-

 

 

 




Optimization Online





 

Unharnessing the power of Schrijver's permanental inequality

Leonid Gurvits(gurvits***at***lanl.gov)

Abstract: Let $A \in \Omega_n$ be doubly-stochastic $n \times n$ matrix. Alexander Schrijver proved in 1998 the following remarkable inequality \begin{equation} \label{le} per(\widetilde{A}) \geq \prod_{1 \leq i,j \leq n} (1- A(i,j)); \widetilde{A}(i,j) =: A(i,j)(1-A(i,j)), 1 \leq i,j \leq n \end{equation} We prove in this paper the following generalization (or just clever reformulation) of (\ref{le}):\\ For all pairs of $n \times n$ matrices $(P,Q)$, where $P$ is nonnegative and $Q$ is doubly-stochastic \begin{equation} \label{st} \log(per(P)) \geq \sum_{1 \leq i,j \leq n} \log(1- Q(i,j)) (1- Q(i,j)) - \sum_{1 \leq i,j \leq n} Q(i,j) \log(\frac{Q(i,j)}{P(i,j)}) \end{equation} The main corrollary of (\ref{st}) is the following inequality for doubly-stochastic matrices: $$ \frac{per(A)}{F(A)} \geq 1; F(A) =: \prod_{1 \leq i,j \leq n} \left(1- A(i,j)\right)^{1- A(i,j)}. $$ We present explicit doubly-stochastic $n \times n$ matrices $A$ with the ratio $\frac{per(A)}{F(A)} = \sqrt{2}^{n}$ and conjecture that $$ \max_{A \in \Omega_n}\frac{per(A)}{F(A)} \approx \left(\sqrt{2} \right)^{n}. $$ If true, it would imply a deterministic poly-time algorithm to approximate the permanent of $n \times n$ nonnegative matrices within the relative factor $\left(\sqrt{2} \right)^{n}$.

Keywords: permanent, Bethe Approximation

Category 1: Global Optimization (Theory )

Category 2: Combinatorial Optimization

Citation:

Download: [PDF]

Entry Submitted: 06/16/2011
Entry Accepted: 06/18/2011
Entry Last Modified: 06/16/2011

Modify/Update this entry


  Visitors Authors More about us Links
  Subscribe, Unsubscribe
Digest Archive
Search, Browse the Repository

 

Submit
Update
Policies
Coordinator's Board
Classification Scheme
Credits
Give us feedback
Optimization Journals, Sites, Societies
Mathematical Optimization Society