The paper highlights that several counting problems are known to be AP-interreducible with #SAT, including #IS (the problem of counting independent sets in a graph) and #BIS (the problem of counting independent sets in bipartite graphs). These problems are significant within the realm of counting complexity, as they represent some of the hardest problems to approximate within the class #P. The relationship between these problems and the complexity of approximately counting maximal independent sets (#MIS) is particularly noteworthy.
Since the authors of the paper have established that #MIS is AP-interreducible with #SAT, it follows that #MIS shares the same level of complexity as these other problems. This means that if an efficient approximation algorithm (FPRAS) were to exist for #MIS, it would imply the existence of such an algorithm for #SAT and consequently for #IS and #BIS as well. The interreducibility indicates that the challenges in approximating #MIS are akin to those faced in approximating #SAT, reinforcing the notion that #MIS is a central problem in the study of counting complexities.