- Unharnessing the power of Schrijver's permanental inequality Leonid Gurvits(gurvitslanl.gov) Abstract: Let $A \in \Omega_n$ be doubly-stochastic $n \times n$ matrix. Alexander Schrijver proved in 1998 the following remarkable inequality $$\label{le} per(\widetilde{A}) \geq \prod_{1 \leq i,j \leq n} (1- A(i,j)); \widetilde{A}(i,j) =: A(i,j)(1-A(i,j)), 1 \leq i,j \leq n$$ 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 $$\label{st} \log(per(P)) \geq \sum_{1 \leq i,j \leq n} \log(1- Q(i,j)) (1- Q(i,j)) - \sum_{1 \leq i,j \leq n} Q(i,j) \log(\frac{Q(i,j)}{P(i,j)})$$ 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: 06/16/2011Entry Accepted: 06/18/2011Entry Last Modified: 06/16/2011Modify/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 Optimization Online is supported by the Mathematical Optmization Society.