A Parallel Algebraic Constraint Solver for Integer Programming

Abstract

Conventional methods for solving integer programming (IP) are based on heuristic searching algorithms. Recently, the tools of commutative algebra and algebraic geometry have bought new insights to integer programming via the theory of Grobner bases. The key idea is to encode IP problems into a special ideal associated with the constraint matrix A and the cost (object) function C. An important property of the ideal is that its Grobner bases correspond directly to the test sets of the IP problem. Using a proper test set, the optimal value of the cost function can be computed by constructing a monotonic path from the initial non-optimal solution of the problem to the optimal solution. Thus, IP can be solved without using intensive heuristic search. This approach is particularly interesting from the point of view of parallelism due to the inherent parallelism of the Buchberger algorithm that can be used to compute the Grobner bases. This paper presents a parallel geometric Buchberger algo..