The permanent, graph gadgets and counting solutions for certain types of planar formulas

Abstract

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.