Undirected determinant, permanent and their complexity

Abstract

We view the determinant and permanent as functions on directed weighted graphs and introduce their analogues for the undirected graphs. We prove that the task of computing the undirected determinants as well as permanents for planar graphs, whose vertices have degree at most 4, is \#P-complete. In the case of planar graphs whose vertices have degree at most 3, the computation of the undirected determinant remains \#P-complete while the permanent can be reduced to the FKT algorithm, and therefore is polynomial. The undirected permanent is a Holant problem and its complexity can be deduced from the existing literature. The concept of the undirected determinant is new. Its introduction is motivated by the formal resemblance to the directed determinant, a property that may inspire generalizations of some of the many algorithms which compute the latter. For a sizable class of planar 3-regular graphs, we are able to compute the undirected determinant in polynomial time.