Question

For each of the two questions below, decide whether the answer is (i) "Yes," (ii) "No," or (iii) "Unknown, because it would resolve the question of whether P = NP." Give a brief explanation of your answer. (a) Let’s define the decision version of the Interval Scheduling Prob- lem from Chapter 4 as follows: Given a collection of intervals on a time-line, and a bound k, does the collection contain a subset of nonoverlapping intervals of size at least k? Question: Is it the case that Interval Scheduling ≤P Vertex Cover? (b) Question: Is it the case that Independent Set ≤P Interval Scheduling?

Answer 1

The original student question pertains to computational complexity and the relationship between different problems within the NP complexity class, seeking to understand polynomial-time reductions between Interval Scheduling, Vertex Cover, and Independent Set.

Step-by-step explanation:

The questions provided revolve around computational complexity, particularly the concepts surrounding the complexity classes P, NP, and the relations between different computational problems. Unfortunately, the rest of the information provided is not related to the P vs NP question raised and therefore not relevant for forming an answer to the question at hand. The original question seems to ask whether 'Interval Scheduling' can be polynomial-time reduced to 'Vertex Cover' (Interval Scheduling ≤P Vertex Cover), and whether 'Independent Set' can be polynomial-time reduced to 'Interval Scheduling' (Independent Set ≤P Interval Scheduling). These questions are seeking to understand the computational complexity relationship between different NP problems.

Answer 2

Check the explanation

Step-by-step explanation:

a) The answer B yes

• Because the greedy algorithm has used to perform the calculation of interval scheduling problem such as 0(n log n)

• In this interval scheduling process can be am in polynomial time without use the block box solves the vertex cover problem

• such that interval scheduling algorithm has solved o the polynomial computation process steps and add a polynomial number of call so black box test solvers of vertex cover

Sp therefore Interval Scheduling ≤P Vertex Cover

B)

The answer is unknown because it resolve the question has P=NP Proof:

• Independent set ≤P interval scheduling

• such that Y<PX value can M solved to the NP time then it can M solved y value in polynomial value

• use the interval schedule process can be solve polynomial time

• such the independent set(IS) has to solve the polynomial time process_ therefore independent set is NP complete process

• If and only if condition has P=NP to polynomial time of x solvable

• use P=NP has the IS is NP

• If P=NP has independent set SP interval scheduling

• Because independent set can M solved to NP to the polynomial number of process call have to be test in black box to solve the interval schedule.