Let $x$ be an NP$-$hard maximization problem. Assume that we have designed an algorithm $A$

which, given an arbitrary instance I of $x$ and an arbitrary positive real number $,$ computes a

$)-$approximate solution for instance I.

$($a$)$ Assume that the worst$-$case running time of algorithm $A$ is $O(|I{|}^{\frac{1}{}}),$ where $|I|$

denotes the length of the binary representation of instance I. Does A correspond to an

FPTAS $($full polynomial$-$time approximation scheme$)$ for X$?$ If not, does A correspond to a

PTAS $($polynomial$-$time approximation scheme$)$ for X$?$ Only your final yes$/$no answer$($s$)$ will

be graded; there is no need to provide justification.

$($b$)$ Repeat part $($a$),$ but now assume that the worst$-$case running time of algorithm

$A$ is $O(|I{|}^3*{}^{-2}).$

$($c$)$ Repeat part $($a$),$ but now assume that the worst$-$case running time of algorithm

$A$ is $O(|I{|}^3+{}^{-2}).$

$($d$)$ Repeat part $($a$),$ but now assume that the worst$-$case running time of algorithm

$A$ is $O(|I{|}^k)$ where $k=2^{\frac{1}{}}.$