Undirected determinant, permanent and their complexity

TL;DR

This paper introduces undirected analogues of the determinant and permanent for graphs, analyzing their computational complexity, and identifies cases where these can be computed efficiently or are P-complete.

Contribution

It defines the undirected determinant, analyzes its complexity, and shows polynomial-time computability for certain classes of graphs, extending the understanding of graph invariants.

Findings

Undirected determinant is P-complete for planar graphs with degree .

Undirected permanent reduces to FKT algorithm for degree  planar graphs.

Polynomial-time computation of undirected determinant for certain planar 3-regular graphs.

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.