Counting satisfying solutions using planar bipartite double covers

🔲 This post is related to a paper of mine (Title: On the construction of graphs with a planar bipartite double cover from boolean formulas and its application to counting satisfying solutions) that will get published later this year in the Journal of Theoretical Computer Science. It is related to [this] blog post which i wrote already four years ago and settles some of its questions.

In a short overview, the paper adresses the following action chain:

1. Assume you have a boolean CNF formula $\Phi$
2. You build the incidence graph $\mathcal{I}$ of $\Phi$ using some techniques from the literature, e.g., each clause is represented by a node and also each variable is represented by a node. Then there is an edge between a clause-node and a variable-node if the variable occurs in that clause in $\Phi$.
3. You replace each node in $\mathcal{I}$ with a gadget, i.e., a small graph that has certain properties, which only have edge weights from 0 and 1. The properties of the gadgets ensure that the permanent of $\mathcal{G}$ contains #$\Phi$, as Valiant did in his paper.
4. You build the bipartite double cover $\mathcal{G}^{**}$ for which it holds that $$\text{PerfMatch}(\mathcal{G}^{**}) = \text{Permanent}(\textbf{A}_\mathcal{G})$$
5. If $\mathcal{G}^{**}$ is planar, then $\text{PerfMatch}(\mathcal{G}^{**})$ could be counted in polynomial time.

        //PerfMatch counts the number of Perfect Matchings of a graph

Sidenote: In [this] last blog post i talked about perfect matchings and setups which allow their efficient computation.