NP

Definition

NP, or Non-deterministic Polynomial time, is a complexity class used in computational theory that contains decision problems for which a proposed solution can be verified in polynomial time by a deterministic Turing machine. This concept is crucial when discussing the difficulty of problems and the feasibility of finding solutions, especially in the context of NP-completeness proofs, which seek to classify problems based on their solvability and verifiability.

5 Must Know Facts For Your Next Test

1. The 'N' in NP stands for 'non-deterministic,' indicating that the solutions can be guessed and verified quickly, even if finding them isn't straightforward.
2. Not all problems in NP are NP-complete; some can be solved in polynomial time and are thus classified as being in P.
3. Many real-world problems, such as the traveling salesman problem and the knapsack problem, are NP-complete, making them particularly important for optimization studies.
4. The famous P vs NP question asks whether every problem whose solution can be verified quickly (in NP) can also be solved quickly (in P).
5. If any NP-complete problem is shown to have a polynomial-time solution, it would imply that P = NP, fundamentally changing our understanding of computational complexity.

Review Questions

• How does the concept of NP relate to decision problems and their verifiability?
  • NP is directly tied to decision problems where a proposed solution can be verified quickly by a deterministic Turing machine. This means that if someone presents a solution to a problem in NP, you can check its correctness in polynomial time. This characteristic is vital for understanding how we approach complex problems, particularly in the realm of optimization.
• What distinguishes NP-complete problems from other problems within the NP class?
  • NP-complete problems are unique because they are the hardest problems within the NP class. While all NP-complete problems can be verified quickly if given a solution, they also have the property that any other problem in NP can be transformed into an NP-complete problem using a polynomial-time reduction. If one NP-complete problem is solvable in polynomial time, then all problems in NP would also be solvable in polynomial time, leading to significant implications for computational theory.
• Evaluate the implications of solving an NP-complete problem efficiently on the broader landscape of computer science and optimization.
  • If a method were found to efficiently solve an NP-complete problem, it would suggest that P = NP, fundamentally altering our approach to many complex optimization tasks. This breakthrough would mean that many currently difficult or impractical problems could suddenly become tractable, changing fields such as cryptography, algorithm design, and operational research. Such an outcome would not only revolutionize theoretical computer science but could also have profound impacts on practical applications across various industries.

Related terms

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

NP-complete: A subset of NP problems that are both in NP and as hard as any problem in NP, meaning if any NP-complete problem can be solved in polynomial time, then every problem in NP can also be solved in polynomial time.

Polynomial time: A time complexity class that represents the computational time taken by an algorithm to solve a problem, which can be expressed as a polynomial function of the size of the input.