Not all divide-and-conquer problems exhibit optimal substructure.
Consider the problem of visiting all vertices of a graph in a simple
cycle, starting from a chosen vertex. The problem is to find the cycle that
yields the smallest total edge cost. We will break our problem down by
finding the last vertex to visit in our tour. The subproblem is then to
find the optimal arrangement of vertices to visit before the chosen last vertex.
Consider this example. The optimal tour is
, but when the
last vertex D is removed, the optimal tour for the subproblem
, which is not part of the
globally optimal solution.