Fig. 8.

Illustration of the proof by contradiction that promise-Toolbox Adaptation is . Assuming that promise-Toolbox Adaptation can be computed in polynomial time, we show that one can solve Satisfiability in polynomial time. We transform the input for Satisfiability into an input for promise-Toolbox Adaptation, then apply the polynomial time algorithm to solve that instance, and using the schema illustrated here infer the solution for Satisfiability in polynomial time. However, since it is known that Satisfiability is , this schema cannot exist (assuming $\mathsf{P}\ne \mathsf{NP}$) and we reject our assumption that promise-Toolbox Adaptation can be solved in polynomial time