NP-complete problems have a version that's hard to approximate

Abstract

It is proved that all of R.M. Karp's (1972) 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. It is further shown that these problems are absurdly hard to approximate. In fact, one cannot even approximate log/sup (k)/ of the magnitude of these problems to within a constant factor, where log/sup (k)/ denotes the iterated logarithm, unless NP is recognized by slightly superpolynomial randomized machines. It is also shown that it is even harder to approximate two counting problems: counting the number of satisfying assignments to a monotone 2-SAT formula and computing the permanent of -1, 0, 1 matrices.<>