I can't understand or imagine some fact about NP-hard problems. If I understand it correctly there is only one polynomial-time algorithm needed – for whichever NP-complete problem – to prove that P = NP.Let's take the subset sum problem, which is NP-complete. It says that given a set such as $\{-7, -3, -2, 5, 8\}$, we're able to find out if there exists a subset of this set summing to zero, within exponential time (and obviously check a solution, for instance $\{-3, -2, 5\}$ within polynomial time).So if someone finds an polynomial-time algorithm for this task, he'll show that P = NP, right?EDIT: I removed:Assuming, that opinions whether P = NP or P  NP amongst computer scientists are about 1:1 (this site claims that they're ~ 52% and ~44% respectively)...As the guys noticed in comments, it's wrong. I should say the proven cases are like 1:1.Okay, so it gets more intuitive now. I mean, P  NP actually seems to be 'more likely now':However, assuming that there are still, say, 5-10% of formally educated people who believes that P = NP and the cited problem is not, say, the most complex one, how is that even possible that no one  of them had found a polynomial-time algorithm yet OR (maybe more likely?) no one of their opponents proved that there's no such an algorithm? Or, does it also mean that (in terms of those people's opinions, again) the 'chances' of there being  such an algorithm are like 1:1, too?From my (maybe naive) point of view, the subset sum problem seems so simple to crack – at least for advanced computer scientists.As you're probably aware of, searching in the Net would not help me much as I just can't deeper into this problem. I've got no such mathematical knowledge to even comprehend it more.
What makes it so difficult to prove P =/ NP?  The subset sum issue
complexity theory;np complete;p vs np