Is an algorithm with an approximation factor of 4000 useful? A paper published in SODA this year (2019) proposed a constant approximation algorithm for the lower bounded facility location problem with general lower bounds.

To my surprise, when reading the paper, I verified that its constant approximation factor is 4000.

Therefore, I was left wondering if this algorithm is really useful for something.
Like, do you know, a solution that is 4000 times worse than the optimal solution can be anything, and probably a solution given by a simple polynomial-time greedy algorithm will be better than that of given by this more complex approximation algorithm.
A very good question! While I think a feasible solution which is 4000 times worse than the optimum is often not very practical, if you have several approximation algorithms to choose from, I would rather implement the one with a better performance guarantee. Well, maybe not always. At the very least, the feasible solutions found are often of much better quality than the guarantee. That is, if an algorithm performs better than terribly in the worst case, it often performs very well in practice. At least I've read something along those lines.

Apart from that, you can think of the result as a mathematical theorem, which may be of interest in its own right. Citing the wikipedia article on approximation algorithm: 


  The desire to understand hard optimization problems from the perspective of approximability is motivated by the discovery of surprising mathematical connections and broadly applicable techniques to design algorithms for hard optimization problems. One well-known example of the former is the Goemans-Williamson algorithm for Maximum Cut which solves a graph theoretic problem using high dimensional geometry.