On the construction of graphs with a planar bipartite double cover from boolean formulas and its application to counting satisfying solutions

Abstract

We address the problem to construct a graph G with binary edge weights from the incidence graph of a boolean formula Φ in a way, that G has a planar bipartite double coverG¨. This allows one to use the identity between the permanent of G's adjacency matrix AG and the number of perfect matchings of G¨, i.e., perm(AG)=pm(G¨). Due to the algorithm of Fisher, Kasteleyn and Temperlay (FKT), the right hand side of the equation can be counted in polynomial time if G¨ is planar.We prove a theorem that describes necessary conditions for the gadgets (i.e., small subgraphs with certain properties) of G to allow a planar bipartite double cover G¨. For arbitrary 3CNF formulas and the known approaches to build G, our theorem shows that gadgets, which enable a planar bipartite double cover, do not exist. However, we show that for certain classes of boolean formulas, e.g., #5Pl-Rtw-2CNF, #7Pl-Rtw-2CNF and #3Forest-3CNF, such gadgets exist, hence these counting problems are in P. The results probably extended to other moduli p and perhaps further counting classes.