Is there an approximation algorithm for Minimum Test Collection Problem?
Yes, there are approximation algorithms for this problem. 


If $k$ is the upper bound of the test case, the authors of http://repository.cmu.edu/cgi/viewcontent.cgi?article=1397&amp;context=tepper
constructed a $O(\log{k})$-approximation algorithm
Another approximation algorithm of the same complexity is derived here: 
https://link.springer.com/chapter/10.1007/3-540-44676-1_13


Hope that helps!