I’ve only scanned through the paper, but here’s a rough summary of how it all hangs together.
From page 86 of the paper.
… polynomial time
algorithms succeed by successively
“breaking up” the problem into
smaller subproblems that are joined to
each other through conditional
independence. Consequently, polynomial
time algorithms cannot solve
problems in regimes where blocks whose
order is the same as the underlying
problem instance require simultaneous
resolution.
Other parts of the paper show that certain NP problems can not be broken up in this manner. Thus NP/= P
Much of the paper is spent defining conditional independence and proving these two points.