language: en

ON UNAPPROXIMABLE VERSIONS OF NP-COMPLETE PROBLEMS

DAVID ZUCKERMAN

Abstract. We prove that all of Karp's 21 original NP -complete problems have a version that's 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 log (k) denotes the logarithm iterated k times, unless N P is recognized by slightly superpolynomial randomized machines. We use the same technique to improve the constant * such that MAX CLIQUE is hard to approximate to within a factor of n * . Finally, we show 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.

Key words. NP-complete, unapproximable, randomized reduction, clique, counting problems, permanent, 2SAT AMS subject classifications. 68Q15, 68Q25, 68Q99

1. Introduction.