Abstract
Holographic reductions between some Holant problems and some #CSP(Hd) problems are built, where Hd is some complex value binary function. By the complexity of these Holant problems, for each integer d ≥ 2, #CSP(Hd) is in P when each variables appears at most d times, while it is #P-hard when each variables appears at most d + 1 times. #CSP(Hd) counts weighted summation of graph homomorphisms from input graph G to graph Hd, and the maximum occurrence of variables is the maximum degree of G. We conjecture the converse of holographic reduction holds for most of #Bi-restriction Constraint Satisfaction Problems, which can be regarded as a generalization of a known result about counting graph homomorphisms. It is proved that the converse of holographic reduction holds for some classes of problems.
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