借助Bethe近似释放Schrijver积和不等式的威力
Unleashing the power of Schrijver's permanental inequality with the help of the Bethe Approximation
摘要：
设$A \in Ω_n$为双随机$n \times n$矩阵。Alexander Schrijver在1998年证明了下面的显着不等式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.我们利用上述Shrijver不等式证明了以下下界：\frac{per(A)}{F(A)} \geq 1; F（A）=：\prod_{1 \leq i,j \leq n}（1- A（i，j））^{1- A（i，j）}。我们利用这个新的下界证明了S.Friedland关于单体-二聚体问题的渐近下匹配猜想。本文利用我们对（LAMC）的证明中的一些思想来反证[Lu，Mohr，Szekely]的正相关猜想.我们提出了具有比率为$\frac{per(A)}{F(A)} = \sqrt{2}^{n}$的显式双随机$n \times n$矩阵$A$，猜想\max_{A \in Ω_n} \frac{per(A)}{F(A)} \approx（\sqrt{2}）^{n}，并给出了支持这个猜想的一些例子.如果是真的，这个猜想（以及论文中提到的其他猜想）将意味着一个确定性的多时间算法来近似相对因子$(\sqrt{2})^{n}$内的$n \times n$非负矩阵的积和式。目前最好的这种因素是$e^n$。
英文摘要：
Let $A \in Ω_n$ be doubly-stochastic $n \times n$ matrix. Alexander Schrijver proved in 1998 the following remarkable inequality 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 use the above Shrijver's inequality to prove the following lower bound: \frac{per(A)}{F(A)} \geq 1; F(A) =: \prod_{1 \leq i,j \leq n} (1- A(i,j))^{1- A(i,j)}. We use this new lower bound to prove S.Friedland's Asymptotic Lower Matching Conjecture(LAMC) on monomer-dimer problem. We use some ideas of our proof of (LAMC) to disprove [Lu,Mohr,Szekely] positive correlation conjecture. We present explicit doubly-stochastic $n \times n$ matrices $A$ with the ratio $\frac{per(A)}{F(A)} = \sqrt{2}^{n}$; conjecture that \max_{A \in Ω_n}\frac{per(A)}{F(A)} \approx (\sqrt{2})^{n} and give some examples supporting the conjecture. If true, the conjecture (and other ones stated in the paper) would imply a deterministic poly-time algorithm to approximate the permanent of $n \times n$ nonnegative matrices within the relative factor $(\sqrt{2})^{n}$. The best current such factor is $e^n$.
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