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H. Crowder, E. L. Johnson, and M. W. Padberg. Solving large-scale zero-one linear programming problems. Operations Research, 31:803--834, 1983.

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H. P. Crowder, E. L. Johnson and M. W. Padberg (1983), Solving Large-Scale Zero-One Linear Programming Problems, Operations Research, 31, pp. 803--834.

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H. Crowder, E.L. Johnson & M.W. Padberg (1983) Solving large-scale zeroone linear programming problems. Oper. Res. 31, 803--834. 21

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Crowder, H., Johnson, E.L., and Padberg, M.: Solving Large-Scale Zero-One Linear Programming Problems. Operations Research 31 803, 1983

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H.P. Crowder, E.L. Johnson, and M.W. Padberg, \Solving Large-Scale Zero-One Linear Programming Problems", Operations Research, 31 (1983) 803-834.

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