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)|
|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