一类平面公式的积式、图形小工具及计数解
The permanent, graph gadgets and counting solutions for certain types of planar formulas
摘要：
本文基于Valiant \cite{Val79a}和Ben-Dor/Halevi \cite{Ben93}的思想，即通过计算一个特殊构造的矩阵的积和式来计算一个布尔公式的满足解的个数.我们发现，Desnanot-Jacobi身份（$\dji$）防止英勇的原始方法，以实现一个简约的减少到永久的领域的特征2。作为下一步，由于永久式的计算是$#\classP$ -完全的，我们利用永久式的相等性和无权图的二部双覆盖中的完美匹配数。只要这个二分双覆盖（BDC）是平面的，完美匹配的数量可以使用Kasteleyn的算法\cite{Kas67}在多项式时间内计数。为了加强BDC的平面性，我们用新的小工具替换Valiant的原始小工具，并描述这些小工具必须具有的属性。我们表明，\textit{circular planarity}的属性起着至关重要的作用，找到正确的小工具计数问题。为了绕过$\dji$屏障，我们切换到字段$\mathbb{Z}/p\mathbb{Z}$，以获得素数$p > 2$。利用这种方法，我们能够在随机多项式时间内计算$\forestdreisat$公式的解的数量。最后，我们提出了一个猜想，国家的广义gadget不能找到，因为否则可以证明$\classRP = \classNP$。该猜想建立了图$\grG$的邻接矩阵的子式的行列式与$\grG$的BDC关于给定节点集的\textit{circular planar}结构之间的关系。
英文摘要：
In this paper, we build on the idea of Valiant \cite{Val79a} and Ben-Dor/Halevi \cite{Ben93}, that is, to count the number of satisfying solutions of a boolean formula via computing the permanent of a specially constructed matrix. We show that the Desnanot-Jacobi identity ($\dji$) prevents Valiant's original approach to achieve a parsimonious reduction to the permanent over a field of characteristic two. As the next step, since the computation of the permanent is $#\classP$-complete, we make use of the equality of the permanent and the number of perfect matchings in an unweighted graph's bipartite double cover. Whenever this bipartite double cover (BDC) is planar, the number of perfect matchings can be counted in polynomial time using Kasteleyn's algorithm \cite{Kas67}. To enforce planarity of the BDC, we replace Valiant's original gadgets with new gadgets and describe what properties these gadgets must have. We show that the property of \textit{circular planarity} plays a crucial role to find the correct gadgets for a counting problem. To circumvent the $\dji$-barrier, we switch over to fields $\mathbb{Z}/p\mathbb{Z}$, for a prime $p > 2$. With this approach we are able to count the number of solutions for $\forestdreisat$ formulas in randomized polynomial time. Finally, we present a conjecture that states which kind of generalized gadgets can not be found, since otherwise one could prove $\classRP = \classNP$. The conjecture establishes a relationship between the determinants of the minors of a graph $\grG$'s adjacency matrix and the \textit{circular planar} structure of $\grG$'s BDC regarding a given set of nodes.
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