Computational Complexity: A Comprehensive Guide for Computer Science Students

Question 1

Which defining property characterizes NP-complete problems?

Accepted Answer

Any NP problem can be efficiently reduced to them

Answer

Solvable in polynomial time

Answer

Equivalent to P problems

Answer

Solvable using brute-force search algorithms

Question 2

Which of the following best characterizes a problem belonging to complexity class P?

Accepted Answer

It can be solved using an algorithm that runs in polynomial time.

Answer

It can be solved in exponential time.

Answer

It cannot be solved using any algorithm.

Answer

Its time complexity is yet to be determined.

Question 3

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

Accepted Answer

Finding the maximum clique in a graph.

Answer

Finding the shortest path in a weighted graph.

Answer

Determining whether a graph is bipartite.

Answer

Finding the minimum spanning tree of a graph.

Question 4

What is the significance of the Cook-Levin theorem in computational complexity theory?

Accepted Answer

It establishes that NP-hard problems are equivalent to NP-complete problems.

Answer

It proves that P problems cannot be solved in exponential time.

Answer

It implies that NP-complete problems are solvable in polynomial time.

Answer

It provides a method for solving NP-hard problems.

Question 5

Time complexity of an algorithm refers to:

Accepted Answer

The worst-case time required for the algorithm to complete.

Answer

The average time taken by the algorithm to complete.

Answer

The best-case time needed for the algorithm to complete.

Answer

The time required for the algorithm to complete for a specific input.

Question 6

The halting problem in computational complexity theory demonstrates that:

Accepted Answer

There exists no general algorithm that can determine whether a given program will eventually halt.

Answer

All problems can be solved in polynomial time.

Answer

NP-complete problems are equivalent to P problems.

Answer

The field of computational complexity is unrelated to the halting problem.

Question 7

The complexity class 'NP' encompasses problems that:

Accepted Answer

Can be solved using non-deterministic algorithms in polynomial time.

Answer

Can be solved using deterministic algorithms in polynomial time.

Answer

Are NP-complete.

Answer

Are NP-hard.

Question 8

The relationship between complexity classes P, NP, and NP-complete can be accurately represented as:

Accepted Answer

P is a subset of NP, which is a subset of NP-complete.

Answer

NP is a subset of P, which is a subset of NP-complete.

Answer

NP-complete is a subset of P, which is a subset of NP.

Answer

P, NP, and NP-complete are all equivalent.

Question 9

According to the Cook-Levin Theorem, what is a polynomial-time verification algorithm?

Accepted Answer

An algorithm that can quickly check if a given solution to a problem is valid.

Answer

An algorithm that can solve any problem in a short amount of time.

Answer

An algorithm that can simplify any problem into a known NP-complete problem.

Answer

An algorithm that can determine if a problem is NP-complete.

Question 10

Answer Options Refined: Which problem is in the P class?

Accepted Answer

Determining if a graph contains a cycle

Answer

Finding the maximum flow in a network

Answer

Traveling salesman problem

Answer

Subset sum problem

Question 11

Precision Enhanced: What's the relationship between P and NP classes?

Accepted Answer

P is a subset of NP

Answer

NP is a subset of P

Answer

P and NP are disjoint sets

Answer

P and NP are equivalent sets

Question 12

What is the time complexity of the brute-force search algorithm for the Traveling Salesman Problem?

Accepted Answer

O(n!)

Answer

O(n)

Answer

O(n^2)

Answer

O(2^n)

Question 13

Which of the following is NOT a class of problems in computational complexity theory?

Accepted Answer

NPC

Answer

P

Answer

NP

Answer

PSPACE

Question 14

What is the time complexity of a problem that can be solved using a divide-and-conquer approach?

Accepted Answer

O(n log n)

Answer

O(n)

Answer

O(n^2)

Answer

O(2^n)

Question 15

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

Accepted Answer

Polynomial-time reduction

Answer

Diagonalization

Answer

Exhaustive search

Answer

Approximation

Question 16

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

Accepted Answer

They are all exponentially difficult to solve exactly

Answer

They are all solvable in polynomial time

Answer

They can be efficiently solved for large instances

Answer

They are all related to graph theory

Question 17

Consider the following statement: 'If a problem is in NP, then it must also be in P.' Is this statement true or false?

Accepted Answer

False

Answer

True

Answer

It depends on the specific problem

Answer

It is unknown

Question 18

Which of the following techniques is commonly utilized to obtain approximate solutions for NP-hard problems?

Accepted Answer

Approximation algorithms

Answer

Dynamic programming

Answer

Divide and conquer

Answer

Greedy algorithms

Question 19

Define the differences between time complexity and space complexity, providing specific algorithm examples that demonstrate these complexities.

Accepted Answer

Time complexity measures the number of steps an algorithm takes, while space complexity measures the amount of memory it requires.

Answer

Time complexity measures the memory usage, while space complexity measures the steps taken by an algorithm.

Answer

Both time and space complexity measure the number of steps an algorithm takes.

Answer

Both time and space complexity measure the memory requirement of an algorithm.

Question 20

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

Accepted Answer

Traveling Salesman Problem

Answer

Primality Testing

Answer

Matrix Multiplication

Answer

Sorting an array in ascending order

Question 21

Which of the following is a famous unsolved problem in computer science?

Accepted Answer

P = NP?

Answer

Fermat's Last Theorem

Answer

Collatz Conjecture

Answer

Goldbach's Conjecture

Question 22

What is the significance of the Cook-Levin theorem?

Accepted Answer

It established the existence of NP-complete problems

Answer

It proved that P = NP

Answer

It introduced the concept of reducibility

Answer

It developed the first polynomial-time algorithm

Question 23

Which of the following is an example of an NP-hard problem?

Accepted Answer

Knapsack Problem

Answer

Binary Search

Answer

Bubble Sort

Answer

Selection Sort

Question 24

What is the difference between asymptotic analysis and exact complexity analysis?

Accepted Answer

Asymptotic analysis provides an approximation of complexity while exact analysis provides a precise value

Answer

Exact analysis is more difficult than asymptotic analysis

Answer

Asymptotic analysis is more useful in practice

Answer

There is no difference between these two types of analysis

Question 25

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

Accepted Answer

Reducibility

Answer

Induction

Answer

Exhaustive search

Answer

Randomized algorithms

Question 26

What are the implications of a problem being NP-hard?

Accepted Answer

It is unlikely that an efficient algorithm can be found

Answer

It means the problem is impossible to solve

Answer

It is always possible to find an approximate solution

Answer

It is only difficult to solve for large problem sizes

Question 27

Which of the following problems has been proven to be NP-complete?

Accepted Answer

Traveling Salesman Problem

Answer

Bubble Sort

Answer

Binary Search

Answer

Finding prime numbers

Question 28

Which of the following is widely recognized as an unsolved problem in computer science?

Accepted Answer

The P versus NP problem

Answer

The Collatz conjecture

Answer

The Fermat's Last Theorem

Answer

The Twin Prime conjecture

Question 29

How does the NP-completeness of a problem affect the performance and behavior of optimization algorithms that seek to find optimal solutions for that problem?

Accepted Answer

NP-complete problems have no known efficient algorithms for finding exact solutions.

Answer

NP-complete problems can be solved efficiently in polynomial time by optimization algorithms.

Answer

Optimization algorithms always find optimal solutions for NP-complete problems.

Answer

Optimization algorithms can solve NP-complete problems in exponential time using brute-force approaches.

Question 30

Which complexity class represents the set of decision problems that can be solved by a deterministic Turing machine in polynomial time?

Accepted Answer

P

Answer

NP

Answer

NP-complete

Question 31

Which complexity class represents the set of decision problems that can be solved by a non-deterministic Turing machine in polynomial time?

Accepted Answer

NP

Answer

NP-complete

Answer

NP-hard

Answer

P

Question 32

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

Accepted Answer

Traveling Salesman Problem

Answer

Finding the maximum flow in a network

Answer

Sorting a list of numbers

Answer

Finding the shortest path in a graph

Question 33

Which of the following statements about NP-hard problems is accurate?

Accepted Answer

They are at least as hard as NP-complete problems

Answer

They can be solved in polynomial time

Answer

They are always solvable by a deterministic Turing machine

Answer

They are equivalent to NP-complete problems

Question 34

What is a potential consequence of a problem being NP-complete?

Accepted Answer

It may be difficult or impossible to find an efficient algorithm to solve the problem exactly

Answer

The problem can be solved quickly using a quantum computer

Answer

The problem is always solvable in exponential time

Question 35

Which problem is believed to be NP-complete, but has not been proven?

Accepted Answer

P versus NP problem

Answer

Vertex Cover Problem

Answer

Knapsack Problem

Answer

Traveling Salesman Problem

Question 36

Which technique involves breaking down a problem into smaller subproblems to reduce its complexity?

Accepted Answer

Divide and conquer

Answer

Breadth-first search

Answer

Backtracking

Answer

Dynamic programming

Question 37

Which aspect of an algorithm measures its efficiency in terms of the time it takes to complete?

Accepted Answer

Time complexity

Answer

Input size

Answer

Space complexity

Answer

Output size

Question 38

What is a common notation used to represent the time complexity of an algorithm?

Accepted Answer

Big O notation

Answer

Pseudocode

Answer

State diagram

Answer

Flowchart

Question 39

Which of the following is the best definition of the time complexity of an algorithm?

Accepted Answer

The asymptotic relationship between the input size and the number of operations the algorithm performs.

Answer

The time it takes for the algorithm to run on a specific computer.

Answer

The average number of operations the algorithm performs across all possible inputs.

Question 40

What is the time complexity of the binary search algorithm, which searches for an element in a sorted array?

Accepted Answer

O(log n)

Answer

O(n log n)

Answer

O(1)

Answer

O(n)

Question 41

Which complexity class contains problems that can be solved in polynomial time by a deterministic Turing machine?

Accepted Answer

P

Answer

NP-complete

Answer

NP-hard

Answer

NP

Question 42

Which of the following is NOT known to be an NP-complete problem?

Accepted Answer

Sorting a list of numbers in ascending order

Answer

Graph Coloring Problem

Answer

Traveling Salesperson Problem

Answer

Boolean Satisfiability Problem (SAT)

Question 43

What is the significance of reducing one problem to another in computational complexity theory?

Accepted Answer

It implies that if the second problem is solvable in polynomial time, then the first problem is also solvable in polynomial time.

Answer

It means that the two problems are equivalent and have the same solutions.

Question 44

What is the primary purpose of using Big-O notation in computational complexity?

Accepted Answer

To describe the asymptotic behavior of algorithms, focusing on their worst-case complexity.

Answer

To calculate the exact number of operations an algorithm will perform for any given input.

Answer

To provide a precise measure of the algorithm's efficiency in all cases.

Question 45

An algorithm has a time complexity of O(n^2). If the input size is doubled, how will the running time likely be affected?

Accepted Answer

Quadrupled

Answer

Doubled

Answer

Remain unchanged

Answer

Increased by a constant factor

Question 46

What is a fundamental distinction between deterministic and non-deterministic algorithms?

Accepted Answer

Deterministic algorithms always produce the same output for the same input, while non-deterministic algorithms may not.

Answer

Deterministic algorithms are always more efficient than non-deterministic algorithms.

Question 47

Why is the study of computational complexity considered important in computer science?

Accepted Answer

It helps us understand the inherent limitations of computation and guides the development of efficient algorithms.

Answer

It allows us to compare the performance of different programming languages.

Question 48

Which of the following problems is **guaranteed** to be solvable in polynomial time and is therefore in the class P?

Accepted Answer

Finding the shortest path between two points in a graph using Dijkstra's algorithm

Answer

Solving the Traveling Salesperson Problem to optimality

Answer

Determining if a graph has a Hamiltonian cycle

Question 49

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

Accepted Answer

If a polynomial-time algorithm is found for any NP-complete problem, it implies that all NP-complete problems can be solved in polynomial time.

Answer

NP-complete problems are known to be solvable in polynomial time.

Question 50

What is the **essence** of the 'P vs. NP' problem?

Accepted Answer

It asks whether every problem whose solution can be **verified** in polynomial time can also be **solved** in polynomial time.

Answer

It determines the speed of the fastest algorithm for a specific problem.

Question 51

Which of the following is **not** a characteristic of a problem in the class NP?

Accepted Answer

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

Answer

It can be verified in polynomial time.

Answer

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

Question 52

A reduction from problem A to problem B essentially means that:

Accepted Answer

If we have an efficient algorithm for problem B, we can use it to efficiently solve problem A as well.

Answer

Problem A is always more difficult than problem B.

Answer

Problem A and problem B are equally difficult to solve.

Question 53

Which of the following is a **well-known** NP-complete problem?

Accepted Answer

The Boolean Satisfiability Problem (SAT)

Answer

Sorting a list of numbers in ascending order

Answer

Calculating the factorial of a number

Answer

Finding the greatest common divisor of two integers

Question 54

What is the significance of the 'Cook-Levin Theorem' in the context of computational complexity?

Accepted Answer

It proved that the SAT problem is NP-complete, implying that any problem in NP can be reduced to SAT.

Answer

It defined the complexity classes P and NP.

Question 55

Which of these scenarios would **most likely** result in an exponential increase in the runtime of an algorithm?

Accepted Answer

A recursive algorithm that explores all possible combinations of elements in a set.

Answer

A linear search algorithm on a sorted array.

Answer

A binary search algorithm on a sorted array.

Question 56

What is the **fundamental** difference between 'P' and 'NP'?

Accepted Answer

'P' consists of problems solvable in polynomial time, while 'NP' consists of problems whose solutions can be verified in polynomial time.

Answer

'P' is a subset of 'NP', meaning all problems in 'P' are also in 'NP'.

Question 57

Which of the following is a **viable** approach for handling an NP-complete problem when an exact solution is not feasible?

Accepted Answer

Developing an approximation algorithm that provides a reasonable solution in polynomial time.

Answer

Using brute force to try all possible solutions.

Answer

Waiting for quantum computers to become powerful enough to solve them efficiently.