P=NP: What are the most promising methods?

Question

I know that P=NP has not been solved up to now, but can anybody tell me something about the following: What are currently the most promising mathematical / computer scientific methods that could be helpful to tackle this problem? Or are there even none such methods known to be potentially helpful up to now? Is there any (free) compendium on this topic where I can find all / most of the research done in this area?

Pregnable

Nitpic: you wrote P minus NP. The great question is whether P=NP (P equals NP). Often written as P=NP? The first promising subset is to consider only NP-complete problems, not all NP problems. I suggest re-phrasing the question to deal only with NP-complete problems. – Pimply

Subjective and off topic, I'm sorry. I won't insult you with the obvious suggestions about where to look instead of here. – Flirtatious

@bmargulies: How is this off topic? – Rondo

@Rondo way off in theoretical computer science, which is not programming. However, I see that my close vote is entirely lonely, so I guess I'm wrong this time. – Flirtatious

I think we won't know which method of proof is most promising until a proof is actually found (either way). If you want to know which methods of approximating an NP-complete problem are currently the most promising approximations, that's a completely different question which does have answers. – Dichroite

This question appears to be off-topic because it is about theoretical computer science, which is more appropriate at cs.stackexchange.com or cstheory.stackexchange.com. – Breastwork

Answer 1

An excellent overview appeared last year in Communications of the ACM. I think it became the most downloaded article of CACM ever, so your question may be relevant after all :-)

The Status of the P=NP Problem, Lance Fortnow, Communications of the ACM, Vol. 52 No. 9, 2009

Gonophore