TITLE: What to do when your research runs into a computationally challenging problem? 
QUESTION [16 upvotes]: 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!

REPLY [15 votes]: Isn't this particular case easier to prove using the topology of the independence complexes of your $K_{n,m}$ and $K_{s,t}$?
$Ind(K_{n,m})$ is the disjoint union of an (n-1)-simplex and an (m-1)-simplex, and $Ind(K_{s,t})$ is the disjoint union of a (s-1)-simplex and a (t-1)-simplex.  So the Stanley-Reisner complex of the disjoint union of those two graphs would be $\Delta=Ind(K_{n,m})\coprod Ind(K_{s,t})$ is the join of those two complexes, which is connected ($\dim\widetilde{H}_0(\Delta)$=0) and has $\dim\widetilde{H}_1(\Delta)=1$.
From Hochster's formula, this gives you a nonzero Betti number at homological stage n+m+s+t-2, so $pd(R/I)\geq n+m+s+t-2$.  As this is the entire Stanley Reisner complex, and as the complex is connected, that's as high as the projective dimension could be.
This isn't quite the complex you wanted - but you'd just need to examine what happens to the join of the independence complexes of $K_{n,m}$ and $K_{s,t}$ after deleting the face $\{v,w\}$ corresponding to the edge you added between the graphs and all of the faces containing it.  This complex still is connected and has $\dim \widetilde{H}_1(\Delta)=1$, so the projective dimension of both complexes is the same.
I guess a more general answer to this particular question though is - instead of computing the resolutions using edge ideals, you might consider using GAP to compute the homology of subcomplexes of your total complex of the appropriate size.  These homology calculations combined with Hochster's formula are often better tools at proving projective dimension or regularity bounds than trying to resolve the ideals themselves.