A Machine Model for NP-Approximation Problems and the Revenge of the Boolean Hierarchy

This column addresses the following questions about the complexity of approximating the CLIQUE problem. 1. Is nding the vertices of the largest clique more diicult than merely nding the vertices of a 2-approximate clique (that is, a clique with at least half the size of the largest clique)? 2. What computational resources can be used to nd the vertices of a 2-approximate clique? 3. Could fewer resources be used? The results of recent studies (on the complexity of NP-approximation problems yield the following answers: (1) Yes, unless the Polynomial Hierarchy collapses; (2) Nondeterministic polynomial time with log log n+O(1) queries to SAT; and (3) Not unless the Polynomial Hierarchy collapses. Surprisingly, the proof of these results depends on the fact that a collapse of the Boolean Hierarchy implies a collapse of the Polynomial Hierarchy.