P NP NP-Hard NP-Complete problems || P Versus NP || Relationship between P NP & NP Complete Problems

Summary

This lecture dives into computational complexity theory, explaining the classification of problems into polynomial-time (P class) and exponential-time (NP class) problems. It distinguishes between deterministic algorithms (used for P class problems) and non-deterministic algorithms (for NP class problems). The lecture also explores NP-complete and NP-hard problems, highlighting their significance in computational theory. It concludes with Cook's Theorem, which proves that P equals NP, suggesting that problems solvable in polynomial time can also be verified in polynomial time. This essential concept helps frame the challenges in algorithmic efficiency and problem-solving.

Takeaways

• 😀 Polynomial time problems can be solved in deterministic algorithms, while non-polynomial time problems are solved with non-deterministic algorithms.
• 😀 P Class problems are solvable in polynomial time using deterministic algorithms, making them computationally efficient.
• 😀 NP Class problems are solvable in polynomial time using non-deterministic algorithms, but they are harder to solve compared to P problems.
• 😀 An example of a polynomial time problem is searching for an element in an array using binary search (O(log n) time complexity).
• 😀 Exponential time problems require more time to solve and include examples like the traveling salesperson problem and graph coloring problem.
• 😀 Deterministic algorithms always produce the same output for a given input, while non-deterministic algorithms have multiple possible outputs for the same input.
• 😀 NP-Complete problems are those that are both in NP and can be reduced to other NP problems in polynomial time.
• 😀 NP-Hard problems are as difficult as NP-Complete problems but may not belong to NP, and solving them requires more complex approaches.
• 😀 Cook’s Theorem proves that the satisfiability problem is NP-Complete and suggests that P = NP if NP-Complete problems can be solved efficiently.
• 😀 The relationship between P and NP is crucial in understanding computational complexity, and Cook’s theorem plays a key role in this theory.
• 😀 Problems classified as NP-Complete are both in NP and can be polynomially reduced to each other, which is significant in the P vs NP debate.

Q & A

What are the two groups of problems based on the time complexity of their solutions?

The two groups of problems are 'Polynomial time problems' and 'Exponential time problems'. Polynomial time problems can be solved in polynomial time using deterministic algorithms, while exponential problems are solved in non-polynomial time using non-deterministic algorithms.

What are some examples of polynomial time problems?

Examples of polynomial time problems include searching an element in an array (Big O of log n using binary search), matrix addition (Big O of n^2), and the all-pair shortest path problem (Big O of n^3).

What is the difference between deterministic and non-deterministic algorithms?

A deterministic algorithm has a uniquely defined operation for each step, meaning given a particular input, it will always produce the same result. A non-deterministic algorithm, on the other hand, has multiple possible operations or outcomes for a given input.

What is the time complexity of the traveling salesperson problem?

The time complexity of the traveling salesperson problem is O(n^2 * 2^n), which makes it an example of an exponential problem.

What does the term 'NP-complete' refer to?

A problem is NP-complete if it belongs to the NP class (problems solvable in non-polynomial time by non-deterministic algorithms) and every other problem in NP can be polynomial-time reducible to it. In simpler terms, NP-complete problems are considered the hardest problems within NP.

What is the relationship between P and NP class problems?

The P class contains problems that can be solved in polynomial time using deterministic algorithms, while NP class problems can be solved in polynomial time using non-deterministic algorithms. Every problem in P is also in NP, but not all NP problems are in P.

What are NP-hard problems?

NP-hard problems are those for which a solution to any NP problem can be reduced to the NP-hard problem in polynomial time. These problems are at least as difficult as the hardest problems in NP, but they are not necessarily in NP themselves.

Can NP-complete problems be solved in polynomial time?

NP-complete problems are not known to have polynomial-time solutions, but they are verifiable in polynomial time. If a polynomial-time solution were found for any NP-complete problem, it would imply that P = NP.

What is the significance of Cook's Theorem?

Cook's Theorem proves that the satisfiability problem (SAT) is NP-complete. This was the first problem shown to be NP-complete, and it established that any problem in NP can be reduced to SAT in polynomial time.

What are some examples of NP-complete problems?

Examples of NP-complete problems include the Hamiltonian cycle problem and the satisfiability problem (SAT). These problems are in NP, and other problems in NP can be reduced to them in polynomial time.