The Integral Basis Method in an Augmentation Framework for Integer Programming

One of the obstacles in a branch-and-cut IP-solving scheme is closing the gap between a known feasible solution and the solution(s) to LP relaxations of the problem. There are, however, a number of real-world problems, where, once an optimal solution is found, its optimality can easily be proved using neither branch-and-bound nor branch-and-cut methods. All one needs to do is find an integral simplex basis for the feasible solution: if none of the reduced cost coefficients promises a further improving simplex step, one immediately has a dual certificate for optimality. The construction of an integral simplex basis is obvious for problems containing only binary variables, or if one can accept a substantial increase in the problem size. Otherwise, if an integral simplex basis for an arbitrary solution is known, one can apply the method of Gomory [Gom63] to obtain an integral simplex basis for any solution. For many instances however, one may not be lucky enough to obtain an optimalityproving simplex basis immediately, even if the current solution is optimal. In that case, one still does not have to resort to branch-and-cut, but may instead decide to use the basis