what is the use of reducing a problem for which no polynomial time algorithm exist into some another problem ?

Question

If I have a problem A for which no polynomial time algo exists then what do we achieve by reducing it to another problem
B and then proving by contradiction that if we could solve B in polynomial time then we could even solve A but since no
polynomial time algorithm exists for A hence no polynomial time algorithm can exist for B also , hence A cannot be solved
in polynomial time

Answer 1

Your final statement is wrong. 

hence A cannot be solved
in polynomial time

It should be "hence B cannot be solved in polynomial time". 

Because we started with A and knows that no polynomial time algorithm exists for A. Now, due to reduction we proved that, no polynomial time algorithm can exist for B also. So, there is no point in looking for a polynomial time algorithm for problem B. 

Comments

Yes Sir thats where I am stuck that when we know that A cannot be solved in polynomial time then what's the point of reducing it to some other algorithm B when it too won't be solved in polynomial time .

---

Well, we are not doing this reduction for A, but for B. The reason is not for making problem B unsolvable in polynomial time but rather to realize that B is unsolvable in polynomial time (using a deterministic TM). 

For example, we didn't had a polynomial time algorithm to check if a number is prime or not for quite a long time. Many researchers were working on it. Had someone reduced a known non NP problem to prime checking, this search could have been stopped a long back. But eventually a polynomial time algorithm was made from IIT Kanpur. 

So, these kind of reductions can help in directing research in proper direction. And one thing I put in above statement is "non NP" and not "non P". Because we still don't know if NP = P and any NP problem becomes a P problem if we find a deterministic polynomial time algorithm for it. So, to be sure a problem is not P, it must be not NP. (We are still not sure about the problems in class outside P but in NP). 

Answer 2

we reduce a problem which is not polynomial to some other  problem because we dont have any solution for  for present time..but in future we may have a polynomial solution....and such problems are called NP problems.We change these problems in some other problem in polynomial time called NP hard problems...so that many NP problms can be converted into one NP hard solution..and we have to find solution for only one NP hard problem inspite of many NP problms....

Comments

Sir , just one confusion that as u said that we convert one problem which is not polynomial to some other problem because we don't have any solution for the present time so then the problem in which we have reduced this problem ,if we do not know about it also that whether it can be solved or not then we say it to be hard and then when will we call this reduced problem to be NP ,since no polynomial time algo is yet known for NP problems as well