### Abstract

Many recent results show the hardness of approximating NP-hard functions. We formalize, in a very simple way, what these results involve: a code-like Levin reduction. Assuming a well-known complexity assumption, we show that such reductions cannot prove the NP-hardness of the following problems, where ε is any positive fraction: (i) achieving an approximation ratio n
^{ 1/2 +ε} for Clique, (ii) achieving an approximation ratio 1.5 + ε for Vertex Cover, and (iii) coloring a 3-colorable graph with O(log n) colors. In fact, we explain why current reductions cannot prove the NP-hardness of coloring 3-colorable graphs with 9 colors. Our formalization of a code-like reduction, together with our justification of why such reductions are natural, also clarifies why current proofs of inapproximability results use error-correcting codes.

Original language | English (US) |
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Pages (from-to) | 404-413 |

Number of pages | 10 |

Journal | Annual Symposium on Foundations of Computer Science - Proceedings |

State | Published - Dec 1 1995 |

### All Science Journal Classification (ASJC) codes

- Hardware and Architecture