Permanent of a matrix

Definition

The permanent of a matrix is a function similar to the determinant but without the alternating signs. For an n x n matrix A, the permanent is defined as the sum of products of the entries of A taken over all possible permutations of its columns. This concept connects closely to counting problems and has significant implications in complexity theory, particularly in relation to counting problems within the class #P and Valiant's theorem.

5 Must Know Facts For Your Next Test

1. The formula for the permanent of an n x n matrix A is given by: $$ ext{perm}(A) = \sum_{\sigma \in S_n} \prod_{i=1}^n a_{i, \sigma(i)}$$ where $S_n$ is the set of all permutations of {1, 2, ..., n}.
2. Computing the permanent is a #P-complete problem, indicating that there is no known polynomial-time algorithm for it unless P = NP.
3. The permanent can be computed using a similar recursive method as for determinants, but the absence of alternating signs leads to vastly different computational complexity.
4. Valiant's theorem shows that computing the permanent is at least as difficult as solving any problem in #P, which includes counting solutions for NP-complete problems.
5. The permanent has applications in various fields such as combinatorics, algebraic geometry, and quantum mechanics, particularly in counting matchings in bipartite graphs.

Review Questions

• How does the computation of the permanent differ from that of the determinant in terms of mathematical structure and complexity?
  • The permanent differs from the determinant primarily in that it sums up products without alternating signs. While both involve permutations of rows and columns, the determinant incorporates negative signs based on the parity of permutations. This structural difference leads to significant complexity implications; specifically, while determinants can be computed in polynomial time using methods like Gaussian elimination, computing permanents is a #P-complete problem with no known efficient algorithm.
• Discuss how Valiant's theorem relates to the class #P-completeness and its implications for computational problems involving permanents.
  • Valiant's theorem establishes that computing the permanent of a matrix is #P-complete, meaning it represents one of the hardest counting problems. This classification implies that if one could find a polynomial-time solution for computing permanents, it would mean all problems in #P could also be solved in polynomial time. The implications are profound since it suggests a potential link between counting problems and decision problems that are already known to be NP-complete.
• Evaluate the significance of recognizing that computing the permanent is #P-complete and how this affects future research in computational complexity.
  • Recognizing that computing the permanent is #P-complete significantly shapes research directions in computational complexity. It sets boundaries on what types of algorithms researchers should pursue for efficient solutions. This understanding encourages exploration into approximation algorithms or randomized algorithms that might offer feasible solutions for practical cases, rather than direct computation. Additionally, it drives investigation into related counting problems, leading to potential breakthroughs in understanding other complex classes within #P.

Related terms

Determinant: A scalar value that is a function of a square matrix, providing important information about the matrix, including whether it is invertible.

#P-completeness: A class of decision problems related to counting the number of solutions to NP problems, which includes the computation of permanents.

Valiant's theorem: A result that establishes that the problem of computing the permanent of a matrix is #P-complete, meaning it is as hard as the hardest problems in #P.