(Equivalently, this can be thought of as a single Turing machine that always guesses correctly)  A binary search on the range of possible distances can convert the decision version of Traveling Salesman to the optimization version, by calling the decision version repeatedly (a polynomial number of times). The decision problem version of the integer factorization problem: given integers "n" and "k", is there a factor "f" with 1 < "f" < "k" and "f" dividing "n"? The Subgraph isomorphism problem of determining whether graph contains a subgraph that is isomorphic to graph . The boolean satisfiability problem, where we want to know whether or not a certain formula in propositional logic with boolean variables is true for some value of the variables.