What is the reverse of greedy algorithm for setcover? A common approach to approximating SETCOVER is the greedy algorithm (Algorithm 2.2 Vazirani). This algorithm greedily picks the most cost-effective subset at each iteration, removes covered elements, until there are no more uncovered elements.

However, one can go from the other end. Start with the full set, and iteratively remove the subset that is least cost-effective, and will not cause any elements to be uncovered, until there are no more such subsets.

In some sense, the second algorithm may be better because there wont be any redundant subsets in the solution. My question is, what is the name of this algorithm? Does it have the same runtime complexity and approximation ratio as the first one? I could not find any mention of this algorithm after a few hours of searching.

(Please include unweighted SETCOVER for comparison)

Edit: I see that the number of steps can in fact be as large as $(n-1)$ where $n$ is the number of subsets.
The approximation guarantee will be significantly worse.

Assume you want to cover the set $U=\{1,\ldots,2n\}$.
For every $i=1,\ldots,n$ define a set with n+1 elements by $S_i=\{i,n+1,\ldots,2n\}$.
Assume we want to cover U with the sets
$C=\{S_1,\ldots,S_n,U\}$, where $c(U)=2, c(S_i)=1$.

We would now start with $C$ ans see that the effectivity of the sets $S_i$ is 
$$\rho(S_i)=\frac{|S_i|}{c(S_i)}=\frac{n+1}{1}=n+1.$$

In contrast to this the effectivity of $U$ is
$$\rho(U)=\frac{|U|}{c(U)}=\frac{2n}{2}=n&lt;n+1.$$

Thus the reverse greedy would throw out the less effective $U$ leaving us with a solution of cost $n$ while there is a solution of cost $2$.