Recently i had a discussion with my old college friends on Facebook, and we were posting technical questions that would tickle our brain and help us re-live our college days…
Following was my questions…. Posting it here as it might help the reader of my blog understand one beautiful characteristic of NP Complete problem.
Q: There are sets of problems, if i we find solution to one of them then we have solutions to all of them, what is the technical name for these problem class?
There were multiple reply to it, one interesting one was design pattern…. Then the discussion went into clue (TSP) then we arrived at NP-Complete. Then the discussion went further into exponential Big-O time etc… But I later sensed the wow factor the question had was gone as part of our discussions… I did some study to bring back the wow factor the question had. Following was my reply…
[Reply]
The real WOW factor that I had when I read these things in my college days is, “WHAT!!!!!! If I solve (in computer terms produce an effective Algo) for one of the problem in the problem class I can solve all of them? How nerdy would be the finder of such things be??? ” [I doubt if I have this level of curiosity now:) ]
These are the current characteristics of NP-Complete problems.
1) There is no known ALGO which can complete in polynomial time (in computational terms no BEST solutions ie. Best in the order of N….- >LogN
2) If we can find solution (Algo –reasonable solution) to one of them you can solve All of them. (the real YAHOO!!! is here!!!}
It means a one NP-Complete problem can be transformed into another NP Complete problem in a polynomial time, isn’t that great. That is what the theorem claims.
It means if I solve one of the NP complete problems I can solve all from the NP Complete list…. http://en.wikipedia.org/wiki/List_of_NP-complete_problems . In other words if I solve the NP complete version of Traveling Sales Person problem I can transform all the other NP complete problems to traveling sales person problem and solve it in polynomial time.
i.e … TSP == Solvable in polynomial time. Then I have solution (best Algo) to NP Complete version of Hamilton Path problem , which no one else has…. Because Hamilton Cycle Solution = Transform2TSP (Hamilton Cycle) (polynomial time) + Solve TSP = (Hence NP complete Hamilton cycle problem is solved in polynomial time) …
To state it practically….if the Subset problem that vels mentioned is NP Complete then it can be converted to the corresponding TSP version in polynomial time and viseversa. (Yahoo!!!!!!!) . And if i have an solution (again a solution for computer programmers is good algo to solve the problem) to the subset problem i can solve TSP…
To make it more fun… see this picture 
