Feasibility

Definition

Feasibility refers to the practicality of solving a problem within a given complexity class, indicating whether a problem can be solved efficiently, often within polynomial time. This concept is essential for distinguishing between different classes of problems, particularly in terms of their computational resource requirements and the likelihood of finding efficient algorithms. Understanding feasibility helps in classifying problems as solvable or unsolvable under certain conditions, influencing how we approach problem-solving in computer science.

5 Must Know Facts For Your Next Test

1. Feasibility directly impacts the classification of problems into categories like P and NP, with P problems being feasible and solvable in polynomial time.
2. NP problems are those for which solutions can be verified quickly, but it remains unknown if they can all be solved quickly, raising questions about their feasibility.
3. NP-complete problems are considered the hardest problems in NP; if one can find a feasible solution for one NP-complete problem, all NP problems can potentially be solved feasibly.
4. NP-hard problems are not necessarily in NP; they may be harder than any NP problem, thus complicating discussions about their feasibility.
5. Determining feasibility is crucial for algorithm design, as it helps prioritize which problems to tackle with existing methods and which may require new approaches.

Review Questions

• How does feasibility influence the classification of problems within different complexity classes?
  • Feasibility plays a vital role in how problems are categorized into complexity classes such as P, NP, NP-complete, and NP-hard. Problems classified as P are deemed feasible since they can be solved in polynomial time, while NP problems may have solutions that can be verified quickly but are not guaranteed to be solvable efficiently. Understanding feasibility helps identify which problems can realistically be addressed with existing algorithms and shapes our expectations regarding computational limits.
• Compare and contrast NP-complete and NP-hard problems in relation to their feasibility.
  • NP-complete problems are characterized by their feasibility in terms of having verifiable solutions within polynomial time, yet no known polynomial-time solution exists for them. Conversely, NP-hard problems do not have this verification property and may even exceed NP problem complexity, meaning they could require more than polynomial time to solve. The distinction is crucial because if any NP-complete problem can be solved feasibly, it implies that all NP problems could also be feasibly solved; however, this relationship does not hold for NP-hard problems.
• Evaluate the implications of proving that an NP-complete problem is feasible on the broader landscape of computational theory.
  • If an NP-complete problem were proven to be feasible by finding a polynomial-time algorithm for its solution, it would have profound implications across computational theory. Such a breakthrough would imply that all problems in NP could also be solved feasibly, fundamentally changing our understanding of computational limits. It would blur the lines between tractable and intractable problems and potentially revolutionize fields relying on complex computations, leading to widespread advancements in technology and algorithm design.

Related terms

Polynomial Time: A class of computational problems that can be solved by an algorithm whose running time is a polynomial function of the size of the input.

Exponential Time:

A class of problems for which the time required to solve them grows exponentially with the input size, making them impractical for large inputs.

Decision Problem:

A problem that can be posed as a yes/no question regarding an input, often used to analyze computational complexity and feasibility.