Marco Cesati and Luca Trevisan.
On the Efficiency of Polynomial Time Approximation Schemes.
Submitted, January 1997.
A polynomial time approximation scheme (PTAS) for an optimization problem $A$ is an algorithm that on input an instance of $A$ and $\epsilon > 0$ finds a $(1+\epsilon)$-approximate solution in time that is polynomial for each fixed $\epsilon$. Typical running times are $n^{O(1/\epsilon)}$ or $2^{1/\epsilon^{O(1)}} n$. While algorithms of the former kind tend to be impractical, the latter ones are more interesting. In several cases, the development of algorithms of the second type required considerably new (and sometimes harder) techniques. For some interesting problems (including Euclidean TSP) only an $n^{O(1/\epsilon)}$ approximation scheme is known. Under likely assumptions, we prove that for some problems (including natural ones) there cannot be approximation schemes running in time $f(1/\epsilon) n^{O(1)}$, no matter how fast function $f$ grows. Our result relies on a connection with Parameterized Complexity Theory. We show that this connection is necessary.