We consider a linear problem over a finite set of integer vectors
and 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 the feasible set is
a set of binary vectors, the length of the constructed path is bounded by the
number of variables, independently of the objective function.