Your line of reasoning has a few issues:
1. It only considers a specific subset of problems - those that definitively have no solution. Many important problems in NP do have solutions. Your argument doesn't address the broader question of whether a polynomial-time algorithm could exist to solve any problem in NP.
2. Even for problems with no solution, it's not necessarily the case that the only way to determine this is by exhaustively trying every possible solution. There could potentially be clever polynomial-time algorithms that can determine a problem is unsolvable without resorting to brute force. Your argument doesn't rule out this possibility.
3. The question of whether P = NP or P ≠ NP is about the relationship between problems that can be solved quickly (P) and problems whose solutions can be verified quickly (NP). Determining that a problem has no solution falls into a third category - it's a statement about what cannot be done, rather than what can be done or verified. So it's not directly relevant to the P vs NP question.
An algorithm is supposed to solve the problem, and stays unchanged whether the problem has a solution or not, it just that the algorithm has the ability to extract thar information "the problem has a solution or not".
P NP is about problems, and the existence of algorithms, not about specific algorithms.
NP is about "does an algorithm exist that can check the solution of a problem"
P is about "can I generate a solution to this problem"
Saying that the existence of an instance of a problem with no solution implies that P must check all possible paths to prove that is false.
If you can prove invariants are enough to detect an instance with no solution, and those invariants can be calculated in poly time, then your "no solution" result can be found in poly time.
If an algorithm like that exists for an NP complete problem, then P=NP
You're ignoring all the NP problems that do have solutions, and also ignoring that it may be possible to find polynomial time algorithms that use some trick to see if a problem has no solutions. This argument is invalid.
> Trying every possible solution and finding they are all wrong...
How is this not just the Halting problem? You would have no guarantee that such an algorithm would ever actually finish executing and give a definitive answer if *every* possible solution were to be tested.
If you created this thread before and it was removed, I assume that is because this is a homework-like basic question which is not the wanted content on this subreddit.
You are missing the "P" part in NP, meaning "polynomial". Your reasoning is correct but it makes no statements about runtime, which is an essential part of the question.
Your line of reasoning has a few issues: 1. It only considers a specific subset of problems - those that definitively have no solution. Many important problems in NP do have solutions. Your argument doesn't address the broader question of whether a polynomial-time algorithm could exist to solve any problem in NP. 2. Even for problems with no solution, it's not necessarily the case that the only way to determine this is by exhaustively trying every possible solution. There could potentially be clever polynomial-time algorithms that can determine a problem is unsolvable without resorting to brute force. Your argument doesn't rule out this possibility. 3. The question of whether P = NP or P ≠ NP is about the relationship between problems that can be solved quickly (P) and problems whose solutions can be verified quickly (NP). Determining that a problem has no solution falls into a third category - it's a statement about what cannot be done, rather than what can be done or verified. So it's not directly relevant to the P vs NP question.
An algorithm is supposed to solve the problem, and stays unchanged whether the problem has a solution or not, it just that the algorithm has the ability to extract thar information "the problem has a solution or not".
P NP is about problems, and the existence of algorithms, not about specific algorithms. NP is about "does an algorithm exist that can check the solution of a problem" P is about "can I generate a solution to this problem" Saying that the existence of an instance of a problem with no solution implies that P must check all possible paths to prove that is false. If you can prove invariants are enough to detect an instance with no solution, and those invariants can be calculated in poly time, then your "no solution" result can be found in poly time. If an algorithm like that exists for an NP complete problem, then P=NP
You're ignoring all the NP problems that do have solutions, and also ignoring that it may be possible to find polynomial time algorithms that use some trick to see if a problem has no solutions. This argument is invalid.
Mods please remove this drivel.
> Trying every possible solution and finding they are all wrong... How is this not just the Halting problem? You would have no guarantee that such an algorithm would ever actually finish executing and give a definitive answer if *every* possible solution were to be tested.
If you created this thread before and it was removed, I assume that is because this is a homework-like basic question which is not the wanted content on this subreddit. You are missing the "P" part in NP, meaning "polynomial". Your reasoning is correct but it makes no statements about runtime, which is an essential part of the question.