Parametric integerprogramming deals with a family of integer programs that is defined by the same constraint matrix but where the right-hand sides are points of a given polyhedron. The question is whether all these i...
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Parametric integerprogramming deals with a family of integer programs that is defined by the same constraint matrix but where the right-hand sides are points of a given polyhedron. The question is whether all these integer programs are feasible. Kannan showed that this can be checked in polynomial time if the number of variables in the integer programs is fixed and the polyhedron of right-hand sides has fixed affine dimension. In this paper, we extend this result by providing a polynomial algorithm for this problem under the only assumption that the number of variables in the integer programs is fixed. We apply this result to deduce a polynomial algorithm to compute the maximum gap between the optimum values of an integer program in fixed dimension and its linear programming relaxation, as the right-hand side is varying over all vectors for which the integer program is feasible.
Given a full-dimensional lattice Lambda subset of z(k) and a cost vector l is an element of Q(>0)(k), we are concerned with the family of the group problems min{ l. x : x r (mod Lambda) x >= 0 } r is an element ...
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Given a full-dimensional lattice Lambda subset of z(k) and a cost vector l is an element of Q(>0)(k), we are concerned with the family of the group problems min{ l. x : x r (mod Lambda) x >= 0 } r is an element of Z(k) The lattice programminggapgap(Lambda, 1) is the largest value of the minima in (0.1) as r varies over Z(k). We show that computing the lattice programminggap is NP-hard when k is a part of input. We also obtain lower and upper bounds for gap(Lambda, 1) in terms of land the determinant of Lambda. (C) 2015 Elsevier B.V. All rights reserved.
Irreducible decomposition of monomial ideals has an increasing number of applications from biology to pure math. This paper presents the Slice Algorithm for computing irreducible decompositions, Alexander duals and so...
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Irreducible decomposition of monomial ideals has an increasing number of applications from biology to pure math. This paper presents the Slice Algorithm for computing irreducible decompositions, Alexander duals and socles of monomial ideals. The paper includes experiments showing good performance in practice. (C) 2008 Elsevier Ltd. All rights reserved.
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