If the solution to a problem is easy to check for correctness, is the problem easy to solve? 1,000,000 prize for the first correct solution. For some questions, there is no known way to find algorithmic problem solving pdf answer quickly, but if one is provided with information showing what the answer is, it is possible to verify the answer quickly.
Given a partially filled-in Sudoku grid, of any size, is there at least one legal solution? Thousands of other problems seem similar, fast to check but slow to solve. Decades of searching have not yielded a fast solution to any of these problems, so most scientists suspect that none of these problems can be solved quickly. However, this has never been proven. Sudoku, can also be solved in polynomial time. 1971, there were previous inklings of the problems involved, the difficulty of proof, and the potential consequences. In 1955, mathematician John Nash wrote a letter to the NSA, where he speculated that cracking a sufficiently complex code would require time exponential in the length of the key.
In such analysis, a model of the computer for which time must be analyzed is required. In 2012, 10 years later, the same poll was repeated. Boolean satisfiability problem in polynomial time. M guaranteed to halt in polynomial time, does there exist a polynomial-size input that M will accept? M that takes the solution to be verified as input. Then the question of whether the instance is a yes or no instance is determined by whether a valid input exists. So a polynomial time solution to Sudoku leads, by a series of mechanical transformations, to a polynomial time solution of satisfiability, which in turn can be used to solve any other NP-complete problem in polynomial time.
Using transformations like this, a vast class of seemingly unrelated problems are all reducible to one another, and are in a sense “the same problem”. Hence, the problem is known to need more than exponential run time. It is also possible to consider questions other than decision problems. How many solutions are there? For these problems, it’s very easy to tell whether solutions exist, but thought to be very hard to tell how many. First, it is not always true in practice. A theoretical polynomial algorithm may have extremely large constant factors or exponents thus rendering it impractical.
It is also intuitively argued that the existence of problems that are hard to solve but for which the solutions are easy to verify matches real-world experience. There would be no special value in “creative leaps,” no fundamental gap between solving a problem and recognizing the solution once it’s found. This is, in my opinion, a very weak argument. The space of algorithms is very large and we are only at the beginning of its exploration.
Being attached to a speculation is not a good guide to research planning. One should always try both directions of every problem. Prejudice has caused famous mathematicians to fail to solve famous problems whose solution was opposite to their expectations, even though they had developed all the methods required. One of the reasons the problem attracts so much attention is the consequences of the answer. Either direction of resolution would advance theory enormously, and perhaps have huge practical consequences as well. Cryptography, for example, relies on certain problems being difficult. NP, then finding a pre-image M can be done in polynomial time, through reduction to SAT.
On the other hand, there are enormous positive consequences that would follow from rendering tractable many currently mathematically intractable problems. Efficient solutions to these problems would have enormous implications for logistics. Yes-or-No questions could be completely replaced by a machine. After all, one would simply have to choose the natural number n so large that when the machine does not deliver a result, it makes no sense to think more about the problem. A method that is guaranteed to find proofs to theorems, should one exist of a “reasonable” size, would essentially end this struggle. NP will turn out to be helpful even if it is proved, because such a proof will almost surely be nonconstructive. It would allow one to show in a formal way that many common problems cannot be solved efficiently, so that the attention of researchers can be focused on partial solutions or solutions to other problems.
For example, it is possible that SAT requires exponential time in the worst case, but that almost all randomly selected instances of it are efficiently solvable. 2009 studied the status of the five worlds. ZFC that such algorithms are correct. In September 2010, Deolalikar was reported to be working on a detailed expansion of his attempted proof.
However, opinions expressed by several notable theoretical computer scientists indicate that the attempted proof is neither correct nor a significant advancement in the understanding of the problem. Effectively, this, in combination with the order, allows the definition of recursive functions. However, these algorithms do not qualify as polynomial time because their running time on rejecting instances are not polynomial. Output: “yes” if any subset of S adds up to 0. Runs forever with no output otherwise.