Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study the complexity of approximate counting Constraint Satisfaction Problems (#CSPs) in a bounded degree setting. Specifically, given a Boolean constraint language $\Gamma$ and a degree bound $\Delta$,  study the complexity of #CSP$_\Delta(\Gamma)$, which is the problem of counting satisfying assignments to CSP instances with constraints from $\Gamma$ and whose variables can appear at most $\Delta$ times. Our main result shows that: (i) if every function in $\Gamma$ is affine, then #CSP$_\Delta(\Gamma)$ is in FP for all $\Delta$, (ii) otherwise, if every function in $\Gamma$ is in a class called IM$_2$, then for all sufficiently large $\Delta$, #CSP$_\Delta(\Gamma)$ is equivalent under approximation-preserving (AP) reductions to the counting problem #BIS (the problem of counting independent sets in bipartite graphs) (iii) otherwise, for all sufficiently large $\Delta$, it is NP-hard to approximate the number of satisfying assignments of an instance of #CSP$_\Delta(\Gamma)$, even within an exponential factor. Our result extends previous results, which apply only in the so-called "conservative" case.
['Andreas Galanis', 'Leslie Ann Goldberg', 'Kuan Yang']
I am writing a research paper on a idea Approximating partition functions of bounded-degree Boolean counting Constraint Satisfaction Problems
[{"from": "human", "value": "I am writing a research paper on a idea Approximating partition functions of bounded-degree Boolean counting Constraint Satisfaction Problems"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study the complexity of approximate counting Constraint Satisfaction Problems (#CSPs) in a bounded degree setting. Specifically, given a Boolean constraint language $\\Gamma$ and a degree bound $\\Delta$,  study the complexity of #CSP$_\\Delta(\\Gamma)$, which is the problem of counting satisfying assignments to CSP instances with constraints from $\\Gamma$ and whose variables can appear at most $\\Delta$ times. Our main result shows that: (i) if every function in $\\Gamma$ is affine, then #CSP$_\\Delta(\\Gamma)$ is in FP for all $\\Delta$, (ii) otherwise, if every function in $\\Gamma$ is in a class called IM$_2$, then for all sufficiently large $\\Delta$, #CSP$_\\Delta(\\Gamma)$ is equivalent under approximation-preserving (AP) reductions to the counting problem #BIS (the problem of counting independent sets in bipartite graphs) (iii) otherwise, for all sufficiently large $\\Delta$, it is NP-hard to approximate the number of satisfying assignments of an instance of #CSP$_\\Delta(\\Gamma)$, even within an exponential factor. Our result extends previous results, which apply only in the so-called \"conservative\" case."}]