Degree two approximate Boolean #CSPs with variable weights

A counting constraint satisfaction problem (#CSP) asks for the number of ways to satisfy a given list of constraints, drawn from a fixed constraint language Γ. We study how hard it is to evaluate this number approximately. There is an interesting partial classification, due to Dyer, Goldberg, Jalsenius and Richerby [DGJR10], of Boolean constraint languages when the degree of instances is bounded by d ≥ 3 every variable appears in at most d constraints under the assumption that “pinning” is allowed as part of the instance. We study the d = 2 case under the stronger assumption that “variable weights” are allowed as part of the instance. We give a dichotomy: in each case, either the #CSP is tractable, or one of two important open problems, #BIS or #PM, reduces to the #CSP. Supported by an EPSRC doctoral training grant.