Algorithms: P and NP

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Questions and Answers

What form of problem is addressed for simplification within the context of the material?

  • Counting problems where the number of solutions is required.
  • Regression problems estimating a continuous output.
  • Optimization problems where the best solution is sought.
  • Decision problems with a yes/no answer. (correct)

If an algorithm is created to solve a problem. What measure is used when assessing the algorithm's efficiency?

  • The worst-case time complexity. (correct)
  • The best-case time complexity.
  • The average-case time complexity.
  • The likelihood of finding any solution.

What does 'n' typically represent when analyzing the time complexity of an algorithm?

  • The number of processors used to run the algorithm.
  • The number of lines of code in the algorithm.
  • The size of the input to the algorithm. (correct)
  • The algorithm's running time in seconds.

A problem is classified as P if what condition is met?

<p>It can be solved by an algorithm whose worst-case time complexity is $O(n^c)$ where c is a constant. (D)</p> Signup and view all the answers

How does the Euler path problem differ from the Hamiltonian cycle problem in terms of complexity?

<p>The Euler path problem can be solved in polynomial time, whereas finding a Hamiltonian cycle is an NP problem. (C)</p> Signup and view all the answers

Given a network G = (V, E) with 'n' nodes and $O(n^2)$ edges, what is the primary challenge in finding a Hamiltonian cycle compared to an Euler cycle?

<p>Hamiltonian cycles involve visiting each node exactly once, and the straightforward approach of checking all permutations is inefficient. (A)</p> Signup and view all the answers

When determining if a number N is prime, why is the time complexity not truly polynomial based on the input size?

<p>Because the input size is measured in bits: the number of bits required to represent <em>N</em> is logarithmic, while the time complexity is related to $\sqrt{N}$ which is exponential in the number of bits. (A)</p> Signup and view all the answers

If a problem is in NP, what key characteristic does it possess?

<p>If the solution to the problem is 'yes', then there exists a proof that can be verified in polynomial time. (B)</p> Signup and view all the answers

What characterizes a co-NP problem?

<p>It is the complement of a problem in NP; if the answer is 'no,' there exists a proof that can be verified in polynomial time. (A)</p> Signup and view all the answers

What is the relationship between the complexity classes P and NP?

<p>P is a subset of NP; all problems in P are also in NP. (C)</p> Signup and view all the answers

According to the context, what condition allows a problem 'A' to belong to the complexity class NP?

<p>If, and only if, a solution to 'A' can be verified in polynomial time. (A)</p> Signup and view all the answers

Among the complexity classes discussed (P, NP, co-NP), which statement accurately describes the relationship between them?

<p>If a problem is in P, it is also in NP. (C)</p> Signup and view all the answers

What is the significance of demonstrating that a problem is in both NP and co-NP?

<p>It suggests that the problem may have a polynomial-time solution, placing it in P. (B)</p> Signup and view all the answers

What is the primary purpose of the CircuitSAT problem in complexity theory?

<p>To test whether an assignment of boolean values to the inputs of a given logic circuit results in a specific output (typically 1). (B)</p> Signup and view all the answers

What is 'worst-case time' complexity?

<p>The longest amount of time that an algorithm could take to complete execution given any possible inputs. (C)</p> Signup and view all the answers

Which of the following is an example of a problem characterized as P?

<p>Determining the shortest path between two nodes in a road-network (C)</p> Signup and view all the answers

How would you describe the difference between NP and co-NP problems?

<p>An <code>NP</code> problem has a proof which can be <em><strong>verified</strong></em> in polynomial time, an <code>co-NP</code> problem has a disproof that can be <em><strong>verified</strong></em> in polynomial time. (D)</p> Signup and view all the answers

What are the parameters one might use to judge if an algorithm is of class P?

<p>The worst case runtime grows polynomially, i.e. $O(n^c)$ (C)</p> Signup and view all the answers

The primes problem belongs to both NP and co-NP, what is the theoretical implication of this point?

<p>Problems in <code>NP</code> and <code>co-NP</code> suggest there may be a polynomial-time solution, placing it in P (D)</p> Signup and view all the answers

How does the difficulty of finding a Hamiltonian Cycle grow as nodes are added to the relevant graph?

<p>The difficulty effectively prevents checking every possible permutation, as it would take $O(n!)$ time. (B)</p> Signup and view all the answers

If a problem has a reasonable method of solution, like DFS (Depth First Search), what can usually be said about that problem?

<p>It can be solved in polynomial time (D)</p> Signup and view all the answers

You are tasked with determining if a graph contains a Euler path. If you used DFS, what is a reasonable runtime for that algorithm to determine a solution?

<p>$O(|V| + |E|)$ (A)</p> Signup and view all the answers

Which of the following most accurately describes the P versus NP problem?

<p>A mathematical conjecture, proposing that problems with solutions verifiable are also solvable with efficient algorithms. (B)</p> Signup and view all the answers

What is the time complexity of efficiently determining if a number N is a Prime?

<p>$\sqrt{N}$ (D)</p> Signup and view all the answers

While Primes are solvable in $\sqrt{N}$ time, are they considered a P problem, and why?

<p>No, because the input size is measured in bits: the number of bits required to represent N is logarithmic, while the time complexity is related to $\sqrt{N}$ which is exponential in the number of bits. (A)</p> Signup and view all the answers

What must be true of all the components of a potential Hamiltonian cycle?

<p>All the components must be unique AND they must all have a path (edge) connecting to each other. (C)</p> Signup and view all the answers

Suppose you are designing an algorithm. When should you consider the worst case time, and not the average case time complexities?

<p>When there may be edge cases with large resource requirements. (B)</p> Signup and view all the answers

What is the main goal when addressing the CircuitSAT problem?

<p>To test whether assignment boolean values result a specific output. (D)</p> Signup and view all the answers

To show one problem is NP-complete, one must prove that the problem is in NP, and...

<p>reduce one known NP-complete problem to it in polynomial time. (C)</p> Signup and view all the answers

A decision problem is said to be polynomial-time reducible to another decision problem if...

<p>there exists a polynomial-time algorithm that transforms instances of one problem into instances of the other. (D)</p> Signup and view all the answers

Which of the following statements is true regarding the relationship between NP-complete and NP-hard problems?

<p>Every NP-complete problem is also NP-hard. (D)</p> Signup and view all the answers

If a polynomial-time algorithm is discovered for an NP-complete problem, what would be the most significant implication?

<p>It would prove that P = NP. (B)</p> Signup and view all the answers

Which of the following best describes the approach known as 'brute force' when applied to problem-solving?

<p>A straightforward approach that exhaustively tries all possible solutions until the correct one is found. (D)</p> Signup and view all the answers

What distinguishes problems belonging to the complexity class NP from those in the complexity class P?

<p>Solutions to problems in NP are easier to verify than solutions to problems in P are to find. (D)</p> Signup and view all the answers

What is the definition of a reduction in the context of complexity theory?

<p>A transformation of one problem into another problem. (D)</p> Signup and view all the answers

Why might a brute-force approach be impractical for solving the Hamiltonian cycle problem on a large graph?

<p>The number of possible paths to check grows factorially with the number of vertices. (C)</p> Signup and view all the answers

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

<p>It demonstrated that the Boolean satisfiability problem (SAT) is NP-complete. (C)</p> Signup and view all the answers

In the context of the "P versus NP" problem, what does it mean for a problem to be in "P"?

<p>The problem can be solved by an algorithm whose worst-case running time is polynomial in the size of the input. (B)</p> Signup and view all the answers

Flashcards

What is problem-solving?

A problem-solving method where an algorithm is found to solve a specific task or challenge.

What is worst-case time analysis?

Measuring the efficiency of an algorithm by considering the worst-case input scenario.

What is Polynomial Time (P)?

Problems for which algorithms exist can solve them in polynomial time, indicating they are relatively easy to solve. T(n) = O(n^c).

What is a Euler path?

A graph traversal algorithm that visits all edges exactly once.

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What is a Hamilton path?

A graph traversal that visits each vertex exactly once.

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What is primality testing?

Determining if a given number N is a prime number involves checking for divisors up to the square root of N.

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What is NP (Nondeterministic Polynomial Time)?

If a problem's solution can be verified in polynomial time, even if finding the solution might be hard.

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What is co-NP?

A problem is in co-NP if a 'no' answer has a polynomial-time verifiable proof.

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What does NP ∩ co-NP imply?

if a problem is in both NP and co-NP, it suggests that both 'yes' and 'no' answers can be verified quickly.

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NP-Completeness

If any problem in NP can be reduced to it in polynomial time, it's among the hardest problems in NP.

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NP-Hardness

If a problem is at least as hard as the hardest problems in NP.

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What is CircuitSAT?

Determining if there is an assignment of inputs that makes the circuit output 1.

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Study Notes

Analysis of Algorithms: P and NP

  • The notes cover the analysis of algorithms, focusing on the complexity classes P and NP

P

  • Most tasks have algorithm(s) to solve the tasks.
  • The issue of whether a given graph G=(V,E) is connected can be determined using Depth-First Search (DFS) in O(|V| + |E|) time.
  • For simplicity, it's assumed that the decision problems being addressed have a yes/no answer.

Measuring the Time Complexity in P

  • The worst-case time complexity is measured as the maximum time it takes for an algorithm to complete for an input of size n.
  • The size, n, generally refers to the number of nodes in a network, or the number of bits required to represent G.

P (Polynomial Time)

  • P includes all problems for which an algorithm exists with a time complexity T(n) = O(n^c), where c ≥ 0.
  • All tasks discussed so far belong to P.

Euler vs. Hamilton

  • Given a network G = (V, E) with n nodes and O(n²) edges, the question is whether there is a cycle that goes through all the nodes (Hamilton) or covers each edge (Euler) exactly once.
  • A "simple" algorithm can determine the Euler cycle, with a complexity of O(|V| + |E|).
  • A brute-force approach to looking for a Hamilton cycle can not work, because the all option calculation is (n-1)! , and n! >= 2^n.

Prime Numbers

  • The "Primes" problem involves determining whether a given number N is prime.
  • A simple algorithm involves checking for divisors from 2 to the square root of N, square root of N is the amount of time it takes.
  • N could be log₂(N) bits in size.

NP

  • NP consists of problems for which, if the answer is "yes", there exists a proof that can be verified in polynomial time.

  • Finding a Hamilton path is in NP, given a graph G=(V,E) and whether there is a path v1, ..., vn where Vi ≠ Vj for i ≠ j, and (Vi,Vi+1) ∈ E.

  • co-NP is similar, except that the "no" answer has a proof.

  • Primality testing is in co-NP.

  • If N is not prime, then N = a * b, where 1 < a, b < N.

  • The proof (witness) is a and b and with can check, in O(1) time, they are smaller than N.

  • Primes is classified as both in NP and co-NP.

  • All problems in P are a part of NP.

  • If it is known if A is in P, then an algorithm R can solve A in polynomial time.

  • A solution for A can found with algorithm R.

  • P⊆NP , and P⊆co-NP

CircuitSAT

  • CircuitSAT involves a circuit K that takes n input bits x₁, ..., xn.
  • Input for the problem are the bits
  • The problem is to decide if there exists an assignment for the input bits x₁ = a₁, ..., xn = an that makes the circuit output 1.
  • The problem is in NP.

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