Suppose $$\varphi$$ and $$\psi$$ are two angles satisfying $$\tan(\varphi) = 2 \tan(\psi) > 0$$ . We prove that under this condition $$\varphi$$ and $$\psi$$ cannot be both rational multiples of π. We use this number theoretic result to prove a classification of the computational complexity of spin systems on k-regular graphs with general (not necessarily symmetric) real valued edge weights. We establish explicit criteria, according to which the partition functions of all such systems are classified into three classes: (1) Polynomial time computable, (2) #P-hard in general but polynomial time computable on planar graphs, and (3) #P-hard on planar graphs. In particular, problems in (2) are precisely those that can be transformed by a holographic reduction to a form solvable by the Fisher-Kasteleyn-Temperley algorithm for counting perfect matchings in a planar graph.

We introduce an idea called anti-gadgets for reductions in complexity theory. These anti-gadgets are presented as graph fragments, but their effect is equivalent to erasing the presence of other graph fragments, as if we had managed to include a negative copy of a certain graph gadget. We use this idea to prove a complexity dichotomy theorem for the partition function Z ( G ) of spin systems over 3-regular directed graphs G , Z ( G ) = ∑ σ : V ( G ) → { 0,1} ∏ ( u , v ) ∈ E ( G ) f (σ ( u ), σ ( v )), where each edge is given a (not necessarily symmetric) complex-valued binary function f : { 0,1} 2 → C. We show that Z ( G ) is either computable in polynomial time or #P-hard, depending explicitly on f . When the input graph G is planar, there is an additional class of polynomial time computable partition functions Z ( G ), while everything else remains #P-hard. Furthermore, this additional class is precisely those that can be transformed by a holographic reduction to matchgates, followed by the Fisher-Kasteleyn-Temperley algorithm via Pfaffians.

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We prove a complexity dichotomy theorem for Holant problems on 3-regular graphs with an arbitrary complex-valued edge function. Three new techniques are introduced: (1) higher dimensional iterations in interpolation; (2) Eigenvalue Shifted Pairs, which allow us to prove that a pair of combinatorial gadgets in combination succeed in proving #P-hardness; and (3) algebraic symmetrization, which significantly lowers the symbolic complexity of the proof for computational complexity. Using holographic reductions the classification theorem also applies to problems beyond the basic model.

We prove a complexity dichotomy theorem for a class of partition functions over k-regular graphs, for any fixed k. These problems can be viewed as graph homomorphisms from an arbitrary k-regular input graph G to the weighted two vertex graph on {0,1} defined by a real-valued symmetric function h. We completely classify the computational complexity of this problem. We show that there are exactly the following alternatives, for any given h. Depending on h, over k-regular graphs, either 1.the problem is #P-hard even for planar graphs,2.the problem is #P-hard for general (non-planar) graphs, but solvable in polynomial time for planar graphs, or3.the problem is solvable in polynomial time for general graphs. The dependence on h is an explicit criterion. Furthermore, we show that in case (2) the problem is solvable in polynomial time over k-regular planar graphs, by exactly the theory of holographic algorithms using matchgates.

Let k≥1 be an integer and let h=[h(0,0)h(0,1)h(1,0)h(1,1)] be a complex-valued symmetric function on domain {0,1} (i.e., where h(0,1)=h(1,0)). We introduce a new technique, called a syzygy, and prove a dichotomy theorem for the following class of problems, specified by k and h: given an arbitrary k-regular graph G=(V,E), where the function h is attached to each edge, compute Z(G)=∑σ:V→{0,1}∏{u,v}∈Eh(σ(u),σ(v)). Z(⋅) is known as the partition function of the spin system, also known as counting graph homomorphisms on domain size two, and is a special case of Holant problems. The dichotomy theorem gives a complete classification of the computational complexity of this problem, depending on k and h. The dependence on k and h is explicit.

We present dichotomy theorems within a class of problems known as holant problems. The holant framework deals with certain counting problems on graphs, and subsumes a wide variety of problems such as COUNTING WEIGHTED H-HOMOMORPHISMS, WEIGHTED #CSP, and also classical problems such as COUNTING VERTEX COVERS. In the absence of any direct knowledge about long-standing open questions such as “P = P#P?”, dichotomy theorems establish classes of problems for which every problem is in one of two complexity classes widely believed not to overlap. In the present work, we show that for a substantial subclass of holant problems, each problem is either in FP or #P-hard. Due to holographic algorithms, some of these #P-hard problems can be solved in polynomial time when the input is restricted to planar graphs, and we derive our dichotomy theorems under this restriction as well.

The Wide Angle Telescope Transit Search (WATTS) is a low-elevation small aperture, wide-field (5.5° × 5.5°), transit search instrument capable of achieving the photometric precision needed to detect giant extrasolar planets. The system is designed to simultaneously observe tens of thousands of stars with R magnitudes between 10 and 13. In just over one year of operation, WATTS has completed five observing campaigns. During this period, 20 candidates have been identified from WATTS data. As the fourth component of the TrES network, WATTS significantly increases the efficiency of the survey when data are combined.

We prove a complexity dichotomy theorem for a class of Holant Problems over k-regular graphs, for any fixed k. These problems can be viewed as graph homomorphisms from an arbitrary k-regular input graph G to the weighted two vertex graph on {0,1} defined by a symmetric function h. We completely classify the computational complexity of this problem. We show that there are exactly the following alternatives, for any given h. Depending on h, over k-regular graphs: Either (1) the problem is #P-hard even for planar graphs; or (2) the problem is #P-hard for general (non-planar) graphs, but solvable in polynomial time for planar graphs; or (3) the problem is solvable in polynomial time for general graphs. The dependence on h is an explicit criterion. Furthermore, we show that in case (2) the problem is solvable in polynomial time over k-regular planar graphs, by exactly the theory of holographic algorithms using matchgates.

The purpose of this work is to prove a generalization of the dichotomy theorem from [6], extending that result to a larger class of counting problems. This is achieved through the use of interpolation and holographic reductions. We also use holographic reductions to establish a close connection between a class of problems which are solvable using Fibonacci gates and the class of problems which can be solved by applying a particular kind of counting argument.