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 expressible in terms of holographic transformations. 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 as both a proof technique and in the statement of the tractability criterion.

We prove a complexity dichotomy theorem for symmetric complex-weighted Boolean #CSP when the constraint graph of the input must be planar. The problems that are #P-hard over general graphs but tractable over planar graphs are precisely those with a holographic reduction to matchgates. This generalizes a theorem of Cai, Lu, and Xia for the case of real weights. We also obtain a dichotomy theorem for a symmetric arity 4 signature with complex weights in the planar Holant framework, which we use in the proof of our #CSP dichotomy. In particular, we reduce the problem of evaluating the Tutte polynomial of a planar graph at the point (3, 3) to counting Eulerian orientations over planar 4-regular graphs to show the latter is #P-hard. This strengthens a theorem by Huang and Lu to the planar setting. Our proof techniques combine new ideas with refinements and extensions of existing techniques. These include planar pairings, the unary recursive construction, the anti-gadget technique, and pinning in the Hadamard basis.

We prove a complexity dichotomy for complex-weighted Holant problems with an arbitrary set of symmetric constraint functions on Boolean variables. This dichotomy is specifically to answer the question: Is the FKT algorithm under a holographic transformation [38] a universal strategy to obtain polynomial-time algorithms for problems over planar graphs that are intractable in general? This dichotomy is a culmination of previous ones, including those for Spin Systems [25], Holant [21,6], and #CSP [20].In the study of counting complexity, such as #CSP, there are problems which are #P-hard over general graphs but polynomial-time solvable over planar graphs. A recurring theme has been that a holographic reduction to FKT precisely captures these problems. Surprisingly, for planar 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.In previous work, an important tool was a dichotomy for #CSP d , which denotes #CSP where every variable appears a multiple of d times. However the very first step in the #CSP d dichotomy proof fundamentally violates planarity. In fact, due to our newly discovered tractable problems, the putative form of a planar #CSP d dichotomy is false when d ≥ 5. Nevertheless, we prove a dichotomy for planar #CSP 2 . In this case, the putative form of the dichotomy is true. We manage to prove the planar Holant dichotomy without relying on a planar #CSP d dichotomy for d ≥ 3, while the dichotomy for planar #CSP 2 plays an essential role.As a special case of our new planar tractable problems, counting perfect matchings (#PM) over k-uniform hypergraphs is polynomial-time computable 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 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 thatis either computable in polynomial time or #P-hard, depending explicitly on f .To state the dichotomy theorem more explicitly, we show that the partition function Z(G) on 3-regular directed graphs G is computable in polynomial time when f belongs to one of four classes, which can be described as (1) degenerate, (2) generalized disequality, (3) generalized equality, and (4) affine after a holographic transformation. In all other cases it is #P-hard. Here class (4), after a holographic transformation, can also be described as an exponential quadratic polynomial of the form i Q(x,y) , where i = √ −1 and the cross term xy in the quadratic polynomial Q(x, y) has an even coefficient. If the input graph G is planar, then an additional class of functions becomes computable in polynomial time, and everything else remains #P-hard. This additional class is precisely those which can be computed by holographic algorithms with matchgates, making use of the Fisher-Kasteleyn-Temperley algorithm via Pfaffians.There is a long history in the study of "Exactly Solved Models" in statistical physics. In the language of complexity theory, physicists' notion of an "Exactly Solvable" system corresponds to a system with a polynomial time computable partition function. A central question is to identify which "systems" can be solved "exactly" and which "systems" are "difficult". While in physics, there is no rigorous definition of being "difficult", complexity theory supplies the proper notion-#P-hardness.The main innovation in this paper is the idea of the anti-gadget. It is analogous to the pairing of a particle and its anti-particle in physics. Coupled with the idea of anti-gadgets, we also introduce a general way of proving #P-hardness by two types of gadgets called recursive gadgets and projector gadgets. We prove a Group Lemma which spells out a general condition for the technique to succeed. This Group Lemma states that as long as the group generated by the transition matrices of the constructed gadgets is infinite, then one can interpolate all unary functions-a key step in the proof of #P-hardness. Interpolation is carried out by forming a Vandermonde system and proving that it is of full rank. The anti-gadget concept makes the transition to group theory very natural and seamless.Not only is the idea of anti-gadgets useful in proving a new complexity dichotomy theorem in counting complexity, we also show that anti-gadgets provide a simple explanation for some miraculous cancellations that were observed in previous results. Furthermore, anti-gadgets can also guide the search for gadget sets m...

Holographic algorithms introduced by Valiant are composed of two ingredients: matchgates, which are gadgets realizing local constraint functions by weighted planar perfect matchings, and holographic reductions, which show equivalences among problems with different descriptions via certain basis transformations. In this paper, we replace matchgates in the paradigm above by the affine type and the product type constraint functions, which are known to be tractable in general (not necessarily planar) graphs. More specifically, we present polynomial-time algorithms to decide if a given counting problem has a holographic reduction to another problem defined by the affine or product-type functions. Our algorithms also find a holographic transformation when one exists. We further present polynomial-time algorithms of the same decision and search problems for symmetric functions, where the complexity is measured in terms of the (exponentially more) succinct representations. The algorithm for the symmetric case also shows that the recent dichotomy theorem for Holant problems with symmetric constraints is efficiently decidable. Our proof techniques are mainly algebraic, e.g., using stabilizers and orbits of group actions.