Abstract 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.

We prove a complexity dichotomy theorem for Holant problems over an arbitrary set of complex-valued symmetric constraint functions $\mathcal{F}$ on Boolean variables. This extends and unifies all previous dichotomies for Holant problems on symmetric constraint functions (taking values without a finite modulus). We define and characterize all symmetric vanishing signatures; they turn out to be essential to the complete classification of Holant problems. The dichotomy theorem has an explicit tractability criterion expressible in terms of holographic transformations. A Holant problem defined by a set of constraint functions $\mathcal{F}$ is solvable in polynomial time if it satisfies this tractability criterion, and is #P-hard otherwise. The tractability criterion can be intuitively stated as follows: A set $\mathcal{F}$ is tractable if (1) every function in $\mathcal{F}$ has arity at most two; or (2) $\mathcal{F}$ is transformable to an affine type; or (3) $\mathcal{F}$ is transformable to a product type; or (4) $\mathcal{F}$ is vanishing, combined with the right type of binary functions; or (5) $\mathcal{F}$ belongs to a special category of vanishing-type Fibonacci gates. The proof of this theorem utilizes many previous dichotomy theorems on Holant problems and Boolean constraint satisfaction problems (#CSP). Holographic transformations play an indispensable role as both a proof technique and in the statement of the tractability criterion.

We prove a complexity dichotomy for complex-weighted Holant problems with an arbitrary set of symmetric constraint functions on Boolean variables. In the study of counting complexity, such as #CSP, there are problems which are #P-hard over general graphs but P-time solvable over planar graphs. A recurring theme has been that a holographic reduction [36] to FKT precisely captures these problems. This dichotomy answers the question: Is this a universal strategy? Surprisingly, we discover new planar tractable problems in the Holant framework (which generalizes #CSP) that are not expressible by a holographic reduction to FKT. In particular, the putative form of a dichotomy for planar Holant problems is false. Nevertheless, we prove a dichotomy for #CSP2, a variant of #CSP where every variable appears even times, that the presumed universality holds for #CSP2. This becomes an important tool in the proof of the full dichotomy, which refutes this universality in general. The full dichotomy says that the new P-time algorithms and the strategy of holographic reductions to FKT together are universal for these locally defined counting problems. As a special case of our new planar tractable problems, counting perfect matchings (#PM) over k-uniform hypergraphs is P-time computable when the incidence graph is planar and k ≥ 5. The same problem is #P-hard when k = 3 or k = 4, also a consequence of the dichotomy. More generally, over hypergraphs with specified hyperedge sizes and the same planarity assumption, #PM is P-time computable if the greatest common divisor (gcd) of all hyperedge sizes is at least 5.

We show that an effective version of Siegel's Theorem on finiteness of integer solutions for a specific algebraic curve and an application of elementary Galois theory are key ingredients in a complexity classification of some Holant problems. These Holant problems, denoted by Holant(f), are defined by a symmetric ternary function f that is invariant under any permutation of the κ ≥ 3 domain elements. We prove that Holant(f) exhibits a complexity dichotomy. The hardness, and thus the dichotomy, holds even when restricted to planar graphs. A special case of this result is that counting edge κ-colorings is #P-hard over planar 3-regular multigraphs for all κ ≥ 3. In fact, we prove that counting edge κ-colorings is #P-hard over planar r-regular multigraphs for all κ ≥ r ≥ 3. The problem is polynomial-time computable in all other parameter settings. The proof of the dichotomy theorem for Holant(f) depends on the fact that a specific polynomial p(x, y) has an explicitly listed finite set of integer solutions, and the determination of the Galois groups of some specific polynomials. In the process, we also encounter the Tutte polynomial, medial graphs, Eulerian partitions, Puiseux series, and a certain lattice condition on the (logarithm of) the roots of polynomials.

We prove a complexity dichotomy theorem for Holant problems over an arbitrary set of complex-valued symmetric constraint functions {F} on Boolean variables. This extends and unifies all previous dichotomies for Holant problems on symmetric constraint functions (taking values without a finite modulus). We define and characterize all symmetric vanishing signatures. They turned out to be essential to the complete classification of Holant problems. The dichotomy theorem has an explicit tractability criterion. A Holant problem defined by a set of constraint functions {F} is solvable in polynomial time if it satisfies this tractability criterion, and is #P-hard otherwise. The tractability criterion can be intuitively stated as follows: A set {F} is tractable if (1) every function in {F} has arity at most two, or (2) {F} is transformable to an affine type, or (3) {F} is transformable to a product type, or (4) {F} is vanishing, combined with the right type of binary functions, or (5) {F} belongs to a special category of vanishing type Fibonacci gates. The proof of this theorem utilizes many previous dichotomy theorems on Holant problems and Boolean #CSP. Holographic transformations play an indispensable role, not only as a proof technique, but also in the statement of the dichotomy criterion.

We introduce an idea called anti-gadgets in complexity reductions. These combinatorial gadgets have the effect of erasing the presence of some other graph fragment, as if we had managed to include a negative copy of a graph gadget. We use this idea to prove a complexity dichotomy theorem for the partition function Z(G) on 3-regular directed graphs G, where each edge is given a complex-valued binary function f: {0,1}2 → C. We show that