Non-deterministic polynomial time

Definition

Non-deterministic polynomial time (NP) refers to the class of decision problems for which a proposed solution can be verified in polynomial time by a deterministic Turing machine. This means that if you have a candidate solution, you can check whether it’s correct within a time frame that scales polynomially with the size of the input. NP is significant because it includes many important problems in computer science, and understanding it helps us grasp the broader categories of complexity classes.

5 Must Know Facts For Your Next Test

1. The term 'non-deterministic' implies that multiple possible solutions can exist for a problem, and the correct one can be verified quickly once identified.
2. If a problem is in NP, it means we can verify solutions quickly, but we may not necessarily be able to find those solutions quickly.
3. All problems in P are also in NP, but it's unknown whether all NP problems can be solved in polynomial time (this is the famous P vs NP question).
4. NP-hard problems are at least as hard as NP-complete problems, but they may not even be decision problems, hence are not in NP.
5. Many real-world problems, like the Traveling Salesman Problem and Knapsack Problem, are NP-complete, making them crucial to fields such as optimization and cryptography.

Review Questions

• How does non-deterministic polynomial time relate to the concept of verifying solutions to decision problems?
  • Non-deterministic polynomial time is centered around the idea that while finding solutions might be hard, verifying them is much easier. When we talk about NP, we mean that if someone gives us a proposed solution to a decision problem, we can check its correctness efficiently using a deterministic algorithm. This verification process being polynomial means that even though the problem might be difficult to solve outright, we can still confirm the validity of potential solutions quickly.
• In what ways do NP-complete problems serve as benchmarks for understanding non-deterministic polynomial time?
  • NP-complete problems act as critical reference points within non-deterministic polynomial time because they represent the hardest challenges in NP. If we could find a polynomial-time algorithm for any single NP-complete problem, it would imply that all problems in NP could also be solved in polynomial time. Therefore, studying these benchmark problems provides insight into whether P equals NP and helps identify the boundaries of computational efficiency.
• Evaluate the implications of proving P = NP on the fields of computer science and mathematics.
  • Proving P = NP would have groundbreaking implications across computer science and mathematics because it would mean that every problem whose solution can be verified quickly could also be solved quickly. This would revolutionize fields like cryptography, optimization, and algorithm design since many cryptographic systems rely on certain problems being hard to solve. Moreover, it would reshape our understanding of computation limits and push forward advancements in technology, potentially making previously intractable problems solvable.

Related terms

P: The class of decision problems that can be solved by a deterministic Turing machine in polynomial time.

NP-complete: A subset of NP problems that are as hard as the hardest problems in NP, meaning if one NP-complete problem can be solved in polynomial time, all NP problems can.

deterministic Turing machine: A theoretical computational model that, given a specific input, will always produce the same output through a defined set of rules.