FPRAS for CompClassProb

Establish whether CompClassProb, the compatibility-constraint generalization of ClassProb that computes the probability of worlds satisfying a prescribed tuple-multiplicity vector while avoiding specified forbidden pairs of block realizations, admits a fully polynomial-time randomized approximation scheme, either in general or under structural restrictions on its augmented incidence graph H^+.

Background

The paper proves that regular ClassProb admits an FPRAS by reducing its class-probability computation to the permanent of a nonnegative matrix. CompClassProb extends the problem by imposing binary forbidden relations between pairs of block realizations, and the paper establishes strong parameterized hardness results for this variant.

It remains unresolved whether the approximation approach based on the permanent can be generalized to CompClassProb. The authors explicitly identify both the unrestricted case and cases subject to structural restrictions on the augmented incidence graph H^+ as open possibilities.