Complexity theory seems only interested in algorithms that shed light on complexity classes.
What has complexity theory done recently that is of practical importance?
What has complexity theory done recently that is of practical importance?
But, for instance, I recently ran into some complexity theory when exploring some of these topics, which are quite practical:
• implementing a parser-generator
• lambda calculus reductions (for a lisp-y interpreter)
• encryption/security (you don't want there to be a
simple function to crack your crypto, right?)
It is probably worth noting that "practical importance" is probably the least useful measure of the value of pure research. In fact, historically, applications emerge from high level mathematics which was began with no practical applications initially in mind.
It's not "on the brink" until I can explain how it functions to a lay person, in one sentence to explain what it does and in one sentence why they should care. E.G: Fourier transform:
"It's something which takes a complicated wave and finds out what simple waves add up to the complicated wave. This is useful, for example, because sounds are complicated waves, if you know what simple waves there are, you can do things like autotuning."
Clearly you have a rather unique idea of what "on the brink" might mean, namely that the field is on the verge of providing breakthroughs that have concrete applications. That might be a rather limiting point of view that somewhat cheapens the value of the article.
Complexity theory seems to lead practice by around a decade...
See the No Free Lunch Theorem.