L. Lovasz, H. E. Scarf (1992). The generalized basis reduction algorithm. Mathematics of Operations Research 17, 751-764.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the polytope P = f 2 Z : l x d X 0 u x d g is empty. The basis reduction algorithm runs in polynomial time. If one wants an overall algorithm that runs in polynomial time for a xed number of variables, one needs to apply the algorithms of H.W. Lenstra, Jr. 7] or of Lov asz and Scarf [9]. Otherwise, one can, for instance, apply integral branching on the unit vectors in space or linear programming based branch and bound. By integral branching in the space we mean the following. Assume we are at node k of the search tree. Take any unit vector e j , i.e. the jth vector of the ....

....choose the unit vectors in our branching scheme. One alternative is to just take them in any predetermined order, and another is to determine, at node k, which unit vector yields the smallest value of bu k c dl k e. This branching scheme is similar to the scheme proposed by Lov asz and Scarf [9] except that we in general are not sure whether the number of branches created at each level is bounded by some constant depending only on the dimension. What we hope for, and what seems to be the case given our computational results, is that bu k c dl k e is small at most nodes of the tree. A ....

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L. Lovasz, H. E. Scarf (1992). The generalized basis reduction algorithm. Mathematics of Operations Research 17, 751-764.

....x lies in : Ax = 0g; the set of integer points in a subspace. Every such set can be shown to form an integer lattice, namely it can be rewritten in the form L = fx : x = B ; 2 Z In Section 1 we introduce lattices and the basis reduction algorithms of Lov asz [62] and of Lov asz Scarf [68] . Every n dimensional lattice can be generated by n linearly independent vectors, called the lattice basis. A reduced basis is a basis in which the vectors are short and nearly orthogonal. The basis reduction algorithm of Lov asz runs in polynomial time and produces basis vectors of short ....

....programming problem can be solved in polynomial time for a xed number of variables. The proof was algorithmic and consisted of two main steps: a linear transformation, and Lov asz basis reduction algorithm [62] Later, Gr otschel, Lov asz, Schrijver [44] Kannan [56] and Lov asz Scarf [68] developed algorithms using similar principles to Lenstra s algorithm. In computational integer programming, however, basis reduction has received less attention. One of the few implementations that we are aware of is reported on by Cook, Rutherford, Scarf, Shallcross [20] in which some dicult, ....

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L. Lovasz and H. E. Scarf (1992), The generalized basis reduction algorithm, Mathematics of Operations Research 17, 751-764.

....if the number of variables, here n, is xed. Lenstra (1983) showed that integer programming in xed dimension is solvable in polynomial time. Lenstra s algorithm relies on results from the geometry of numbers like Khintchine s atness theorem, lattice basis reduction, and the ellipsoid method. Lov asz Scarf (1992) found a way to avoid the ellipsoid method. However, present algorithms for integer programming in xed dimension are still far from being elementary. The cutting plane method pioneered by Gomory (1958) computes iteratively tighter approximations of the integer hull P I of a polyhedron P , until P ....

....P I can be polynomially bounded by size(P ) in xed dimension. This follows from a generalization of a result by Hayes Larman (1983) see (Schrijver 1986, p. 256) The following upper bound on the number of vertices of P I was proved by Cook, Hartmann, Kannan McDiarmid (1992) B ar any, Howe Lov asz (1992) show that this bound is tight. Theorem 2. If P R n is a rational polyhedron which is the solution set of a system of at most m linear inequalities whose size is at most , then the number of vertices of P I is at most 2m d (6n 2 ) d 1 , where d = dim(P I ) is the dimension of the ....

Lovasz, L. & Scarf, H. E. (1992), `The generalized basis reduction algorithm', Mathematics of Operations Research 17(3), 751 - 764.

....fx 2 Z n : Ax = 0g; the set of integer points in a subspace. Every such set can be shown to form an integer lattice, namely it can be rewritten in the form L = fx : x = B ; 2 Z p g: In Section 1 we introduce lattices and the basis reduction algorithms of Lov asz [62] and of Lov asz Scarf [68] . Every n dimensional lattice can be generated by n linearly independent vectors, called the lattice basis. A reduced basis is a basis in which the vectors are short and nearly orthogonal. The basis reduction algorithm of Lov asz runs in polynomial time and produces basis vectors of short ....

....programming problem can be solved in polynomial time for a xed number of variables. The proof was algorithmic and consisted of two main steps: a linear transformation, and Lov asz basis reduction algorithm [62] Later, Gr otschel, Lov asz, Schrijver [44] Kannan [56] and Lov asz Scarf [68] developed algorithms using similar principles to Lenstra s algorithm. In computational integer programming, however, basis reduction has received less attention. One of the few implementations that we are aware of is reported on by Cook, Rutherford, Scarf, Shallcross [20] in which some dicult, ....

[Article contains additional citation context not shown here]

L. Lovasz and H. E. Scarf (1992), The generalized basis reduction algorithm, Mathematics of Operations Research 17, 751-764.

....programming problem can be solved in polynomial time for a xed number of variables. The proof was algorithmic and consisted of two main steps: a linear transformation, and Lov asz basis reduction algorithm [31] Later, Gr otschel, Lov asz, Schrijver [17] Kannan [25] and Lov asz Scarf [35] developed algorithms using similar principles to Lenstra s algorithm. In computational integer programming, however, basis reduction has received less attention. One of the few implementations that we are aware of is reported on by Cook, Rutherford, Scarf, Shallcross [10] in which some dicult, ....

....vectors. Two algorithms for nding a reduced basis are described in Section 1. First, the reduction algorithm by Lov asz, as presented by Lenstra, Lenstra, Lov asz [31] is described in Section 1.1. Lov asz algorithm works with Euclidean norms, whereas the algorithm by Lov asz Scarf [35], presented in Section 1.2, works with a norm related to a given convex set. In Section 1 we also discuss some recent implementations. The main ideas behind the integer programming algorithms by Lenstra [32] Gr otschel, Lov asz, Schrijver [17] Kannan [25] and Lov asz Scarf [35] described ....

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L. Lovasz, H. E. Scarf (1992). The generalized basis reduction algorithm. Mathematics of Operations Research 17, 751-764.

....introduced by Balas (1979) and used this relation to develop a class of finitely converging cutting plane algorithms, called lift and project algorithms, for mixed 0 1 linear programming problems. Cook et al. 1993) presented an implementation of the generalized basis reduction algorithm by Lov asz and Scarf (1992) for solving general integer programming problems. Basis reduction was first introduced to integer programming by H.W. Lenstra, Jr. 1983) who showed that the problem: does there exist a vector x 2 ZZ n such that Ax b can be solved in polynomial time for fixed n. The proof was algorithmic. ....

L. Lov' asz and H.E. Scarf (1992) "The generalized basis reduction algorithm", Mathematics of Operations Research 17 751--764.

....polytope P = f 2 Z (n m) l x d X 0 u x d g is empty. The basis reduction algorithm runs in polynomial time. If one wants an overall algorithm that runs in polynomial time for a xed number of variables, one needs to apply the algorithms of H.W. Lenstra, Jr. 7] or of Lov asz and Scarf [9]. Otherwise, one can, for instance, apply integral branching on the unit vectors in space or linear programming based branch and bound. By integral branching in the space we mean the following. Assume we are at node k of the search tree. Take any unit vector e j , i.e. the jth vector of the ....

....choose the unit vectors in our branching scheme. One alternative is to just take them in any predetermined order, and another is to determine, at node k, which unit vector yields the smallest value of bu k c dl k e. This branching scheme is similar to the scheme proposed by Lov asz and Scarf [9] except that we in general are not sure whether the number of branches created at each level is bounded by some constant depending only on the dimension. What we hope for, and what seems to be the case given our computational results, is that bu k c dl k e is small at most nodes of the tree. A ....

[Article contains additional citation context not shown here]

L. Lovasz, H. E. Scarf (1992). The generalized basis reduction algorithm. Mathematics of Operations Research 17, 751-764.

....programming has been focusing on problems with zero one variables. In contrast, there are relatively few numerical results on dealing with general integer programming problems. Nevertheless, Cook et al. [1] have recently implemented the generalized basis reduction algorithm of Lov asz and Scarf [4] and reported the solution of a number of difficult problems with up to 100 integer variables. In this paper we consider the following problem: Given an arbitrary simplex P , for example, the convex hull of m 1 (0 = m = n) affinely independent vectors of R n , determine whether P ....

L.Lov'asz and H.Scarf,The generalized basis reduction algorithm. Mathematics of Operations Research 17(1992) 751764

....of our instances we do not use a transformation in our algorithm. We can simply write down an initial basis for a lattice that contains all vectors of interest for our problem, and then apply the L 3 algorithm directly to this basis. For the integer programming problem P , Lov asz and Scarf [12] developed an algorithm that, as Lenstra s algorithm, uses branching on hyperplanes. Instead of using a transformation to transform the convex body and the initial basis vectors, and then applying a basis reduction algorithm, their algorithm produces a Lov asz Scarf reduced basis by measuring ....

L. Lov'asz, H.E. Scarf (1992). The generalized basis reduction algorithm. Mathematics of Operations Research 17, 751--764.

....Such a description could also be obtained by for instance deriving the Hermite normal form of the matrix A, but that typically creates large (but polynomially bounded) numbers. A full dimensional polytope is useful if one wants to apply the algorithm of Lenstra [12] or of Lov asz and Scarf [14]. 3 Basis reduction and its use in integer programming We begin by giving the definition of a lattice and a reduced basis. Definition 1 A subset L ae IR n is called a lattice if there exists a basis b 1 ; b 2 ; b l of IR n such that L = f l X j=1 ff j b j : ff j 2 ZZ; 1 j lg: ....

....with dimension one less than the dimension of its predecessor. In Figure 3 we show how the distance between hyperplanes H k b n increases if we use a basis with nearly orthogonal vectors instead of a basis with non orthogonal ones. For the integer programming problem P , Lov asz and Scarf [14] developed an algorithm that, as Lenstra s algorithm, uses branching on hyperplanes. Instead of using a transformation to transform the polytope and the initial basis vectors, and then applying a basis reduction algorithm, their algorithm produces a Lov asz Scarf reduced basis by measuring the ....

[Article contains additional citation context not shown here]

L. Lov'asz, H.E. Scarf (1992). The generalized basis reduction algorithm. Mathematics of Operations Research 17, 751--764.

....of parallel hyperplanes. The number of such hyperplanes can be proved to be bounded by a constant depending only on n. For any lattice such a basis exists and can be found in polynomial time starting from an arbitrary basis by using basis reduction; see Lenstra et al. 34] Lov asz and Scarf [36] designed a generalized basis reduction algorithm, which works directly on the polyhedron instead of using approximations such as Lenstra does. The advantage of their method is that less information is lost, the disadvantage is that it uses considerably more computational steps. Cook et al. 20] ....

L. Lov'asz, H.E. Scarf (1992). The generalized basis reduction algorithm. Math. Oper. Res. 17, 751--764.

....Such a description could also be obtained by for instance deriving the Hermite normal form of the matrix A, but that typically creates large (but polynomially bounded) numbers. A full dimensional polytope is useful if one wants to apply the algorithm of Lenstra [12] or of Lov asz and Scarf [14]. 3 Basis reduction and its use in integer programming We begin by giving the definition of a lattice and a reduced basis. Definition 1 A subset L ae IR n is called a lattice if there exist linearly independent vectors b 1 ; b 2 ; b l in IR n such that L = f l X j=1 ff j b j : ....

....between hyperplanes H k b n increases if we use a basis with nearly orthogonal vectors instead of a basis with non orthogonal ones. b 2 b 1 b 2 b 1 X X (a) b) Figure 3: a) Non orthogonal basis. b) Nearly orthogonal basis. For the integer programming problem P , Lov asz and Scarf [14] developed an algorithm that, as Lenstra s algorithm, uses branching on hyperplanes. Instead of using a transformation to transform the polytope and the initial basis vectors, and then applying a basis reduction algorithm, their algorithm produces a Lov asz Scarf reduced basis by measuring ....

[Article contains additional citation context not shown here]

L. Lov'asz, H.E. Scarf (1992). The generalized basis reduction algorithm. Mathematics of Operations Research 17, 751--764.

....2 Pgc. Since the direction d is flat the number of subproblems created at each level of the search tree is limited by a constant depending only on n. Moreover, we have no more than n levels in the tree. To find flat directions Cook et al. use the generalized basis reduction technique developed by Lov asz and Scarf (1992). One of the main drawbacks of polyhedral techniques, as described in Section 2, is that the separation problem based on several facet defining inequalities is hard to solve, or sometimes even hard to formulate. Boyd (1994) developed a cutting plane algorithm for general integer programming that ....

L. Lov' asz and H.E. Scarf (1992) "The generalized basis reduction algorithm", Mathematics of Operations Research 17 751--764.

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L. Lov'asz and H. Scarf. The generalized basis reduction algorithm. Mathematics of Operations Research 17 (1992) 751-764. 14

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