This paper makes three contributions to estimating the number of perfect matching in bipartite graphs. First, we prove that the popular sequential importance sampling algorithm works in polynomial time for dense bipartite graphs. More carefully, our algorithm gives a $(1-\epsilon)$-approximation for the number of perfect matchings of a $\lambda$-dense bipartite graph, using $O(n^{\frac{1-2\lambda}{8\lambda}+\epsilon^{-2}})$ samples. With size $n$ on each side and for $\frac{1}{2}>\lambda>0$, a $\lambda$-dense bipartite graph has all degrees greater than $(\lambda+\frac{1}{2})n$. Second, practical applications of the algorithm requires many calls to matching algorithms. A novel preprocessing step is provided which makes significant improvements. Third, three applications are provided. The first is for counting Latin squares, the second is a practical way of computing the greedy algorithm for a card guessing game with feedback, and the third is for stochastic block models. In all three examples, sequential importance sampling allows treating practical problems of reasonably large sizes.
翻译:本文为估算双面图中完美匹配的数量做出了三点贡献。 首先, 我们证明流行的连续重要抽样算法在密集双面图的多元时段中起作用。 更仔细地说, 我们的算法为 $( 1-\\ epsilon) 提供美元- emppartite 图形的完美匹配值。 其次, 算法的实际应用需要用 $( no\frac{ 1-2\\ lambda} 8\lambda ⁇ ⁇ ⁇ ⁇ 2\ ⁇ ) 来匹配。 提供了一个新的预处理步骤, 从而大大改进了 。 第一个是计算拉丁方 $( $\ frac { 1\\\\ \ \ \ \ \ limbda> lambda> 0$ ), 第二个是计算贪婪双面图的实用方法, 其最高度超过 $( lambda) {frac{ { 1\\\ 2} $。 其次, 实际应用算法应用方法需要许多匹配算算算算算算算算算法的大小。 。 3 的模型中, 需要大量的模型。