A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries

@article{Jerrum2004APA,
  title={A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries},
  author={M. Jerrum and A. Sinclair and Eric Vigoda},
  journal={J. ACM},
  year={2004},
  volume={51},
  pages={671-697}
}
We present a polynomial-time randomized algorithm for estimating the permanent of an arbitrary n × n matrix with nonnegative entries. This algorithm---technically a "fully-polynomial randomized approximation scheme"---computes an approximation that is, with high probability, within arbitrarily small specified relative error of the true value of the permanent. 
A remark on approximating permanents of positive definite matrices
TLDR
There is a fully polynomial randomized approximation scheme (FPRAS) for the permanent of A as the integral of some explicit log-concave function on ${Bbb R}^{2n}$. Expand
Simply Exponential Approximation of the Permanent of Positive Semidefinite Matrices
TLDR
A deterministic polynomial time cn approximation algorithm for the permanent of positive semidefinite matrices is designed and it is shown that the permanent is within a cn factor of the top eigenvalue of the Schur power matrix. Expand
A Tight Analysis of Bethe Approximation for Permanent
  • Nima Anari, A. Rezaei
  • Computer Science, Physics
  • 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
  • 2019
We prove that the permanent of nonnegative matrices can be deterministically approximated within a factor of square root 2 n in polynomial time, improving upon the previous deterministicExpand
A deterministic approximation algorithm for computing the permanent of a 0, 1 matrix
TLDR
These results use a recently developed deterministic approximation algorithm for counting partial matchings of a graph and Jerrum-Vazirani method of approximating permanent by near perfect matchings to construct a deterministic polynomial time approximation algorithm. Expand
An Almost Linear Time Approximation Algorithm for the Permanen of a Random (0-1) Matrix
TLDR
The algorithm with inputs A, ∈ > 0 produces an output X A with (1-∈)per(A) 0, and almost all (0-1) matrices the algorithm runs in time O(n 2 ω), i.e., almost linear in the size of the matrix. Expand
A Note on Deterministic Poly-Time Algorithms for Partition Functions Associated with Boolean Matrices with Prescribed Row and Column Sums
TLDR
The main application is the first poly-time deterministic algorithm which approximates the partition functions associated with boolean matrices with prescribed row and column sums within simply exponential multiplicative factor. Expand
A polynomial-time approximation algorithm for the number of k-matchings in bipartite graphs
We show that the number of k-matching in a given undirected graph G is equal to the number of perfect matching of the corresponding graph Gk on an even number of vertices divided by a suitableExpand
Boolean matrices with prescribed row/column sums and stable homogeneous polynomials: Combinatorial and algorithmic applications
  • L. Gurvits
  • Computer Science, Mathematics
  • Inf. Comput.
  • 2013
TLDR
The main application is the first poly-time deterministic algorithm which approximates the partition functions associated with boolean matrices with prescribed row and column sums within simply exponential multiplicative factor. Expand
An asymptotic approximation for the permanent of a doubly stochastic matrix
A determinantal approximation is obtained for the permanent of a doubly stochastic matrix. For moderate-deviation matrix sequences, the asymptotic relative error is of order O(n−1).
Approximating the Permanent of a Random Matrix with Vanishing Mean
  • Lior Eldar, S. Mehraban
  • Mathematics, Computer Science
  • 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)
  • 2018
TLDR
A quasi-polynomial time deterministic algorithm for approximating the permanent of a typical n × n random matrix with unit variance and vanishing mean µ to within inverse polynomial multiplicative error to counter the common intuition that the difficulty of computing the permanent stems merely from the authors' inability to treat matrices with many opposing signs. Expand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 39 REFERENCES
A Deterministic Strongly Polynomial Algorithm for Matrix Scaling and Approximate Permanents
TLDR
This work develops the first strongly polynomial-time algorithm for matrix scaling –– an important nonlinear optimization problem with many applications and suggests a simple new (slow)Polynomial time decision algorithm for bipartite perfect matching. Expand
A mildly exponential approximation algorithm for the permanent
A new approximation algorithm for the permanent of ann ×n 0,1-matrix is presented. The algorithm is shown to have worst-case time complexity exp(O(n1/2 log2n)). Asymptotically, this represents aExpand
A deterministic strongly polynomial algorithm for matrix scaling and approximate permanents
We present a deterministic strongly polynomial algorithm that computes the permanent of a nonnegative n × n matrix to within a multiplicative factor of en. To this end we develop the first stronglyExpand
Polynomial Time Algorithms to Approximate Permanents and Mixed Discriminants Within a Simply Exponential Factor
  • A. Barvinok
  • Computer Science
  • Random Struct. Algorithms
  • 1999
We present real, complex, and quaternionic versions of a simple ran-domized polynomial time algorithm to approximate the permanent of a non-negative matrix and, more generally, the mixed discriminantExpand
The Complexity of Computing the Permanent
  • L. Valiant
  • Computer Science, Mathematics
  • Theor. Comput. Sci.
  • 1979
Abstract It is shown that the permanent function of (0, 1)-matrices is a complete problem for the class of counting problems associated with nondeterministic polynomial time computations. RelatedExpand
Approximating the Permanent
TLDR
A randomised approximation scheme for the permanent of a 0–1s presented, demonstrating that the matchings chain is rapidly mixing, apparently the first such result for a Markov chain with genuinely c... Expand
Geometric Bounds for Eigenvalues of Markov Chains
We develop bounds for the second largest eigenvalue and spectral gap of a reversible Markov chain. The bounds depend on geometric quantities such as the maximum degree, diameter and covering numberExpand
Clifford algebras and approximating the permanent
TLDR
The first main result shows how to compute in polynomial time an estimator with the same mean and variance over the 4-dimensional algebra (which is the quaternions, and is non-commutative); in addition to providing some hope that the computations can be performed in higher dimensions, this quaternion algorithm provides an exponential improvement in the variance over that of the 2-dimensional complex version studied by Karmarkar et al. Expand
Chernoff-type bound for finite Markov chains
This paper develops bounds on the distribution function of the empirical mean for irreducible finite-state Markov chains. One approach, explored by D. Gillman, reduces this problem to bounding theExpand
Fast Uniform Generation of Regular Graphs
TLDR
The algorithm is based on simulation of a rapidly convergent stochastic process, and runs in polynomial time for a wide class of degree sequences, including all regular sequences and all n -vertex sequences with no degree exceeding √ n /2. Expand
...
1
2
3
4
...