A Generalized Simplex Method for Integer Problems given by Verification Oracles

We consider a linear problem over a finite set of integer vectors. We assume that there is a verification oracle, which is an algorithm being able to verify whether a given vector optimizes a given linear function over the feasible set. Given an initial solution, the algorithm proposed in this paper finds an optimal solution of the problem together with a path, in the 1-skeleton of the convex hull of the feasible set, from the initial solution to the optimal solution found. The length of this path is bounded by the sum of distinct values which can be taken by the components of feasible solutions, minus the dimension of the problem. In particular, in the case when S is a set of binary vectors, the length of the constructed path is bounded by n.