There are a set of courses, each of them requiring a set of disjoint time intervals. For example, a course could require the time from 9am to 11am and 2pm to 3pm and 4pm to 5pm. You want to know, given a number K, if it’s possible to take at least K courses. You can only take one course at any single point in time (i.e. any two courses you choose can’t overlap). Show that the problem is NP-complete, which means that choosing courses is indeed a difficult thing in our life. Use a reduction from the Independent set problem. There are a set of courses in USC, each of them requiring a set of disjointtime intervals. For example, a course could require the time from 9amto 11am and 2pm to 3pm and 4pm to 5pm. You want to know, given anumber K, if it's possible to take at least K courses. Since you want tostudy hard and take courses carefully, you can only take one course at anysingle point in time (i.e. any two courses you choose can't overlap). Showthat the problem is NP-complete, which means that choosing courses isindeed a difficult thing in our life. Use a reduction from the Independentset problem.

The main goal of the course selection problem is to find out whether it is possible to enrol in a specific number of courses without any time overlap. Each course is represented by a series of non-overlapping time intervals and the goal is to determine whether it is possible to take at least K courses.Here, we will prove that the course selection problem is NP-complete, indicating its computational difficulty. In order to do this, we will create a polynomial reduction from the well-known Independent Set problem to the course selection problem and demonstrate that the problem belongs to the class of problems verifiable in non-deterministic polynomial time (NP).Course selection problemNP : We are able to prove that the course selection problem is verifiable in NP. We can iterate through the classes in a list of courses to find out in polynomial time whether the set of classes contains overlapping intervals, and we can also check if the number of courses is equal to K.Course selection problem NP-Hard : We can perform a polynomial reduction from the independent set problem to the course selection problem in order to prove that the problem is NP-Hard.Polynomial mapping: IS Course selection (where IS is an independent set)The mapping involves building a graph with N vertices, where N courses are total, and each vertex represents a course. An edge is drawn between the vertices of two courses if there is a time overlap. By allocating time intervals according to these edges, schedule overlap is ensured for courses that share an edge. The maximum set of classes with non-overlapping schedules is represented by the independent set in this graph.Claim: An independent set has a size of k if and only if there are K courses that can be taken at the same time. Proof:=> Given an independent set with a size of K, we can conclude that K courses can be selected. When courses are part of the independent set, it means there are no edges between them, which means that there are no time overlaps.=> An independent set of size K can be obtained if we are given a set of K courses that can be taken simultaneously. Since the schedules of these classes do not overlap, none of them will have any edges connecting them, creating an independent set with K vertices.The provided answer shows that the course selection problem is NP-complete since it is verifiable in both NP and NP-Hard. We prove that the course selection problem is in NP and show that it is verifiable in polynomial time. Also, a polynomial mapping from the independent set problem is used to demonstrate its NP-hardness, indicating the difficulty in figuring out how many courses can be filled without causing schedule overlaps.

Engineering

Data Structures and Algorithms

There are a set of courses, each of them requiring a set of disjoint time intervals. For example, a course could require the time from 9am to 11am and 2pm to 3pm and 4pm to 5pm. You want to know, given a number K, if it’s possible to take at least K courses. You can only take one course at any single point in time (i.e. any two courses you choose can’t overlap). Show that the problem is NP-complete, which means that choosing courses is indeed a difficult thing in our life. Use a reduction from the Independent set problem.

There are a set of courses, each of them requiring a set of disjoint time intervals. For example, a course could require the time from 9am to 11am and 2pm to 3pm and 4pm to 5pm. You want to know, given a number K, if it’s possible to take at least K courses. You can only take one course at any single point in time (i.e. any two courses you choose can’t overlap). Show that the problem is NP-complete, which means that choosing courses is indeed a difficult thing in our life. Use a reduction from the Independent set problem.

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