Occasionally, but more frequently lately, I would like to perform some hard computations. As an example, yesterday the following question came up:

What is the projective dimension of the edge ideal of the graph $G$ which is two complete bipartite graphs $K_{a,b}, K_{c,d}$ joined by another edge?

(I believe I can get the answer by hand, but a confirmation would be nice. Also, I wish to compute other similar examples).

For the above question, my personal computer crashed when $a=b=c=d=6$. I used Macaulay 2 with a special package for such situations: EdgeIdeals.

I have a few vague ideas on how to solve this: email people who are better at computations, try to find access to more powerful computers (a small fee is OK), or *carefullly* use MO (but may be that only works for people like Kevin Buzzard). Still:

What can one do in such situations?

I am looking for more generic answers (that can apply not only for the examples above, but in other situations). For example, a pointer to what powerful computers one can get access to would be helpful. Thank you!