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- We mentioned that if we want to prove P = NP, we only need to pick up any one NPC problem and design a polynomial-time algorithm for the problem. If you want to prove P = NP, select one NPC problem based on your preference and describe your idea of a polynomial-time algorithm that solves the problem. It does not have to be a formal algorithm or pseudo-code, a description of your idea of designing such an algorithm would be fine.Suppose Professor Weise described a deterministic algorithm that runs in polynomial time for a known NP-Complete problem such as the Satisfiability problem. How would this impact the class NP (the class of problems that can be solved nondeterministically in polynomial time)? OL The class P would be strictly larger than the class NP Ⓒ. The class NP would be equal to the class P O.It would only impact P but not NP IV. It would have no impact on the class NPA Norman window has the shape of a rectangle surmounted by a semicircle. Suppose the outer perimeter of such a window must be 600 cm. In this problem you will find the base length x which will maximize the area of such a window. Use calculus to find an exact answer. When the base length is zero, the area of the window will be zero. There is also a limb on how large x can her when x is large enough, the rectangular portion of the window shrinks down to zero height. What is the exact largest value of x when this occurs?
- In computational complexity theory, we study mostly decision problems (e.g., determine if aBoolean formula has a satisfying assignment), instead of the search problems that are typicallywhat practicing computer scientists really want to solve (e.g., determine if a Boolean formula hasa satisfying assignment, and then output the assignment). It is natural to wonder whether we arestudying the wrong thing. In this exercise, you will show that the difficulty of decision problems isin fact linked closely to that of search problems.Show that if P = NP, then every NP search problem can be solved in polynomial time. In otherwords, for each language A ∈ NP, with a polynomial-time verifier algorithm VA taking inputs x, w,where x ∈ A ⇐⇒ (∃w) VA(x, w) accepts, then there is a polynomial-time algorithm S that, oninput x, does the following. If x ̸∈ A, then S(x) outputs “no”. If x ∈ A, then S(x) outputs w suchthat VA(x, w) accepts.Hint: Build up the witness w bit by bit, by asking NP questions about…Wireless sensor networks are a special kind of network that facilitates communication. Sensor nodes in WSNs relay information to and from a central base station. The computing resources and storage space of a sensor node are limited. Take, for example, an algorithm that can be solved by solving subproblems. Is it preferable to use dynamic programming or divide and conquer to solve these subproblems independently at various sensor nodes? Try to keep your writing short and sweet.Suppose Professor Weine described a deterministic algorithm that runs in polynomial time for a known NP-Complete problem such as the Satisfiability problem How would this impact the class NP (the class of problems that can be solved nondeterministically in polynomial time)? 0. The class P would be strictly larger than the class NP 0. The class NP would be equal to the class P O.It would only impact P but not NP O.It would have no impact on the class NP
- If we want to prove P = NP, we only need to pick up any one NPC problem and design a polynomial-time algorithm for the problem. If you want to prove P = NP, select one NPC problem based on your preference and describe your idea of a polynomial-time algorithm that solves the problem. It does not have to be a formal algorithm or pseudo-code, a description of your idea of designing such an algorithm would be fine.Imagine that you have a problem P that you know is N P-complete. For this problem you have two algorithms to solve it. For each algorithm, some problem instances of P run in polynomial time and others run in exponential time (there are lots of heuristic-based algorithms for real N P-complete problems with this behavior). You can’t tell beforehand for any given problem instance whether it will run in polynomial or exponential time on either algorithm. However, you do know that for every problem instance, at least one of the two algorithms will solve it in polynomial time. (a) What should you do? (b) What is the running time of your solution? 564 Chap. 17 Limits to Computation (c) What does it say about the question of P = N P if the conditions described in this problem existed?Explain the difference between Machine Dependent and In-Dependent Optimization by examining the Replacement Rules. Discuss the Operator Strength Reduction as well.
- In the context of evolutionary computing the goal function is known as the fitnessfunction and the problem is to maximize it. The typical formulation has to be changedin a simple way.min f (x) = − max[− f (x)] (4.9)Another requirement is that the goal function is positive.Phenotype evolution treats x as a phenotype and the goal function as the fitnessfunction. The typical framework for the method is as follows:Correct answer will be upvoted else downvoted. Computer science. You are given two integers n and k. You ought to make a variety of n positive integers a1,a2,… ,a to such an extent that the total (a1+a2+⋯+an) is distinguishable by k and greatest component in an is least conceivable. What is the base conceivable most extreme component in a? Input The primary line contains a solitary integer t (1≤t≤1000) — the number of experiments. The solitary line of each experiment contains two integers n and k (1≤n≤109; 1≤k≤109). Output For each experiment, print one integer — the base conceivable most extreme component in cluster a to such an extent that the aggregate (a1+⋯+an) is distinct by k.If five integers are chosen from the set {1, 2, 3, 4, 5, 6, 7, 8}, must there be at least two integers with the property that the larger minus the smaller is 2? Write an answer that would convince a good but skeptical fellow student who has learned the statement of the pigeonhole principle but not seen an application like this one. Describe the pigeons, the pigeonholes, and how the pigeons get to the pigeonholes.