K. Aardal, C. Hurkens, A. K. Lenstra (1998). Solving a system of diophantine equations with lower and upper bounds on the variables. Research report UU-CS-1998-36, Department of Computer Science, Utrecht University, to appear in Mathematics of Operations Research.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....I# C) if the following property holds: A vector (z; y) is an optimal solution to (3.1) if and only if T (z; y) is an optimal solution to (2.1) 3. 2) The idea of linearly reformulating integer programs has recently occurred in the work of Aardal, Hurkens, and Lenstra [1], where lattice basis reduction is employed to reformulate integer programs of a certain class in a way such that the number of branch and bound nodes is reduced. The Integral Basis Method by Haus, Kppe, and Weismantel [7, 8] is a primal integer programming method that constructs a sequence of ....

....feasible region given by 1, z 1 R . 4.13) The feasible region is shown in Figure 3. It is easy to see that the mixed integral basis of the solutions to (4. 13) with respect to the sets R Z = e of reduction vectors, consists of all the vectors (z 1 ; y 2 , y 3 ) [0, 1]. This set is certainly not finite. 4 2 1 2 y 2 3 Figure 3: The projection of the feasible region of Example 4.5 into the (y 2 , y 3 ) plane. The arrows near the origin are representatives of the infinite mixed integral basis that consists of the vectors (z 1 ; y 2 , y 3 ) 0, 1] ....

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K. Aardal, C. A. J. Hurkens, and A. K. Lenstra. Solving a system of diophantine equations with lower and upper bounds on the variables. Mathematics of Operations Research, 25:427--442, 2000.

....Problem Structure with Basis Reduction to solve a Class of Hard Integer Programs Quentin Louveaux CORE, UCL, Belgium Laurence A. Wolsey October 2000 Abstract Recently Aardal et al. [2] have successfully solved some small dif ficult equality constrained integer programs by using basis reduction to reformulate the problems as inequality constrained integer programs in a different space. Here we adapt their method to solve integer programs that are larger, but have special ....

....to the vector (d,a, d) T. We now develop another approach allowing us to tackle the problem. Consider the problem of finding feasible solutions of equality constrained integer programs, or more specifically of finding points in the set z: where A Z 4xv and b Z 4. For this problem, Aardal et al. [2] have recently developed a successful two step approach based on basis reduction [3, 6, 11] In step i they use basis reduction on the associated lattice 0 i of A b dimension N M 1 to construct an alternative representation of the feasible set Z of the form Z x: x q q PA, A Z N M, X ....

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K. Aardal, C. Hurkens, and A. K. Lenstra. Solving a system of diophantine equations with lower and upper bounds on the variables. Technical Report UU-CS-1998-36, Departement of Computer Science, Utrecht University, 1998.

....# #C # ) if the following property holds: A vector (z; y) is an optimal solution to (3.1) if and only if T(z; y) is an optimal solution to (2.1) 3. 2) The idea of linearly reformulating integer programs has recently occurred in the work of Aardal, Hurkens, and Lenstra [1], where lattice basis reduction is employed to reformulate integer programs of a certain class in a way such that the number of branch and bound nodes is reduced. The Integral Basis Method by Haus, K oppe, and Weismantel [6, 7] is a primal integer programming method that constructs a sequence of ....

....feasible region given by R . 4.11) The feasible region is shown in Figure 2. It is easy to see that the mixed integral basis of the solutions to (4. 11) with respect to the sets R Z = e vectors, consists of all the vectors (z 1 ; y 2 , y 3 ) 1; #, for # [0, 1]. This set is certainly not finite. The way to overcome this di#culty is to resort to a mixed integer generating set that is not contained in the positive orthant. Under this relaxed condition, one can resort to the construction for the case of one negative continuous variable given in the proof ....

K. Aardal, C. A. J. Hurkens, and A. K. Lenstra. Solving a system of diophantine equations with lower and upper bounds on the variables. Mathematics of Operations Research, 25:427--442, 2000.

....3 lattice basis reduction. Being able to solve this subproblem is at the core of Lenstra s algorithm for IP in fixed dimension [Len83] it has furthermore been applied to guide a branching on hyperplanes algorithm [CRSS93] Short lattice vectors are also important for problem reformulation, see [AHL00]. We are, however, not aware of an earlier application of lattice basis reduction to find short augmentation vectors. The combination of the augmentation heuristic with the Integral Basis Method yields an augmentation framework, which we successfully apply to solve MIPLIB problems from scratch . ....

K. Aardal, C. A. J. Hurkens, and A. K. Lenstra, Solving a system of diophantine equations with lower and upper bounds on the variables, Mathematics of Operations Research 25 (2000), 427-- 442.

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K. Aardal, C. Hurkens, A. K. Lenstra (1998). Solving a system of diophantine equations with lower and upper bounds on the variables. Research report UU-CS-1998-36, Department of Computer Science, Utrecht University, to appear in Mathematics of Operations Research.

....behavior of branch and bound that was observed by Cornu ejols and Dawande was that 2 nodes needed to be evaluated, where typically takes values between 0.6 and 0.7. The algorithm we use in our study is described brie y in Section 2. The algorithm was developed by Aardal, Hurkens, and Lenstra [1] for solving a system of linear diophantine equations with bounds on the variables, such as problem (2) and is based on Lov asz lattice basis reduction algorithm as described by Lenstra, Lenstra, and Lov asz [6] Aardal et al. motivate their choice of basis reduction as the main ingredient of ....

....= 10(m 1) keeping all other parameters the same) in order to generate infeasible instances with high probability. We present our analysis together with numerical support in Section 4. 2 An Outline of the Algorithm Here we give a summary of the algorithm developed by Aardal, Hurkens, and Lenstra [1] to solve problem (2) They also give a brief overview of the basis reduction algorithm and the use of basis reduction in integer programming. For a detailed description of the basis reduction algorithm we refer to Lenstra, Lenstra, and Lov asz [6] The main idea behind the algorithm is to use an ....

[Article contains additional citation context not shown here]

K. Aardal, C. Hurkens, A. K. Lenstra (1998). Solving a system of diophantine equations with lower and upper bounds on the variables. Research report UU-CS-1998-36, Department of Computer Science, Utrecht University.

....the few implementations that we are aware of is reported on by Cook, Rutherford, Scarf, Shallcross [20] in which some di#cult, not previously solved, network design problems were solved using the generalized basis reduction algorithm of Lovasz and Scarf. Recently Aardal, Hurkens, Lenstra [2] [3] developed an algorithm for solving a system of diophantine equations with bounds on the variables. They used basis reduction to reformulate a certain integer relaxation of the problem, and were able to solve several integer programming instances that proved hard, or even unsolvable, for several ....

....in Subsection 2. 2 since several interesting theoretical and computational results have been obtained in this area using basis reduction, and since the lattices and the bases that have been used in attacking knapsack cryptosystems are related to the lattice used by Aardal, Hurkens, Lenstra [2] [3]. Their algorithm is outlined in Subsection 2.3. The basic idea behind the algorithms discussed in Subsections 2.2 and 2.3 is to reformulate the problem as a problem of finding a short vector in a certain lattice. One therefore needs to construct a lattice in which any feasible vector is provably ....

[Article contains additional citation context not shown here]

K. Aardal, C. Hurkens and A. K. Lenstra (1998), Solving a system of diophantine equations with lower and upper bounds on the variables, Research report UU-CS-1998-36, Department of Computer Science, Utrecht University, to appear in Mathematics of Operations Research.

....the few implementations that we are aware of is reported on by Cook, Rutherford, Scarf, Shallcross [20] in which some dicult, not previously solved, network design problems were solved using the generalized basis reduction algorithm of Lov asz and Scarf. Recently Aardal, Hurkens, Lenstra [2] [3] developed an algorithm for solving a system of diophantine equations with bounds on the variables. They used basis reduction to reformulate a certain integer relaxation of the problem, and were able to solve several integer programming instances that proved hard, or even unsolvable, for several ....

....in Subsection 2. 2 since several interesting theoretical and computational results have been obtained in this area using basis reduction, and since the lattices and the bases that have been used in attacking knapsack cryptosystems are related to the lattice used by Aardal, Hurkens, Lenstra [2] [3]. Their algorithm is outlined in Subsection 2.3. The basic idea behind the algorithms discussed in Subsections 2.2 and 2.3 is to reformulate the problem as a problem of nding a short vector in a certain lattice. One therefore needs to construct a lattice in which any feasible vector is provably ....

[Article contains additional citation context not shown here]

K. Aardal, C. Hurkens and A. K. Lenstra (1998), Solving a system of diophantine equations with lower and upper bounds on the variables, Research report UU-CS-1998-36, Department of Computer Science, Utrecht University, to appear in Mathematics of Operations Research.

....the few implementations that we are aware of is reported on by Cook, Rutherford, Scarf, Shallcross [10] in which some dicult, not previously solved, network design problems were solved using the generalized basis reduction algorithm of Lov asz and Scarf. Recently Aardal, Hurkens, Lenstra [2] [3] developed an algorithm for solving a system of diophantine equations with bounds on the variables. They used basis reduction to reformulate a certain integer relaxation of the problem, and were able to solve several integer programming instances that proved hard, or even unsolvable, for several ....

.... be brie y discussed in Section 3 since several interesting theoretical and computational results have been obtained in this area using basis reduction, and since the lattices and the bases that have been used in attacking knapsack cryptosystems are related to the lattice used by Aardal et al. 2] [3]. Their algorithm is outlined in Section 4. The basic idea behind the algorithms discussed in Sections 3 and 4 is to reformulate the problem as a problem of nding a short vector in a certain lattice. One therefore needs to construct a lattice in which any feasible vector is provably short. It ....

[Article contains additional citation context not shown here]

K. Aardal, C. Hurkens, A. K. Lenstra (1998). Solving a system of diophantine equations with lower and upper bounds on the variables. Research report UU-CS-1998-36, Department of Computer Science, Utrecht University, to appear in Mathematics of Operations Research.

....of branch and bound that was observed by Cornu ejols and Dawande was that 2 n nodes needed to be evaluated, where typically takes values between 0.6 and 0.7. The algorithm we use in our study is described brie y in Section 2. The algorithm was developed by Aardal, Hurkens, and Lenstra [1] for solving a system of linear diophantine equations with bounds on the variables, such as problem (2) and is based on Lov asz lattice basis reduction algorithm as described by Lenstra, Lenstra, and Lov asz [6] Aardal et al. motivate their choice of basis reduction as the main ingredient of ....

....= 10(m 1) keeping all other parameters the same) in order to generate infeasible instances with high probability. We present our analysis together with numerical support in Section 4. 2 An Outline of the Algorithm Here we give a summary of the algorithm developed by Aardal, Hurkens, and Lenstra [1] to solve problem (2) They also give a brief overview of the basis reduction algorithm and the use of basis reduction in integer programming. For a detailed description of the basis reduction algorithm we refer to Lenstra, Lenstra, and Lov asz [6] The main idea behind the algorithm is to use an ....

[Article contains additional citation context not shown here]

K. Aardal, C. Hurkens, A. K. Lenstra (1998). Solving a system of diophantine equations with lower and upper bounds on the variables. Research report UU-CS-1998-36, Department of Computer Science, Utrecht University.

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