FKT is Not Universal — A Planar Holant Dichotomy for Symmetric Constraints

We prove a complexity classification for Holant problems defined by an arbitrary set of complex-valued symmetric constraint functions on Boolean variables. This is to specifically answer the question: Is the Fisher-Kasteleyn-Temperley (FKT) algorithm under a holographic transformation (Valiant, SIAM J. Comput. 37(5), 1565–1594 2008) a universal strategy to obtain polynomial-time algorithms for problems over planar graphs that are intractable on general graphs? There are problems that are #P-hard on general graphs but polynomial-time solvable on planar graphs. For spin systems (Kowalczyk 2010) and counting constraint satisfaction problems (#CSP) (Guo and Williams, J. Comput. Syst. Sci. 107, 1–27 2020), a recurring theme has emerged that a holographic reduction to FKT precisely captures these problems. Surprisingly, for Holant, we discover new planar tractable problems that are not expressible by a holographic reduction to FKT. In particular, a straightforward formulation of a dichotomy for planar Holant problems along the above recurring theme is false. A dichotomy theorem for #CSPd, which denotes #CSP where every variable appears a multiple of d times, has been an important tool in previous work. However the proof for the #CSPd dichotomy violates planarity, and it does not generalize to the planar case easily. In fact, due to our newly discovered tractable problems, the putative form of a planar #CSPd dichotomy is false when d ≥ 5. Nevertheless, we prove a dichotomy for planar #CSP2. In this case, the putative form of the dichotomy is true. (This is presented in Part II of the paper.) We manage to prove the planar Holant dichotomy relying only on this planar #CSP2 dichotomy, without resorting to a more general planar #CSPd dichotomy for d ≥ 3. A special case of the new polynomial-time computable problems is counting perfect matchings (#PM) over k-uniform hypergraphs when the incidence graph is planar and k ≥ 5. The same problem is #P-hard when k = 3 or k = 4, which is also a consequence of our dichotomy. When k = 2, it becomes #PM over planar graphs and is tractable again. More generally, over hypergraphs with specified hyperedge sizes and the same planarity assumption, #PM is polynomial-time computable if the greatest common divisor (gcd) of all hyperedge sizes is at least 5. It is worth noting that it is the gcd, and not a bound on hyperedge sizes, that is the criterion for tractability.

Perfect matchings, rank of connection tensors and graph homomorphisms

We develop a theory of graph algebras over general fields. This is modelled after the theory developed by Freedman et al. (2007, J. Amer. Math. Soc.20 37–51) for connection matrices, in the study of graph homomorphism functions over real edge weight and positive vertex weight. We introduce connection tensors for graph properties. This notion naturally generalizes the concept of connection matrices. It is shown that counting perfect matchings, and a host of other graph properties naturally defined as Holant problems (edge models), cannot be expressed by graph homomorphism functions with both complex vertex and edge weights (or even from more general fields). Our necessary and sufficient condition in terms of connection tensors is a simple exponential rank bound. It shows that positive semidefiniteness is not needed in the more general setting.