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 prove a complexity classification theorem that classifies all counting constraint satisfaction problems (#CSP) over Boolean variables into exactly three categories: (1) Polynomial-time tractable;(2) #P-hard for general instances, but solvable in polynomial-time over planar graphs; and (3) #P-hard over planar graphs. The classification applies to all sets of local, not necessarily symmetric, constraint functions on Boolean variables that take complex values. It is shown that Valiant's holographic algorithm with matchgates is a universal strategy for all problems in category (2).Proof. One direction is trivial, since (= 4 ) ∈ EQ 2 . For the other direction we prove by induction. For k = 1, we have (= 2 ) ∈ EQ. For k = 2, we have (= 4 ) given. Assume that we have (= 2(k−1) ). Then connecting (= 2(k−1) ) and (= 4 ) by one edge we get (= 2k ).By (2.

Holographic algorithms with matchgates are a novel approach to design polynomial time computation. It uses Kasteleyn's algorithm for perfect matchings, and more importantly a holographic reduction. The two fundamental parameters of a holographic reduction are the domain size k of the underlying problem, and the basis size ℓ. A holographic reduction transforms the computation to matchgates by a linear transformation that maps to (a tensor product space of) a linear space of dimension 2 ℓ . We prove a sharp basis collapse theorem, that shows that for domain size 3 and 4, all non-trivial holographic reductions have basis size ℓ collapse to 1 and 2 respectively. The main proof techniques are Matchgate Identities, and a Group Property of matchgate signatures.