arXiv · cs/0405067
We show that the problem of counting the number of Eulerian circuits in an undirected graph is complete for the class #P.
Graham R. Brightwell, Peter Winkler. 2004-05-18. Note on Counting Eulerian Circuits. https://arxiv.org/abs/cs/0405067
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The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.
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