The minimised geometric Buchberger algorithm

Abstract

IP problems characterise combinatorial optimisation problems where conventional numerical methods based on the hill-climbing technique can not be directly applied. Conventional methods for solving integer programming are based on searching algorithms where heuristics such as branch and bound are applied to reduce the search space. Recently, various algebraic IP solvers have been proposed based on the theory of Grobner bases. The key idea is to encode an IP problem IPA;C into a special ideal associated with the constraint matrix A and the cost (object) function C. An important property of such an encoding is that its Grobner basis corresponds directly to the test set of the IP problem. The main di culty of these new methods is the size of the Grobner bases generated. In the proposed algorithms, large Grobner bases are caused by either introducing additional variables or by considering the generic IP problem IPA;C . Some improvements have been proposed such as the Hosten and Sturmfels method (GRIN) designed to avoid additional variables and the truncated Grobner basis method of Thomas which computes the Grobner basis for a speci c IP problem IPA;C(b) (rather than its generalisation IPA;C). In this paper we propose a new algebraic algorithm for solving integer programming problems. The new algorithm, called the Minimised Geometric Buchberger Algorithm (MGBA), combines the Hosten and Sturmfels method(GRIN) and Thomas's truncated GBA to compute the fundamental segments of a IP problem IPA;C directly in its original space and also the truncated Grobner basis for a speci c IP problem IPA;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 shows that the new algorithm o ers signi cant performance improvement.