IntractabilityTestSol

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Jan 9, 2024

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Intractability Test Question 1: NP-Completeness Explain what it means for a problem to be NP-complete. Give an example of an NP-complete problem and briefly describe why it is in NP and why it is NP-hard. Answer: A problem is NP-complete if it is both in NP (non-deterministic polynomial time) and NP-hard (every problem in NP can be reduced to it in polynomial time). An example is the Traveling Salesman Problem (TSP). TSP is in NP because a proposed solution can be verified in polynomial time. It is NP-hard because problems like Hamiltonian Cycle can be polynomially reduced to it. Question 2: Reducibility Describe how reducibility is used to prove that problems are NP-complete. Give an example of a reduction from one NP-complete problem to another. Answer: Reducibility involves showing that a known NP-complete problem can be efficiently transformed into the problem in question. For example, 3-SAT (a known NP-complete problem) can be reduced to the Vertex Cover problem, thus proving Vertex Cover is NP-complete. Question 3: Approximation Algorithms What is an approximation algorithm? Discuss the concept with respect to the Vertex Cover problem and explain how the approximation ratio is determined. Answer: Approximation algorithms are used for NP-hard problems to find near-optimal solutions in polynomial time. For Vertex Cover, a simple 2-approximation algorithm chooses all vertices of edges in a maximal matching. The approximation ratio, the algorithm’s solution cost over the optimal solution cost, is at most 2 for this algorithm. Question 4: P vs NP Problem Explain the significance of the P vs NP problem in computational complexity theory. What are the implications if P = NP? What if P NP? Answer: The P vs NP problem asks whether every problem whose solution can be quickly verified can also be quickly solved. If P = NP, it would mean that complex problems like TSP can be solved in polynomial time. If P NP, it implies that there are problems in NP (like NP-complete problems) that cannot be solved in polynomial time. 1

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