The Complexity of Optimization Problems

MAX-CLIQUE (finding the optimal value) “reduces” to instances G, k of CLIQUE: try all values of k (or, even better, binary search to find optimal k). Obviously, CLIQUE reduces to MAX-CLIQUE. One could say that CLIQUE is the decision version of MAX-CLIQUE and they are computationally equivalent up to polynomial time reductions. More generally, given optimization problem MAX-f above, its decision version is D-f = {〈C, f, k〉 : (∃x ∈ C)f(x) ≥ k}