On Unapproximable Versions of $NP$-Complete Problems

We prove that all of Karp's 21 original $NP$-complete problems have a version that is hard to approximate. These versions are obtained from the original problems by adding essentially the same simple constraint. We further show that these problems are absurdly hard to approximate. In fact, no polynomial-time algorithm can even approximate $\log^{(k)}$ of the magnitude of these problems to within any constant factor, where $\logk$ denotes the logarithm iterated $k$ times, unless $NP$ is recognized by slightly superpolynomial randomized machines. We use the same technique to improve the constant $\epsilon$ such that MAX CLIQUE is hard to approximate to within a factor of $n^\epsilon$. Finally, we show that it is even harder to approximate two counting problems: counting the number of satisfying assignments to a monotone 2SAT formula and computing the permanent of $-1$, $0$, $1$ matrices.