A. Caprara and M. Fischetti, \f0; 2 g-Chvatal-Gomory cuts," Mathematical Programming, vol. 74, pp. 221-235, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... [13] Schrijver [15] Chv atal Gomory cuts are valid for the integral polyhedron P I : convfx 2 P Z g and, indeed, many important facet inducing inequalities, for polyhedra associated with many important combinatorial optimization problems, are Chv atal Gomory cuts, see Caprara et al. [1, 2]. For this reason (see, e.g. Gr otschel, Lov asz Schrijver [10] we might want to nd an ecient algorithm to solve the following problem: The Chv atal Gomory Separation Problem (CG SEP) Given 2 P : fx 2 IR : Ax bg, nd a Chv atal Gomory cut which is violated by x , or prove that ....

....can be decomposed into an in nite number of subproblems: for each k, solve The Mod k Separation Problem (Mod k SEP) Given some x : Ax bg and some integer k 2, nd a mod k cut which is violated by x , or prove that none exists. Unfortunately, this does not seem to help much. In [1] it is shown that mod kSEP is strongly NP hard even for k = 2. Some more useful concepts can be found in [2] Given some x 2 P , the vector s : b Ax is called the slack vector. Note that the components of s are non negative. It is not dicult to show that the slack of a ....

A. Caprara & M. Fischetti (1996) f0; g-Chvatal-Gomory cuts. Math. Program. , 74, 221-235.

....icts of this relaxation are based on pairs of incompatible arcs, or, as we like to see it, on con icts of degree constraints. The cycle inequalities of this relaxation are known as the odd closed alternating trail inequalities. The separation problem has been solved by Caprara Fischetti (1996) [4]. We are now going to suggest a relaxation that is based on con icts of cuts. We consider the scheme : R A R V de ned as (x) 1 x( 8 (W ) 6= W ( V: Here, we take V : f (W ) 6= W ( V g as the set of all subsets of cuts (W ) 6= W ( V . The scheme ....

Caprara & Fischetti (1996). f0; 1 2 g-Chvatal-Gomory Cuts. Math. Prog. 74(3), 221-235.

.... ) of the fractional point x is planar. It is well known that comb inequalities can be obtained by adding up and rounding a convenient set of TSP degree equations and subtour elimination constraints weighed by 1=2, i.e. they are f0; 1 2 g cuts in the terminology of Caprara and Fischetti [7]. These authors studied f0; 1 2 g cuts in the context of general ILP s. They showed that the associated separation problem is equivalent to the problem of nding a minimum weight member of a binary clutter, i.e. a minimum weight f0; 1g vector satisfying a certain set of mod 2 congruences. This ....

....cuts which can be obtained through multiplier vectors belonging to f0; 1=k; k 1) kg m for any given integer k 2. We call them mod k cuts, as their validity relies on mod k rounding arguments. Note that mod 2 cuts are in fact the f0; 1 2 g cuts studied in Caprara and Fischetti [7]. Any Chv atal Gomory cut is a mod k cut for some integer k 0, as it is well known that undominated Chv atal Gomory cuts only arise for 2 [0; 1) m , since replacing any i by its fractional part i b i c always leads to an equivalent or stronger cut. Moreover, can always be assumed to ....

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A. Caprara, M. Fischetti (1996). f0; 1 2 g-Chvatal-Gomory cuts. Math. Program. (A) 74, 221-235.

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Caprara, A., and Fischetti, M. (1996). f0; 1=2g-Chvatal-Gomory cuts. Mathematical Programming (A), 74, 221-235.

....capacity cuts for each max ow problem, and use parametric techniques to reduce the computational e ort. 20 4.2. 3 Comb inequalities We rst describe an exact separation procedure for matching inequalities (20) which follows the general framework recently proposed by Caprara and Fischetti [2]. As shown in the proof of Theorem 3.1, matching inequalities admit a Chv atal Gomory derivation obtained by combining fan inequalities (14) and bound constraints x e 1 after multiplication for 0 or 1 2, i.e. they are f0; 1=2g Chv atal Gomory cuts in the terminology used in [2] Now let M x ....

....and Fischetti [2] As shown in the proof of Theorem 3. 1, matching inequalities admit a Chv atal Gomory derivation obtained by combining fan inequalities (14) and bound constraints x e 1 after multiplication for 0 or 1 2, i.e. they are f0; 1=2g Chv atal Gomory cuts in the terminology used in [2] . Now let M x 0 represent the inequality system (14) where each row of M is associated with a fan inequality of the type x e x( v) n feg) 0. According to [2] the separation of f0; 1=2g Chv atal Gomory cuts only needs to consider the parity support M of M , where M is de ned as the 0 1 ....

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A. Caprara, M. Fischetti, \0-1/2 Chvatal-Gomory Cuts", Mathematical Programming, 74 (1996) 211-235.

....we also show NP completeness of separation for several other classes of inequalities, including the MIR inequalities of Nemhauser and Wolsey and some new inequalities which we call balanced split cuts and binary split cuts. We also strengthen NP completeness results of Caprara Fischetti [5] (for f0; 1 2 g cuts) and Eisenbrand [12] for Chv atal Gomory cuts) To compensate for this bleak picture, we also give a positive result for the Symmetric Travelling Salesman Problem. We show how to separate in polynomial time over a class of split cuts which includes all comb inequalities ....

.... [24] the disjunctive cuts (see Balas [2] the split cuts of Cook, Kannan Schrijver [9] the MIR inequalities of Nemhauser Wolsey [25] the matrix cuts of Lov asz Schrijver [23] the lift and project cuts of Balas, Ceria Cornu ejols [4] and the f0; 1 2 g cuts of Caprara Fischetti [5]. Although this array of inequalities is rather bewildering, it is known that many of them are essentially the same. For example, the Gomory fractional cuts are equivalent to Chv atalGomory cuts, and the Gomory mixed integer cuts are equivalent to both split cuts and MIR inequalities. The ....

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A. Caprara & M. Fischetti, \f0; 1 2 g-Chvatal-Gomory cuts", Math. Program., vol. 74, pp. 221-235, 1996.

....otherwise no violated odd hole inequality exists. The minimumweight odd cycle problem can be solved as a shortest path problem on a suitably de ned bipartite graph (see, e.g. Gr otschel, Lov asz and Schrijver (1988) 14] Page 235) For UFLP, it was observed by Caprara and Fischetti (1993) [3] that the separation of odd hole inequalities corresponds to the computation of a minimum weight odd cycle on a graph having one node for each (fractional) variable only, obtained from the original intersection graph by simple preprocessing operations which eliminate x nodes. For GUFLP the same ....

....2 ) time in the worst case. Therefore, the overall complexity is O(n 3 n 2 p mn 2 ) The previous procedure gives (if any) violated odd hole constraints of type (3.6) In order to obtain lifted constraints of type (3. 7) we use the following method (see Caprara and Fischetti (1993) [3] for details) Every edge of an odd hole is either a pair (x ik ; x hl ) with i = h, or a pair (x ik ; y j ) with k 2 P j , and is associated with the constraint (3.2) for client i in the rst case, and with the constraint (3.3) for client i and facility j in the second case. A lifted odd hole ....

A. Caprara, M. Fischetti, \Odd-Cut Sets, Odd Cycles, and 0-1/2 Chvatal-Gomory Cuts", working paper, University of Bologna, November 1993.

.... corresponding CGH inequality is obtained by applying a Chv atalGomory derivation, combining with coecients 1 2 the constraints associated with each edge of the odd hole, and rounding down the left hand side coecients and the righthand side of the resulting inequality, see Caprara and Fischetti [7] for details. As an example, Figure 3.1 (a) depicts a subgraph induced by a subset of the nodes in V I . In particular, black and white circles denote, respectively, y and x nodes with a coecient of 1 in the odd hole inequality associated with node set, say, W V I , whereas the boxes denote ....

...., and is associated with the two members of (2.3) corresponding to i = h; j = s and to i = h; j = t, respectively. Notice that by de nition the edge weights in H are nonnegative, and correspond to the sum of the slack values of a suitably de ned weakened version of the associated inequalities, see [7] for details. Paths and cycles of H are called odd if they contain an odd number of odd edges, even otherwise. The inequality corresponding to a cycle of H is obtained by applying the Chv atal Gomory derivation, with coecients 1 2, to the constraints associated with the edges of . Notice that the ....

[Article contains additional citation context not shown here]

A. Caprara and M. Fischetti, \0-1/2 Chvatal-Gomory Cuts", Mathematical Programming 74 (1996) 221-235. 24

....capacity cuts for each max ow problem, and use parametric techniques to reduce the computational e ort. 16 4.2. 3 Comb inequalities We rst describe an exact separation procedure for matching inequalities (19) which follows the general framework recently proposed by Caprara and Fischetti [2]. As shown in the proof of Theorem 3.1, matching inequalities admit a Chv atal Gomory derivation obtained by combining fan inequalities (13) and bound constraints x e 1 after multiplication by 0 or 1 2, i.e. they are f0; 1=2g Chv atal Gomory cuts in the terminology used in [2] Now let M x 0 ....

....and Fischetti [2] As shown in the proof of Theorem 3. 1, matching inequalities admit a Chv atal Gomory derivation obtained by combining fan inequalities (13) and bound constraints x e 1 after multiplication by 0 or 1 2, i.e. they are f0; 1=2g Chv atal Gomory cuts in the terminology used in [2] . Now let M x 0 represent the inequality system (13) where each row of M is associated with a fan inequality of the type x e x( v) n feg) 0. According to [2] the separation of f0; 1=2g Chv atal Gomory cuts only needs to consider the parity support M of M , where M is de ned as the 0 1 ....

[Article contains additional citation context not shown here]

A. Caprara, M. Fischetti, \0-1/2 Chvatal-Gomory Cuts", Mathematical Programming, 74 (1996) 211-235.

No context found.

A. Caprara and M. Fischetti, \f0; 2 g-Chvatal-Gomory cuts," Mathematical Programming, vol. 74, pp. 221-235, 1996.

No context found.

A. Caprara, M Fischetti, f0; 1=2g-Chvatal-Gomory cuts, Mathematical Programming, 74, (1996), 221-235.

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