A Machine Model for the Complexity of NP-Approximation Problems

Many introductions to the theory of NP-completeness make some mention of approximating NP-complete problems. The usual story line says that even though solving an NP-complete problem exactly may be intractable, it is sometimes possible to find an approximate solution in polynomial time. The intuitive assumption is that finding an approximate solution should be easier than finding the exact solution. While this assumption holds for a long list of approximable problems (Bin Packing, Euclidean Travelling Salesman Problem, Set Cover, Subset Sum, etc.), recent works have shown that for the CLIQUE problem, even coarse approximations are not achievable in polynomial time unless P = NP. Furthermore, these non-approximability results extend to other NP-complete problems; most notably to Graph Coloring, Set Cover, MAX3SAT and all MAXSNP-complete problems. Nevertheless, showing that an NP-approximation problem cannot be solved in polynomial time only shows that the problem is difficult to compute; it does not show that solving the approximation problem is just as hard as finding the exact solution. Thus, it remains possible that approximating CLIQUE is in fact easier than finding the exact solution to CLIQUE. In this paper, we point out that for several NP-optimization problems, we can prove, not just intuit, that finding the exact solution is harder than finding an approximate solution, under the assumption that the Polynomial Hierarchy (PH) does not collapse. To keep the exposition simple, we focus on determining the precise complexity of finding approximations to the CLIQUE problem, but the results extend to many other NP-approximation problems.