Optimization, approximation, and complexity classes

We define a natural variant of NP, MAX NP, and also a subclass called MAX SNP. These are classes of optimization problems, and in fact contain several natural, well-studied ones. We show that problems in these classes can be approximated with some bounded error. Furthermore, we show that a number of common optimization problems are complete under a kind of careful transformation (called L-reduction) that preserves approximability. It follows that such a complete problem has a polynomial-time approximation scheme iff the whole class does. These results may help explain the lack of progress on the approximability of a host of optimization problems.

[1]  Shlomo Moran,et al.  Non Deterministic Polynomial Optimization Problems and their Approximations , 1977, Theoretical Computer Science.

[2]  M. Garey.  Johnson: computers and intractability: a guide to the theory of np- completeness (freeman , 1979 .

[3]  Ronald Fagin.  Generalized first-order spectra, and polynomial. time recognizable sets , 1974 .

[4]  Phokion G. Kolaitis,et al.  The decision problem for the probabilities of higher-order properties , 1987, STOC.

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

[6]  Miklós Ajtai,et al.  Recursive construction for 3-regular expanders , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[7]  David S. Johnson,et al.  Approximation algorithms for combinatorial problems , 1973, STOC.

[8]  P. Orponen,et al.  On Approximation Preserving Reductions: Complete Problems and Robust Measures (Revised Version) , 1987 .

[9]  Mark W. Krentel.  The complexity of optimization problems , 1986, STOC '86.

[10]  Giorgio Ausiello,et al.  Toward a Unified Approach for the Classification of NP-Complete Optimization Problems , 1979, Theor. Comput. Sci..

[11]  Giorgio Ausiello,et al.  Structure Preserving Reductions among Convex Optimization Problems , 1980, J. Comput. Syst. Sci..

[12]  David S. Johnson,et al.  The Complexity of Near-Optimal Graph Coloring , 1976, J. ACM.

[13]  Kenneth Steiglitz,et al.  Some Examples of Difficult Traveling Salesman Problems , 1978, Oper. Res..

[14]  Larry Stockmeyer,et al.  Planar 3-colorability is polynomial complete , 1973, SIGA.

[15]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[16]  Giorgio Ausiello,et al.  On the Structure of Combinatorial Problems and Structure Preserving Reductions , 1977, ICALP.

[17]  Paul M. B. Vitányi.  How well can a graph be n-colored? , 1981, Discret. Math..

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

[19]  Richard M. Karp,et al.  Reducibility among combinatorial problems" in complexity of computer computations , 1972 .

[20]  Shlomo Moran,et al.  Non Deterministic Polynomial Optimization Problems and their Approximations , 1977, Theor. Comput. Sci..

[21]  Piotr Berman,et al.  On the Complexity of Approximating the Independent Set Problem , 1989, Inf. Comput..