On syntactic versus computational views of approximability

We attempt to reconcile the two distinct views of approximation classes: syntactic and computational. Syntactic classes such as MAX SNP permit structural results and have natural complete problems, while computational classes such as APX allow us to work with classes of problems whose approximability is well understood. Our results provide a syntactic characterization of computational classes and give a computational framework for syntactic classes. We compare the syntactically defined class MAX SNP with the computationally defined class APX and show that every problem in APX can be "placed" (i.e., has approximation-preserving reduction to a problem) in MAX SNP. Our methods introduce a simple, yet general, technique for creating approximation-preserving reductions which shows that any "well"-approximable problem can be reduced in an approximation-preserving manner to a problem which is hard to approximate to corresponding factors. The reduction then follows easily from the recent nonapproximability results for MAX SNP-hard problems. We demonstrate the generality of this technique by applying it to other classes such as MAX SNP-RMAX(2) and MIN F$^{+}\Pi_2(1)$ which have the clique problem and the set cover problem, respectively, as complete problems. The syntactic nature of MAX SNP was used by Papadimitriou and Yannakakis [J. Comput. System Sci., 43 (1991), pp. 425--440] to provide approximation algorithms for every problem in the class. We provide an alternate approach to demonstrating this result using the syntactic nature of MAX SNP. We develop a general paradigm, nonoblivious local search, useful for developing simple yet efficient approximation algorithms. We show that such algorithms can find good approximations for all MAX SNP problems, yielding approximation ratios comparable to the best known for a variety of specific MAX SNP-hard problems. Nonoblivious local search provably outperforms standard local search in both the degree of approximation achieved and the efficiency of the resulting algorithms.