Abstract
Let k ≥ 1 be an integer and \(h = {\scriptsize \left[\begin{array} {cc} h(00) & h(01) \\ h(10) & h(11) \end{array}\right] }\), where h(01) = h(10), be a complex-valued (symmetric) function h on domain {0,1}. 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 each edge is attached the function h, compute Z(G) = ∑ σ: V → {0,1} ∏ {u,v} ∈ E h (σ(u), σ(v)). Z(·) is known as the partition function of the spin system, also known as 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 also extend this classification to graphs with deg(v), for all v ∈ V, belonging to a specified set of degrees.
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Cai, JY., Kowalczyk, M. (2011). Spin Systems on Graphs with Complex Edge Functions and Specified Degree Regularities. In: Fu, B., Du, DZ. (eds) Computing and Combinatorics. COCOON 2011. Lecture Notes in Computer Science, vol 6842. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22685-4_13
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DOI: https://doi.org/10.1007/978-3-642-22685-4_13
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