Computational Complexity: Delve into the Limits of Algorithms

Question 1

In the context of computational complexity theory, which statement best describes a problem in the P complexity class?

Accepted Answer

It can be solved in polynomial time by a deterministic Turing machine.

Answer

It can be solved in polynomial time by a non-deterministic Turing machine.

Answer

It can be solved in exponential time by a deterministic Turing machine.

Answer

It can be solved in polynomial time by a non-deterministic Turing machine with polynomial space.

Question 2

What is a significant practical implication of NP-completeness?

Accepted Answer

It helps identify inherently difficult problems.

Answer

It provides efficient solutions to all NP-complete problems.

Answer

It proves that all NP-hard problems are NP-complete.

Answer

It has no practical implications.

Question 3

Which of the following is a necessary condition for a problem to be NP-hard?

Accepted Answer

It can be reduced from another NP-hard problem in polynomial time

Answer

It is in NP

Answer

It is NP-complete

Answer

It has a known polynomial-time algorithm

Question 4

Consider a computational problem that can be solved in 2^n time. Which of the following is the best way to classify its complexity?

Accepted Answer

EXP

Answer

P

Answer

NP

Answer

NP-complete

Question 5

Which of the following is an example of a famous unsolved problem in computational complexity theory?

Accepted Answer

The P = NP problem

Answer

Fermat's Last Theorem

Answer

Goldbach's Conjecture

Answer

The Riemann Hypothesis

Question 6

Consider the following problem: Given a set of n points in a plane, find the closest pair of points. Which of the following is the best time complexity achievable for this problem?

Accepted Answer

O(n log n)

Answer

O(n^2)

Answer

O(n)

Answer

O(log n)

Question 7

Which of the following problems is known to be NP-complete?

Accepted Answer

Determining whether a graph is Hamiltonian

Answer

Finding the shortest path in a graph

Answer

Checking if a number is prime

Answer

Sorting a list of numbers

Question 8

What is the time complexity of an algorithm that finds all subsets of a set of size n?

Accepted Answer

O(2^n)

Answer

O(n)

Answer

O(n log n)

Answer

O(n^2)

Question 9

Which of the following is NOT a defining characteristic of an NP problem?

Accepted Answer

It has a polynomial-time algorithm to solve it

Answer

Its solution can be verified in polynomial time

Answer

It can be solved by a non-deterministic Turing machine in polynomial time

Answer

It can be decided in finite time

Question 10

What is the primary proof technique used to establish that a problem is NP-complete?

Accepted Answer

Reduction

Answer

Direct proof

Answer

Proof by contradiction

Answer

Mathematical induction

Question 11

Which of the following is a characteristic of polynomial-time (P) problems?

Accepted Answer

Solvable in polynomial time

Answer

Feasible to solve

Answer

Solvable in exponential time

Answer

Always NP-hard

Question 12

Which of the following is a potential implication of a problem being proven NP-complete?

Accepted Answer

It is unlikely that there exists a polynomial-time algorithm to solve the problem.

Answer

It means that the problem can be solved in linear time.

Answer

It guarantees that there exists a polynomial-time algorithm to solve the problem.

Answer

It suggests that the problem is unsolvable.

Question 13

Which of the following is a limitation of computational complexity theory?

Accepted Answer

It cannot predict the exact time required to solve a specific instance of a problem.

Answer

It only applies to deterministic algorithms.

Answer

It does not provide practical algorithms for solving problems.

Answer

It is only applicable to theoretical problems.

Question 14

Which of the following statements is true about NP-complete problems?

Accepted Answer

They are at least as hard as any problem in NP.

Answer

They can always be solved in exponential time.

Answer

They are only found in the field of theoretical computer science.

Answer

They all have practical applications.

Question 15

What is the significance of the Cook-Levin theorem?

Accepted Answer

It provides a method for proving that a problem is NP-complete.

Answer

It determines the time complexity of an algorithm.

Answer

It generates random instances of NP-complete problems.

Answer

It solves NP-complete problems in polynomial time.

Question 16

True or False: All NP-complete problems have the same time complexity.

Accepted Answer

False

Answer

True

Question 17

Which fundamental class of computational problems, defined as those solvable in polynomial time, is commonly denoted as P?

Accepted Answer

P

Answer

NP

Answer

NP-complete

Answer

PSPACE

Question 18

What is the significance of NP-completeness in computational complexity theory?

Accepted Answer

It classifies problems that are inherently difficult to solve.

Answer

It provides efficient algorithms for solving all NP-complete problems.

Answer

It proves that NP-complete problems are unsolvable.

Answer

It helps estimate the running time of algorithms.

Question 19

What is the fundamental difference between polynomial and exponential time algorithms?

Accepted Answer

Polynomial time algorithms always terminate in a finite number of steps, while exponential time algorithms may not.

Answer

Polynomial time algorithms have a runtime that grows faster than exponential time algorithms.

Answer

Exponential time algorithms have a runtime that grows slower than polynomial time algorithms.

Answer

Exponential time algorithms use more memory than polynomial time algorithms.

Question 20

Which technique is commonly used to prove the NP-completeness of a problem?

Accepted Answer

Polynomial reduction

Answer

Diagonalization

Answer

Mathematical induction

Answer

Simulation

Question 21

What is the practical implication of knowing that a problem is NP-hard?

Accepted Answer

It suggests finding an efficient algorithm for the problem is highly unlikely.

Answer

It means the problem cannot be solved efficiently on any computer.

Answer

It implies that the problem can only be solved for small inputs.

Answer

It guarantees that there is no algorithm that can solve the problem.

Question 22

Consider the problem of determining whether a Hamiltonian path exists in a given graph. What is the current understanding of its complexity?

Accepted Answer

It is an open problem whether it is in P or NP-complete.

Answer

It is known to be NP-complete.

Answer

It is known to be in P.

Answer

It is known to be NP-hard but not NP-complete.

Question 23

Which of the following problems has been proven to be NP-complete and serves as a benchmark for NP-completeness proofs?

Accepted Answer

Satisfiability (SAT)

Answer

Integer factorization

Answer

Prime number generation

Answer

Graph coloring

Question 24

Which of the following complexity classes contains problems that can be solved in polynomial time?

Accepted Answer

P

Answer

NP

Answer

NPC

Answer

EXPTIME

Question 25

Which of the following is a possible implication of the P = NP conjecture?

Accepted Answer

All NP-complete problems have polynomial-time algorithms

Answer

All problems can be solved efficiently

Answer

All P problems can be solved in exponential time

Answer

All NP problems cannot be solved in polynomial time

Question 26

Which of the following is a reduction used to prove that a problem is NP-complete?

Accepted Answer

Polynomial-time reduction

Answer

Exponential-time reduction

Answer

Logarithmic-time reduction

Answer

Constant-time reduction

Question 27

What is the time complexity of the bubble sort algorithm?

Accepted Answer

O(n^2)

Answer

O(n)

Answer

O(n log n)

Answer

O(n^3)

Question 28

Describe the relationship between NP and NP-hard problems.

Accepted Answer

NP-hard problems are a subset of NP problems

Answer

NP problems are a subset of NP-hard problems

Answer

NP and NP-hard problems are disjoint sets

Answer

NP and NP-hard problems are equivalent sets

Question 29

A problem that can be solved in polynomial time using a deterministic Turing machine belongs to the complexity class:

Accepted Answer

P

Answer

NP

Answer

EXP

Answer

NP-complete

Question 30

Select the well-known NP-complete problem from the list:

Accepted Answer

Travelling Salesman Problem

Answer

Primality Testing

Answer

Binary Search

Question 31

Classify the halting problem for Turing machines based on its decidability:

Accepted Answer

Undecidable

Answer

Decidable in polynomial time

Answer

P-complete

Answer

NP-complete

Question 32

What is the implication of a reduction from problem A to problem B?

Accepted Answer

A is at least as hard as B

Answer

A and B have equivalent complexity

Answer

B is at least as hard as A

Question 33

Define computational complexity of a problem:

Accepted Answer

The amount of resources needed to solve the problem

Answer

The level of difficulty in understanding the problem

Answer

The number of steps required to solve the problem

Question 34

What describes the time complexity of an algorithm most accurately?

Accepted Answer

The amount of time an algorithm takes to complete, expressed as a function of the input size.

Answer

The amount of memory used by the algorithm.

Answer

The number of lines of code in the algorithm.

Question 35

What is the time complexity of searching for an element in a sorted array using binary search?

Accepted Answer

O(log n)

Answer

O(n)

Answer

O(n log n)

Answer

O(1)

Question 36

Which of these problems is known to be solvable in polynomial time (i.e., belongs to the P complexity class)?

Accepted Answer

Sorting a list of numbers.

Answer

The Travelling Salesperson Problem.

Answer

Finding the shortest path between two points on a map.

Answer

The Boolean Satisfiability Problem (SAT).

Question 37

What does it mean for a problem to be in NP?

Accepted Answer

A proposed solution can be verified in polynomial time.

Answer

The problem can be solved using a non-deterministic Turing machine.

Answer

No known algorithm can solve the problem in polynomial time.

Answer

A solution can be found in polynomial time.

Question 38

What is a key characteristic of NP-complete problems?

Accepted Answer

If one NP-complete problem can be solved in polynomial time, then all NP problems can be solved in polynomial time.

Answer

All NP-complete problems are in P.

Question 39

In computational complexity, what is a 'reduction'?

Accepted Answer

A method to transform one problem into another problem.

Answer

A technique for solving NP-complete problems.

Answer

A method for proving that a problem is in P.

Question 40

If problem A can be reduced to problem B in polynomial time, and problem B is in P, what can we conclude about problem A?

Accepted Answer

Problem A is also in P.

Answer

Problem A is harder than problem B.

Answer

Problem A is NP-complete.

Answer

We cannot conclude anything about problem A.

Question 41

Which of the following is a common approach to tackle NP-hard problems in practical scenarios?

Accepted Answer

Approximation algorithms.

Answer

Quantum computing.

Answer

Brute-force search.

Answer

Ignoring the problem altogether.

Question 42

What is the significance of the P vs NP problem?

Accepted Answer

It has significant implications for fields like cryptography and optimization.

Answer

It has been definitively proven that P = NP.

Answer

It is purely a theoretical problem with no real-world applications.

Question 43

What is a widely used resource for finding a comprehensive list of known NP-complete problems?

Accepted Answer

Garey and Johnson's book: Computers and Intractability: A Guide to the Theory of NP-Completeness

Answer

The Wikipedia page for 'P versus NP problem'

Answer

Stack Overflow