this post was submitted on 12 May 2024
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If the solution to a problem is easy to check for correctness, must the problem be easy to solve?
For instance, it is easy to check if a filled sudoku grid is a valid solution. Must it therefore be easy to solve sudokus?
Most people would probably intuitively answer "no", and most computer scientists agree, but this has still not been proven, so we actually don't know.
Isn't it proof enough? Using the Sudoku example: there are certainly different levels of difficulties, depending on how many numbers are set in the beginning and other parameters. Checking if the solved answer is correct, is always the same "difficulty" - thus there is no correlation between the difficulty of the puzzle at the beginning and checking the Correctness. Some people might not be able to solve it, but they certainly can check if the solution is right
For the purposes of OPs problem (P v NP), it considers not particular solutions, but general algorithmic approaches. Thus, we consider things as either Hard (exponential time, by size of input), or Easy (only polynomial time, by size of input).
A number of important problems fall into this general class of Hard problems: Sudoku, Traveling Salesman, Bin Packing, etc. These all have initial setups where solving them takes exponential time.
On the other hand, as an example of an easy problem, consider sorting a list of numbers. It's really easy to determine if a lost is sorted, and it's always relatively fast/easy to sort the list, no matter what setup it had initially.