Decomposition of integer programs and of generating sets

Abstract

In this paper we investigate techniques for decomposing the matrix of coefficients of a family of integer programs. From a more practical point of view, these techniques are useful to design a primal algorithm that solves the integer program via generating sets. In this context our approach is applied to the Frobenius problem and to integer programming instances that seem to be difficult for LP-based integer programming codes. From a theoretical point of view, the techniques for decomposing a matrix that we present in this paper give rise to bounds on the L1-norm of all the elements in the Hilbert basis of a pointed cone. Moreover, applying our decomposition techniques we can show that any 0/1 linear integer program with a fixed number of constraints and a fixed number of digits to encode each coefficient in the matrix can be solved in polynomial time. A relation of our method to the group theoretic approach exists and is discussed as well.