The Minimised Geometric Buchberger Algorithm: An Optimal Algebraic Algorithm for Integer Programming

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 diculty 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 Sturmfel...

• Type: Conference
• Year: 1997
• Where: ISSAC
• Authors: Qiang Li, Yike Guo, Tetsuo Ida, John Darlington
• Keywords: Geometric Buchberger Algorithm, Grobner Basis, Ip Problem, ISSAC 1997, Mathematics