Truncated Gröbner Bases for Integer Programming

Abstract.

 The toric ideal I _A of a matrix A=(a ₁, . . . ,  a _n )∈ℤd ^× n is the kernel of the monoid algebra map π^ _A  : k[x ₁, . . . ,  x _n ]→k[t ^±1 ₁, . . . ,  t ^±1 _d ], defined as x _j ?t ^aj. It was shown in [4] that the reduced Gröbner basis of I _A , with respect to the weight vector c, can be used to solve all integer programs minimize {cx : Ax=b, x∈ℕⁿ}, denoted IP _A,b,c,= , as b varies. In this paper we describe the construction of a truncated Gröbner basis of I _A with respect to c, that solves IP _A,b,c,= for a fixed b. This is achieved by establishing the homogeneity of I _A with respect to a multivariate grading induced by A. Depending on b, the truncated Gröbner basis may be considerably smaller than the entire Gröbner basis of I _A with respect to c. For programs of the form maximize{cx : Ax≦b, x≦u, x∈ℕ ⁿ} in which all data are non-negative, this algebraic method gives rise to a combinatorial algorithm presented in [17].