An Algorithm for Integer Linear Programming: A Combined Algebraic and Enumeration Approach

This paper develops an algorithm for pure integer programming problems. It first transforms the integer programming problem to an algebraically equivalent Hermite canonical problem, and then employs the Fourier-Motzkin elimination. These algebraic operations transform the problem into a form that leads to an efficient implicit enumeration scheme to calculate an optimal solution. The algorithm constructs, in a finite number of operations, an optimal solution to an integer program with n variables and n or n + 1 inequality constraints. If the original problem has more than n + 1 constraints, then the integer program with only the constraints that are binding at an optimal linear programming solution is solved in place of the original problem. Computational results are presented.