Computer scientists arrange problems solved by computers into categori

Question

Computer scientists arrange problems solved by computers into categories based on the time it takes to solve them.

The time that it takes to solve a polynomial-time problem increases by a constant multiplier each time the size of the problem doubles. For instance, a polynomial time problem might take 5,000 calculations to optimize choices from a selection of 15 objects. Doubling the number of objects to 30 could increase the number of calculations by a factor of 5 to 25,000 calculations. This pattern would continue, with 60 objects requiring 125,000 calculations.

Problems that are solvable in the more complex exponential time increase in complexity by a constant multiplier with each additional item in the problem. If a computer could solve a 30-object exponential problem in 1 hour, adding a 31st object could increase the time required to 2 hours, and a 40-object version would then take that computer 1,024 hours—more than 6 weeks.NP-complete problems are a special class of mathematical puzzles. An NP-complete problem can be converted efficiently to any other NPcomplete problem. It is generally believed, but not proven, that NPcomplete problems require exponential time to solve. A polynomial-time solution for any (and thus every) NP-complete problem would revolutionize mathematics and computer science.A  Traveling Salesman Problem (TSP)  involves connecting multiple points with a single continuous path of the shortest possible distance, without backtracking. TSP's namesake is a common real-life scenario: a sales representative must visit every city on a list while minimizing the distance traveled. The TSP is NP-complete, but certain heuristics can quickly produce routes believed to be within 2–3 percent of the actual minimum.

A  Bounded Knapsack Problem (BKP)  involves selecting the most valuable set of items, constrained by a weight limit, from among a collection of items with different values and weights. The problem's name comes from campers, who must optimize the "value" of their camping equipment within the limits of their knapsacks' capacities. Finding an exact solution to the BKP is NP-complete.

The  Greatest Common Divisor (GCD)  of a set of numbers is the greatest integer that divides into each term of the set without remainder. Other problems involving factoring are notoriously difficult, but finding a common divisor is comparatively easy. The euclidean algorithm finds the GCD in polynomial time.

asanhu · Accepted Answer

On question 1: 
A) NO. First look at tab 3, it mentions "finding an exact solution to the BKP is NP-complete"; Then comes to tab 2 where it presents the characteristics of NP-complete "a polynomial-time solution for any NP-complete problem would be revolutionary", which indicates that NP-complete is not essentially a "polynomial-time" problem, and only "polynomial-time" problem has the features mentioned in question A.
B) YES. First look at tab 3, it mentions "finding an exact solution to the BKP is NP-complete"; Then comes to tab 2 where it presents the characteristics of NP-complete "a polynomial-time solution for any NP-complete problem would be revolutionary".
C) YES. First locate "heuristic" in tab 3's TSP, which is a NP-complete, but certain heuristics can quickly produce approximate solution, and since "finding an exact solution to the BKP is NP-complete", thus, heuristic methods can potentially find an approximate answer (but not exact) to BKP.

On question 2:
We need to determine what is the essence of this "microchip" problem, by looking at " in the minimum distance possible", we can infer that this problem is a derivative of the TSP. And the characteristics of TSP is "using certain heuristics can quickly produce approximate answers".

On question 3：
The question says "assuming the current beliefs about NP-complete problems are true", meaning we can use "polynomial-time solution" to solve the NP-complete problem. Now let's go to tab 1 to find out the method of time calculation. 
"The time that it takes to solve this problem increases by a constant multiplier each time the size of the problem doubles".
Now we need to find out the constant multiplier each time when we add one object into the BKP.
10 objects - 1 min, 11 object - 3 min, so the constant multiplier is 2 each time when we add one object in.
So 12 object = 3*3=9 min, 13 object = 9*3=27min

andreagonzalez2k · Answer

1- No, No, Yes
2- C
3- C, 27 = 3^3

Sajjad1994 · Answer

OA of this question has been updated, It is:

Q.1

No, No and Yes

Q.2

C

Q.3

C

mjsgmat · Answer

In Problem 2, since we can say the scenario is similar to TSP. Can't we say that the problem can be reduced to GCD as well? Since one NP can be converted to other?

Why isn't (ii) option correct?

Sajjad1994 · Answer

Official Explanation

2. A microchip manufacturer wants a continuous electronic circuit to pass once through each semiconductor device on a chip in the minimum distance possible. According to the tabbed information, which of the following is true of the problem faced by the manufacturer?

Explanation

The problem appears to be similar to the TSP. We are told the TSP can be approximated closely by using efficient heuristics. Thus, we can infer that heuristics would also be able to give an efficient approximation for this problem, so  choice (C)  is correct. Answer  choice (A)  is too extreme. A polynomial solution would be revolutionary, but the information given does not suggest it is impossible. The other answer choices are all outside the scope of the information given; there is no way to know that any of them are true based on this information.

Answer: C

SamMenon · Answer

For Question 1 - How can we be sure that heuristics can potentially produce solution to BKP ? It can be true for TSP. Can some one help out?

DmitryFarberMPrep · Answer

The key is in the 2nd tab: " An NP-complete problem can be converted efficiently to any other NP-complete problem." Since the BKP and the TSP are both NP-complete, one can be converted to the other. So any solution for the TSP would also work for the BKP.

kartickdey · Answer

In Q 1.3 How can we be sure that heuristics can potentially produce solution to BKP ? It can be true for TSP but not for BKP

Bunuel · Answer

Your doubt is addressed here: https://gmatclub.com/forum/computer-sci ... l#p3614527

Yes	No
		Doubling the number of items in the problem increases the processing time to find the exact solution by a constant multiplier.
		An exponential-time solution to this problem, if discovered, would be revolutionary.
		A heuristic method could potentially produce an approximate solution to this problem.

	It is mathematically impossible to find a solution in polynomial time.
	The problem can be mathematically reduced to finding a GCD.
	There are efficient ways to get close approximations to the actual minimum solution.
	The manufacturer cannot solve the problem without the help of expert mathematicians or computer theoreticians.
	Adding 10 more semiconducting devices to the chip would make the problem impossible to solve in a timely manner.

	7 minutes
	9 minutes
	27 minutes
	54 minutes
	81 minutes

Computer scientists arrange problems solved by computers into categori

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Computer scientists arrange problems solved by computers into categori

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