Optimization problems and the polynomial hierarchy

Abstract

It is demonstrated that such problems as the symmetric Travelling Salesman Problem, Chromatic Number Problem, Maximal Clique Problem and a Knapsack Packing Problem are in the Δ P 2 level of PH and no lower if ∑ P 1 ≠ Π P 1, or NP≠co-NP. This shows that these problems cannot be solved by polynomial reductions that use only positive information from an NP oracle, if NP≠co-NP. It is then shown how to extend these results to prove that interesting problems are properly in Δ P, X i+1 for all X, k where ∑ P, X k ≠ Π P, X k in PH X .