How can you determine whether a problem is solvable in polynomial time?

Every problem in NP is polynomially reducible to SAT, and SAT is reducible to every NP-hard problem. Therefore, a polynomial time solution to any NP-hard problem (such as 3Col) implies that every problem in NP can be solved in polynomial time.

"Polynomial-time" means the time requirement for an algorithm is bound to grow polynomially for its input size or workload. With several inputs, a polynomial-time algorithm will execute in a timeframe bounded by some exponential size of each input.

Polynomial time refers to the efficiency of an algorithm in solving a problem. An algorithm is said to run in polynomial time if its worst-case time complexity is a polynomial function of the input size. This implies that the algorithm's running time grows at most polynomially with the size of the input, indicating a manageable and computationally efficient solution.

Polynomial time refers to the computational complexity of an algorithm or problem-solving method, where the time required for execution grows polynomially with the size of the input. In simpler terms, an algorithm runs in polynomial time if its time complexity is bounded by a polynomial function of the input size. For example, algorithms with time complexities like ? ( ? ) O(n), ? ( ? 2 ) O(n 2 ), ? ( ? 3 ) O(n 3 ), etc., are considered to run in polynomial time. This means that as the size of the input increases, the time required for the algorithm to complete its execution increases at a manageable rate according to a polynomial function.

Polynomial time defines an algorithm's efficiency by bounding its worst-case running time with a polynomial function of the input size. Efficient polynomial time algorithms grow at a manageable rate, while non-polynomial time algorithms are considered inefficient. The concept is fundamental in computational complexity theory, categorizing problems into classes like P and NP. It's essential for understanding algorithmic efficiency and problem-solving feasibility.

NP stands for Non Polynomial time - Problems which cannot be solved in polynomial time but can be verified in polynomial time. It might be a bit confusing for you, it was for me too - but I will simplify it - 1. In NP Problem cannot be solved in polynomial time. 1st statement it is clear to everyone. Travelling Salesman problem can not be solved under polynomial time like O(n^2), O(n^3), O(n^4), O(n^5)… so on 2. But can be verified in Polynomial Time. You can verify whether the solution is correct or not once you got the solution of the problem. If by anyhow you managed to calculate the solution of Travelling Salesman problem then we can verify whether the solution is correct or not in polynomial time. Note: P ? NP

NP, or nondeterministic polynomial time, encompasses a class of decision problems for which a solution can be verified quickly. While solutions to NP problems may not be found efficiently, if a potential solution is provided, its correctness can be verified in polynomial time. Examples include the traveling salesman problem and the knapsack problem.

Polynomial time serves as a cornerstone in theoretical computer science, aiding in the classification and understanding of computational problems.

Non-deterministic Polynomial (NP) time refers to the set of problems for which a solution can be verified in polynomial time given the correct answer. This class encompasses problems where finding a solution might be complex and potentially require exponential time, but once a solution is proposed, its correctness can be checked efficiently within polynomial time. NP includes both problems that can be solved in polynomial time (P) and those that are believed not to have polynomial-time solutions but have polynomial-time verifiers. This distinction is crucial for understanding the complexity landscape, especially when exploring the boundaries between tractable and intractable problems in computational theory.

A problem which can not solved in polynomial time i.e worst case complexity of a algorithm is not x to the power costant,then it is called non polynomial problem. From perspective of Automata it feasible to have problem polynomial,some of example of polynomial problems are traveling salesman problem etc.

NP-completeness identifies a special set of problems within NP that are particularly challenging. A problem is NP-complete if it belongs to NP, and any problem in NP can be transformed (in polynomial time) into an instance of the NP-complete problem. Solving one NP-complete problem efficiently would imply an efficient solution to all problems in NP, making them key in complexity theory. Common Mistake: Thinking NP-Complete Implies Intractable Misunderstanding: While NP-complete problems are challenging, being NP-complete doesn't automatically mean intractable. It signifies a specific relationship within NP, and some NP-complete problems might have efficient algorithms yet to be discovered.

* NP-complete problems are among the hardest in the NP set. These problems are “hard” in the sense that if any NP-complete problem can be solved quickly (in polynomial time), then every problem in NP can also be solved quickly. * Whether P equals NP is one of the most important open problems in CS and is 1 of the 7 Millennium Prize Problems for which there is a prize of 1 million USD. * One fundamental property of NP-complete problems is that any problem in NP can be reduced to any NP-complete problem in polynomial time. This means if you have a polynomial-time solution for one NP-complete problem, you can transform any problem in NP into that NP-complete problem and solve it in polynomial time too. Note: P ? NP

Many years ago I had the pleasure of writing a deadlock detection algorithm. It was a lesson in graph searching, recursion, and realizing intractable means intractable. OSs used to take shortcuts when resolving deadlocks. One I remember would just randomly kill processes until the deadlock went away. I learned why they do stuff like that. I thought I got it down to an n-cubed problem but was fooling myself. Interesting stuff...

It is a problem which can not be solved in polynomial time but can reduce to a np problem in polynomial time.It makes a sense to prove a problem is non polynomial without solving it.

NP-completeness defines a class of decision problems in computational complexity theory. A problem is NP-complete if it's in NP (solutions can be verified in polynomial time) and it's as hard as the hardest problems in NP, meaning all NP problems can be reduced to it in polynomial time. This makes NP-complete problems among the most challenging within NP. Finding a polynomial-time solution for any NP-complete problem would imply P = NP, solving a major open question in computer science. However, no such efficient algorithm has been found, marking NP-complete problems as likely intractable for exact solutions within polynomial time.

* NP-hard is a broader classification that encompasses problems as hard as the hardest problems in NP, but they are not necessarily in NP themselves. NP-complete problems are a subset of NP-hard problems. * NP-hard problems may include problems that cannot be verified in polynomial time (hence, possibly not in NP), but solving them would still imply a solution for all NP problems due to their comparative difficulty. * This means if a polynomial-time solution exists for an NP-hard problem, it would also solve all NP-complete (and consequently all NP) problems, but the converse is not necessarily true.

NP-hardness characterizes problems as hard as the toughest problems in NP, regardless of whether they belong to NP. These problems, including decision, optimization, and search problems, are such that any NP problem can be reduced to them in polynomial time. However, unlike NP-complete problems, NP-hard problems do not require a solution to be verifiable in polynomial time. Solving an NP-hard problem efficiently would solve all NP problems efficiently, but NP-hard problems may not even have a definitive solution to verify. This makes NP-hardness a measure of computational intractability.

We can use this reduction to solve a lot of problems and save time whether it is a NP hard, NP complete or Polynomial time.

The best way is the direct way, i.e., finding a polynomial-time algorithm or splitting a problem so that each sub-problem can be solved in the polynomial time or space. Two, I would cast a problem as an optimization problem, define a clear objective function, and see if a deep neural network can find a solution.

In my work on polynomial solvability, I focus on identifying whether problems can be solved within time frames that scale polynomially with input size. This involves developing algorithms and analyzing their time complexity to ensure a polynomial relationship. My research spans P vs. NP and fractal mathematics, aiming to demonstrate polynomial solvability for problems in P and exploring the boundaries of NP to find or disprove efficient solutions. For P-class problems, I've contributed algorithms that confirm polynomial solvability. In the NP realm, my work seeks to either discover efficient algorithms or highlight inherent complexities that challenge polynomial-time resolution, advancing the field of computational complexity theory.

It does matter alot take example from the movie imitation game where enigma does not stop initially when Ben was decoding the message although problem is np, but still efficiency matter while designing the solution. It is like burning fuel slowly for starting the engine so that it will the too much time to start giving output as not using bike.

Even if your algorithm has a polynomial time complexity, consider the practical implications. Large constants or high-degree polynomials may still result in inefficiencies for large inputs.

Consider you have a polynomial algorithm, and now consider the input size to a problem size N where N you can't fit any memory, GPU, or store on any storage. In many cases, if a task requires some invariant. Order, for example. i.e., premutation invariant, even in a distributed system, it would be hard to split a problem into many subproblems where each is polynomial time complexity. Hence, you need to scale a system where the algorithm becomes intractable as the problem grows. For example, consider Vector size (N,1), and each vector element is a token for a vocab; consider that you have a polynomial time algorithm in this problem. The order of tokens matters. If the size of N is large, you need to deal with complexity.

While designing any solution for automation we do consider time and space as main constraints.Hence there is part of code optimisation in the industry. From realtime perspective we do use dynamic programming solution in the coding for optimisation of space,even we reduce prints or debug statement in the solution.