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

....have been derived [1] 11] 20] 21] For such sets, the concepts of projection, lifting and cover have played an important role. Using separation heuristics, the resulting inequalities have been successfully used as cutting planes in various branch and bound and branch and cut systems [6] for problems containing constraints involving only binary variables. In mixed 0 1 programming, so called single node flow sets, of the form Z = x, y) # R n B n : X j#N x j X j#N x j # b, l j y j # x j # u j y j j # N #N , where n = N #N , ....

H. Crowder, E.L. Johnson and M.W. Padberg, Solving Large Scale Zero-One Linear Programming Problems, Operations Research 31, 803834 (1983).

....or possibly, on relaxations of the problem. This approach was first used to solve pure 0 1 programs by considering the constraints of the formulation separately and using facet inducing valid inequalities for the knapsack problems associated with each constraint (see Crowder, Johnson and Padberg [10]) One important example of this approach is the mixed integer rounding (MIR) inequality (see Nemhauser and Wolsey [15] and (3) below) which is derived from a single mixed integer constraint. The MIR inequality can also be conseidered as a generalization of Chvatal s integer rounding inequality ....

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

....may sometimes improve the quality of relaxations considerably. MSC 2000: 90C22; 90C25, 90C27, 90C09, 90C20, 90C06 Keywords: bisection, equicut, max cut, quadratic 0 1 programming, semide nite programming, spectral bundle method, subgradient method 1 Introduction Crowder, Johnson, and Padberg [7] initiated the rise of general mixed integer programming by solving several large scale, unstructured 0 1 linear programming problems via a uni ed cutting plane framework. Can we set up a similar framework for large scale quadratic 0 1 programming problems It seems likely, that this question ....

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

....Poles of Attraction initiated by the Belgian State, Prime Minister s Oce, Science Policy Programming. The scienti c responsibility is assumed by the authors. This research was also supported by NSF Grant No. DMI 9700285 and by Philips Electronics North America. work in this area includes Crowder, Johnson, and Padberg [1983], who showed that valid inequalities for knapsack sets de ned by single rows can be used to solve IP s, and Padberg, Van Roy, and Wolsey [1985] who showed that valid inequalities for single node xed charge (SNFC) ow models can be used to solve structured MIP s. In this paper we introduce a ....

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

.... the convex hull of T then find a vector a and a scalar b such that T ae fx : a T x bg and a T OE(x ) b; add the cut a T OE(x) b to L; end end return L; The trick of trying to separate x from S by separating OE(x ) from T was used previously by Crowder, Johnson, and Padberg [17] in the context of integer linear programming, where S consists of all integer solutions of some explicitly recorded system Ax = b; x u (31) and x satisfies (31) in place of x. Crowder, Johnson, and Padberg consider systems (31) such that A is sparse and = 0; u = e; for each equation ff ....

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

....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 ki 0 for all i = 1; n; k = 1; m, a vector b 2 I Q m and define IP : convfx 2 f0; 1g n j Ax bg. With each constraint k the authors associate the single knapsack ....

...., a ki 0 for all i = 1; n; k = 1; m, a vector b 2 I Q m and define IP : convfx 2 f0; 1g n j Ax bg. With each constraint k the authors associate the single knapsack polytope SK k : SK(f1; ng; a k1 ; a kn ) T ; b k ) Clearly, IP T m k=1 SK k . In [CJP83] large scale 0 1 linear programs are solved by using single knapsack inequalities for the polytopes SK k in order to chop off fractional solutions that are obtained during the run of a cutting plane algorithm. Gottlieb and Rao ( GR90] GR90a] study the generalized assignment problem, a ....

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

....e x t e b; 1 x 1 e x 2 e Delta Delta Delta x Te e 0 for all e 2 F , x integral g: 7) 10 The polytope ICOV(g; b) can be viewed as a knapsack polytope with additional ordering constraints. Facial properties of knapsack polytopes have been studied in recent years, see e.g. [5, 3]. Knapsack polytopes with the additional constraints P t x t e 1 (for all e 2 F ) and x 0, have been studied in [14, 20] under the names multiple choice knapsack problem or knapsack problem with generalized upper bounds (which is actually a larger class of problems) The polytope ....

H. Crowder, E. L. Johnson, and M. Padberg. Solving large-scale zeroone linear programming problems. Operations Research, 31(5):803--834, 1983.

....; dK ) 4 For all v 2 T check whether ct 0 and x v is feasible for (IP ) In this case set x : x t. Note that the knapsack relaxation that we derive in Step 1 of the scheme above can simultaneously be used to derive valid inequalities for the integer programming problem (IP ) see [CJP83]. 4.2 Example: Minimal cover vectors There is a one subclass of irreducible elements in every test set of (IP ) that we want to introduce now. With each such element as we will see we can associate an inequality that is satisfied by all optimal points of the given instance. We call these ....

H. Crowder, E. L. Johnson and M. W. Padberg, Solving large-scale zero-one linear programming problems, Operations Research 31, No. 5, 1983, 803 - 834.

....in science, technology, business, and environment, and their number is tremendous and increasing. Such developments go side by side with the development of tools for tackling mixed integer programs. One important breakthrough that still drives the current activities in this field was the code of Crowder, Johnson, and Padberg [1983] that was able to solve real world 0=1 integer programs that many researchers considered untracktable by this time. Their main idea was to interpret each single row of the constraint matrix as a knapsack constraint and to strengthen the original integer program by adding inequalities associated ....

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

....a network flow problem with a pseudopolynomial number of vertices and edges. By applying network flow techniques the problem can be solved with a pseudopolynomial time and space complexity. In our code we have integrated a heuristic to separate minimal cover inequalities that was introduced in [CJP83]. This procedure can be described as follows. Separation heuristic for minimal cover inequalities Input: An instance of the single knapsack problem (N; f; C) and a vector x 0 2 [0; 1] N . Output: A violated minimal cover inequality or the procedure fails. Solve the following linear ....

....knapsack problem is given and that x 0 2 [0; 1] N ThetaM is the fractional point to be cut off. Separation of (1; d) Configuration Inequalities In order to find violated (1; d) configuration inequalities we have been implementing and evaluating two heuristics. One of these was introduced in [CJP83] and can be described as follows. Heuristic 1 Choose a minimal cover S (with respect to some knapsack k) from the pool. Let z 2 S be an item with maximum weight. Set N 0 : S n fzg and d : jN 0 j. For every item i 2 N n S such that P j2N 0 [fig f j F k , if P j2N 0 [fig f j C ....

H. Crowder, E. L. Johnson and M. W. Padberg, "Solving Large-Scale Zero-One Linear Programming Problems", Operations Research 31, 803 - 844 (1983).

....that (simultaneous) lifting (see [P75] Z74] and complementing (see [W75] of minimal cover inequalities yields all the facets of the 0 1 knapsack polytope (cf. BZ78] BZ84] a theoretical machinery became available to attack knapsack problems from a polyhedral point of view. In fact, since [CJP83] several papers have been written that are based on this polyhedral theory for the 0 1 knapsack problem and are meant to turn the theory into an algorithmic tool for the solution of practical problems (see for instance [RW87] FMW93] Moreover, the last decade has brought a wide range of ....

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).

....k) configuration inequalities, but are derived by means of a weight reduction principle (see [10] One reason why many researchers are interested in new polyhedral results for 2 knapsack problems is that such results often apply to more general cases. In fact, Crowder, Johnson and Padberg [3] have first shown that general 0 1 integer programs can be solved quite efficiently via branch and cut algorithms. The cutting plane phase of their code is essentially based on valid inequalities for the 0 1 knapsack polytopes associated with the rows of the given problem. Other applications ....

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).

....is called fixed. If l i j u i j , we say that x j is free in iteration i (node v i ) Variable setting based on reduced cost belongs to the folklore and is used by many authors to improve the formulation of zero one integer programming problems (for example by Crowder, Johnson, and Padberg [9] and Padberg and Rinaldi [33] Note that the new bounds are a function of #, z LP , and z # . As a consequence, each time that we find a new best integer feasible solution in the branch and bound algorithm we can recompute the bounds. Suppose we find an improved primal solution in iteration k. ....

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

....that we have to pack arcs a # A that take positive time to complete (namely, t a ) into a working day of length T . Hence, solutions satisfying (3e) can be interpreted 18 knapsack sets (see Padberg [29] Balas [3] Wolsey [38] Hammer, Johnson, and Peled [20] Crowder, Johnson and Padberg [12], Weismantel [37] and Wolsey [40] Hence, lifted cover inequalities derived from (3e) are valid for the MSP. We use a separation algorithm as described by Van Roy and Wolsey [35] and Gu, Nemhauser and Savelsbergh [19] to identify violated lifted cover inequalities. 4.4.5 Combined Separation ....

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

....i 0; i = 1; P , which ensures that in every optimal solution, y i = 1 if and only if x i 0. During the last two decades researchers have often used strong valid inequalities for structured MIP relaxations to solve more complicated models. Among those who pioneered such methods were Crowder, Johnson, and Padberg [1983], who demonstrated that the use of inequalities for knapsack relaxations de ned by single constraints of 0 1 IP problems can be e ective in solving IP s. Today the use of valid inequalities for the knapsack polytope is a common feature in IP solvers. Padberg, Van Roy, and Wolsey [1985] derived ow ....

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

....by Johnson and Padberg [20] Bourjolly [3] Sewell [29] Ikebe and Tamura [19] Tamura [30, 31] and others. Since the inequalities of (GSSP) represent binary relations of pairs of variables, results on (GSSP) are used in preprocessing for integer programming problems including 0 1 variables [2, 5, 16, 17, 28]. SSP) is known to be NP hard in general, even if w = e, the vector of all ones. On the other hand, it is also known that for some classes of graphs, including the perfect graph, SSP) can be solved in polynomial time. Grotschel, Lov asz and Schrijver [11] showed that the number #(G; w) which is ....

H. Crowder, E. L. Johnson and M. Padberg, Solving large-scale zero-one linear programming problems, Oper. Res., 31 (1983), pp. 803--834.

....programming [12] In Canon and Hoffman [9] a complex branch and cut algorithm was run on a network of 9 DECstations, joined to form a Local Area VAXcluster. Data, such as the global queue of active nodes, were shared through disk files. The test set was a subset of those used in Crowder et al. [10]. In Applegate, Bixby, Chv atal and Cook [1] the computations were very coarse grained, with individual tasks often running for a large fraction of a day on the hardest instances. The parallelism, which employed a rather complex list of tasks, was implemented using the master slave paradigm. ....

....x j 2 f0; 1g (j = 1; p) M) where A 2 R m Thetan , b 2 R m ; c 2 R n , and p n. 2 Basic Features of the Algorithm 2. 1 Preprocessing Problem preprocessing has been shown to be a very effective way of improving integer programming formulations prior to and during branch and bound [7, 10, 15, 19]. Rather than writing our own preprocessor, we have simply employed the CPLEX 3.0 3 preprocessor, invoking it not only once, but repeatedly until no further reductions result. In addition to applying standard linear programming (LP) reductions, also valid for integer programs, CPLEX applies ....

[Article contains additional citation context not shown here]

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

....on the knapsack coefficients for conv(P ) to be described by these inequalities. It can easily be shown that (1; k) configuration inequalities are in fact LCIs, although not simple LCIs. LCIs have been used successfully in branch and cut algorithms for the solution of 0 Gamma 1 integer programs (Crowder, Johnson, and Padberg [1983], Hoffman and Padberg [1991] Gu, Nemhauser, and Savelsbergh [1994] However, there are still many interesting open questions associated with LCIs. It is well known that in general LCIs do not completely describe conv(P ) Weismantel [1994] studies the 0 1 knapsack polytopes P i for i 2 f1; ....

H. Crowder, E.L. Johnson and M.W. Padberg (1983). Solving Large Scale ZeroOne Linear Programming Problems. Oper. Res. 31, 803-834.

....other combinatorial optimization problems. These subproblems can yield bounds, as a result of Lagrangian relaxation (see Nemhauser and Wolsey [13, xII.3. 6] or improved formulations, as a result of constraint strengthening (see Dietrich and Escudero [6] Hoffman and Padberg [9] Crowder et al. [5] proposed a method for solving sparse 0 1 integer programming problems maximize P n j=1 p j x j subject to P n j=1 w ij x j c i (i = 1; 2; m) 2) x j 2 f0; 1g (j = 1; 2; n) where the objective function coefficients p j , constraint coefficients w ij , and right hand sides ....

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

....(Cordier et al. 1999] which is a general purpose branch and cut MIP solver. bc opt uses several classes of inequalities in its branch and cut algorithm, including, among others, ow cover inequalities (see Padberg, et al. 1985] and Gu, et al. 1999] lifted knapsack cover inequalities (see Crowder, et al. 1983], Weismantel [1997] and Gu, et al. 1998] and path inequalities (Van Roy and Wolsey [1987] recall that there are a generalization of the (l; S) inequalities) bc prod includes all of the cutting plane features of bc opt. In addition, it has a number of cutting plane routines designed speci ....

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

....L 1 Delta Delta Delta c nLn d (5) of (2) with d c 1 Delta Delta Delta c n 1. Obviously, an assignment ff satisfies (2) if and only if ff satisfies (3) if and only if ff satisfies (5) The last step of the normalization process as described by (4) is also called coefficient reduction [CJP83]. Note that a linear pseudo Boolean inequality in normal form cL d is satisfiable if and only if P c d, because c i 0 for all 1 i n. Hence, cL d is unsatisfiable if and only if P c d, and we can easily decide whether cL d is . From now on we assume that all linear pseudo Boolean ....

H. Crowder, E. L. Johnson, and M. Padberg. Solving large-scale zero-one linear programming problems. Operations Research, 31(5):803--834, September 1983.

....to the relaxation can result in an improved lower bound for the relaxation, which in turn may mean that the linear subproblem can be fathomed without having to resort to branching. There are many di#erent classes of cutting planes. FATCOP 2. 0 includes two classes knapsack cover inequalities [3] and flow cover inequalities [15] Knapsack covers and flow covers inequalities are derived from structures that are present in many, but not all, MIP instances. This implies that for some instances, FATCOP 2.0 will be able to generate useful cutting planes, and for other instances it will not. ....

....was enforced for the task. Unless explicitly stated otherwise, all the advanced features of FATCOP 2.0 described in Section 2 were employed. 4. 1 Assessing node preprocessing It is well known that lifted knapsack covers, flow covers and diving heuristics are e#ective in solving MIP problems [3, 15, 12]. However, the reported overall benefits of node preprocessing are less clear due to the amount of computing time they may take. A key issue is that node preprocessing is too expensive to carry out at every node. Since our tasks now correspond to subtrees of the brand and bound tree, it makes ....

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

....the relaxed problem is achieved by p4 = 0:503067, p5 = 0:496933, p6 = 0:496933, with an objective function value of DEM 54,6891, and an integrality gap of 14.2 . Again, all passengers are transported. 5. 3 Min Cover Inequalities Lifted cover inequalities were introduced by Crowder et al. [9] and applied to pure binary programs. Padberg et al. 22] and van Roy and Wolsey [27] extended these inequalities to generalized ow cover inequalities for mixed binary programs. Our cutting planes are related to the cuts described in these references. We will compare the results of our ....

H. Crowder, E.L. Johnson, and M.W. Padberg. Solving Large Scale Zero-One Linear Programming Problems. Operations Research, 31:803-834, 1983. 20

.... and bound tree in an efficient manner [5] 16] 28] 7] 14] 15] With a few exceptions, notably [30] and [19] research in past decades has strayed from developing and examining effective search techniques and instead focused on improving the bound obtained by the linear programming relaxation [9] [33] 34] The goal of this paper is to reexamine search techniques in light of the many advances made in the field over the years. One major change over the past decades is in the hardware on which MIPs are solved. Computers of today are orders of magnitudes faster than their counterparts of ....

....have several nonbasic variables with zero reduced costs, when these methods are largely useless. Over the years since Beale made this statement, much work has been done in the area of generating strong valid inequalities for MIP and incorporating these inequalities into a branch and cut scheme [9] [17] 31] 33] With the advent of these sophisticated cutting planes, one may suspect to have fewer nonbasic variables with zero reduced cost. The rationale for this statement is as follows. Non basic variables with zero reduced cost correspond to alternative optimal solutions to the linear ....

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

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