 
Summary: Reductions, Codes, PCPs, and Inapproximability
Sanjeev Arora \Lambda
Princeton University
Abstract
Many recent results show the hardness of approxi
mating NPhard functions. We formalize, in a very
simple way, what these results involve: a codelike
Levin reduction.
Assuming a wellknown complexity assumption, we
show that such reductions cannot prove the NP
hardness of the following problems, where ffl is any
positive fraction: (i) achieving an approximation ra
tio n 1=2+ffl for Clique, (ii) achieving an approximation
ratio 1:5 + ffl for Vertex Cover, and (iii) coloring a 3
colorable graph with O(logn) colors. In fact, we ex
plain why current reductions cannot prove the NP
hardness of coloring 3colorable graphs with 9 colors.
Our formalization of a codelike reduction, together
with our justification of why such reductions are nat
ural, also clarifies why current proofs of inapproxima
