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Electronic Colloquium on Computational Complexity

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TR95-023 | 16th May 1995 00:00

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 allow for clean complexity-theoretic
results and natural complete problems, while computational classes such
as APX allow us to work with problems whose approximability is
well-understood. Our results give a computational framework for
studying syntactic classes, and provide a syntactic characterization
of computational classes, as follows.

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., L-reduced to a problem) in MAX SNP. Our methods
introduce a general techique for creating approximation preserving
reductions which
show that any ``well'' approximable problem can be reduced (in an
approximation preserving manner) to a problem which is hard to
approximate to corresponding factors. We demonstrate
this technique by applying it to the classes RMAX(2) and MIN F^+ \Pi_2$
which have the Set Cover problem and the Clique problem as complete
problems, respectively.

We use the syntactic nature of MAX SNP to define a general paradigm,
non-oblivious 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-complete problems.
Non-oblivious local search provably
out-performs standard local search in both the degree of approximation
achieved, and the efficiency of the resulting algorithms.

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