TITLE: Algorithm to count the number of perfect matchings in non planar graph
QUESTION [7 upvotes]: I need to count the number of perfect matchings of a certain family of graphs. This family of graph is non planar and a type of snark. For the initial cases, it seems that this number is growing exponentially. My request is different from the one here because right now, I have no interest in knowing what these perfect matchings are.
The FKT algorithm gives a polynomial time algorithm for a planar graph and I am wondering if there is anything similar for non planar graphs. Does anyone have any ideas?

REPLY [8 votes]: Counting the number of perfect matchings in arbitrary graphs (i.e. non planar, non bipartite...) seems to be quite more difficult than the restricted cases that FKT-type algorithms can handle.
In particular, Valiant proved that the problem is in $\mathrm{\text{#}P}$.
With that in mind, you can find information on approximation algorithms, see for example:

Steve Chien, A determinant-based algorithm for counting matchings in a general graph
Martin Fürer, Shiva Prasad Kasiviswanathan, Approximately Counting Perfect Matchings in General Graphs