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

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.<<ETX>>

[1]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[2]  Carsten Lund,et al.  On the hardness of approximating minimization problems , 1993, STOC.

[3]  Avi Wigderson,et al.  Dispersers, deterministic amplification, and weak random sources , 1989, 30th Annual Symposium on Foundations of Computer Science.

[4]  Richard M. Karp,et al.  Monte-Carlo Approximation Algorithms for Enumeration Problems , 1989, J. Algorithms.

[5]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[6]  Desh Ranjan,et al.  Quantifiers and Approximation , 1993, Theor. Comput. Sci..

[7]  Leslie G. Valiant,et al.  The Complexity of Computing the Permanent , 1979, Theor. Comput. Sci..

[8]  Michael Sipser,et al.  Expanders, Randomness, or Time versus Space , 1988, J. Comput. Syst. Sci..

[9]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[10]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[11]  Carsten Lund,et al.  Proof verification and the intractability of approximation problems , 1992, FOCS 1992.

[12]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[13]  David Zuckerman,et al.  Simulating BPP using a general weak random source , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[14]  Mark Jerrum,et al.  A mildly exponential approximation algorithm for the permanent , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[15]  László Lovász,et al.  Approximating clique is almost NP-complete , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[16]  Carsten Lund,et al.  Nondeterministic exponential time has two-prover interactive protocols , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.