FKT algorithm

History

The problem of counting planar perfect matchings has its roots in statistical mechanics and chemistry, where the original question was: If diatomic molecules are adsorbed on a surface, forming a single layer, how many ways can they be arranged?{{Citation | author-link = Brian Hayes (scientist) | author-link = Rodney J. Baxter | orig-year = 1982 | conference-url = http://www.egr.unlv.edu/~larmore/FOCS/focs2010/

Another type of physical system to consider is composed of dimers, which is a polymer with two atoms. The dimer model counts the number of dimer coverings of a graph. Another physical system to consider is the bonding of H2O molecules in the form of ice. This can be modelled as a directed, 3-regular graph where the orientation of the edges at each vertex cannot all be the same. How many edge orientations does this model have?

Motivated by physical systems involving dimers, in 1961, Pieter Kasteleyn{{citation | author-link = Pieter Kasteleyn | author-link = Pieter Kasteleyn | author-link = Pieter Kasteleyn | editor-last = Harary | editor-first = F. | editor-link = Frank Harary

Algorithm

Explanation

The main insight is that every non-zero term in the Pfaffian of the adjacency matrix of a graph G corresponds to a perfect matching. Thus, if one can find an orientation of G to align all signs of the terms in Pfaffian (no matter + or - ), then the absolute value of the Pfaffian is just the number of perfect matchings in G. The FKT algorithm does such a task for a planar graph G. The orientation it finds is called a Pfaffian orientation.

Let G = (V, E) be an undirected graph with adjacency matrix A. Define PM(n) to be the set of partitions of n elements into pairs, then the number of perfect matchings in G is :\operatorname{PerfMatch}(G) = \sum_{M \in PM(|V|)} \prod_{(i,j) \in M} A_{i j}. Closely related to this is the Pfaffian for an n by n matrix A :\operatorname{pf}(A) = \sum_{M \in PM(n)} \operatorname{sgn}(M) \prod_{(i,j) \in M} A_{i j}, where sgn(M) is the sign of the permutation M. A Pfaffian orientation of G is a directed graph H with adjacency matrix B such that pf(B) = PerfMatch(G).{{cite conference | author-link = Robin Thomas (mathematician) | conference-url = http://www.icm2006.org/

Finally, for any skew-symmetric matrix A, :\operatorname{pf}(A)^2 = \det(A), where det(A) is the determinant of A. This result is due to Arthur Cayley.{{cite journal | author-link1 = Arthur Cayley |trans-title= On skew determinants

High-level description

1. Compute a planar embedding of G.
2. Compute a spanning tree T1 of the input graph G.
3. Give an arbitrary orientation to each edge in G that is also in T1.
4. Use the planar embedding to create an (undirected) graph T2 with the same vertex set as the dual graph of G.
5. Create an edge in T2 between two vertices if their corresponding faces in G share an edge in G that is not in T1. (Note that T2 is a tree.)
6. For each leaf v in T2 (that is not also the root):
7. Let e be the lone edge of G in the face corresponding to v that does not yet have an orientation.
8. Give e an orientation such that the number of edges oriented clock-wise is odd.
9. Remove v from T2.
10. Return the absolute value of the Pfaffian of the adjacency matrix of G, which is the square root of the determinant.

Generalizations

The sum of weighted perfect matchings can also be computed by using the Tutte matrix for the adjacency matrix in the last step.

Kuratowski's theorem states that : a finite graph is planar if and only if it contains no subgraph homeomorphic to K5 (complete graph on five vertices) or K3,3 (complete bipartite graph on two partitions of size three). Vijay Vazirani generalized the FKT algorithm to graphs that do not contain a subgraph homeomorphic to K3,3.{{cite journal | author-link = Vijay Vazirani | doi-access= free | hdl-access= free Since counting the number of perfect matchings in a general graph is also #P-complete, some restriction on the input graph is required unless FP, the function version of P, is equal to #P. Counting matchings, which is known as the Hosoya index, is also #P-complete even for planar graphs.{{citation

Applications

The FKT algorithm has seen extensive use in holographic algorithms on planar graphs via matchgates. For example, consider the planar version of the ice model mentioned above, which has the technical name #PL-3-NAE-SAT (where NAE stands for "not all equal"). Valiant found a polynomial time algorithm for this problem which uses matchgates.{{cite journal

References

References

1. (2005). "What is a Dimer?". AMS.
2. (1961). "Dimer problem in statistical mechanics-an exact result". Philosophical Magazine.