Minimised Geometric Buchberger Algorithm for Integer Programming

Recently, various algebraic integer programming (IP) solvers have been proposed based on the theory of Gröbner bases. The main difficulty of these solvers is the size of the Gröbner bases generated. In algorithms proposed so far, large Gröbner bases are generated by either introducing additional variables or by considering the generic IP problem IP _{A, C·} Some improvements have been proposed such as Hosten and Sturmfels' method (GRIN) designed to avoid additional variables and Thomas' truncated Gröbner basis method which computes the reduced Gröbner basis for a specific IP problem IP_{A, C}(b) (rather than its generalisation IP_{A, C}). In this paper we propose a new algebraic algorithm for solving IP problems. The new algorithm, called Minimised Geometric Buchberger Algorithm, combines Hosten and Sturmfels' GRIN and Thomas' truncated Gröbner basis method to compute the fundamental segments of an IP problem IP_{A, C} directly in its original space and also the truncated Gröbner basis for a specific IP problem IP_{A, C}(b). We have carried out experiments to compare this algorithm with others such as the geometric Buchberger algorithm, the truncated geometric Buchberger algorithm and the algorithm in GRIN. These experiments show that the new algorithm offers significant performance improvement.