On Weighted Maximum Model Counting: Complexity and Fragments

Maximum model counting $(\text{MAX}\#\exists \text{SAT})$ is a recently introduced extension of projected model counting $(\#\exists \text{SAT})$ that maximizes over a set of variables $X$ the number of assignments over a set $Y$ that can be extended to a satisfying assignment over $Z$. It is known that $\text{MAX}\#\exists \text{SAT}$ also generalizes weighted $\#\exists \text{SAT}$ and MAXSAT if weights are introduced to the problem. However, for the latter a non-trivial gadget is needed. We propose a more generic weighting scheme that evaluates a fitness term and a probability term simultaneously. In this setting, $\text{MAX}\#\exists \text{SAT}$ extends weighted MAXSAT and $\#\exists \text{SAT}$ without the need of gadgets. As MAXSAT is the canonical problem of cost-optimal reasoning and $\#\exists \text{SAT}$ can be seen as canonical problem of probabilistic reasoning, $\text{MAX}\# 3\text{SAT}$ with the proposed weighting scheme naturally fills the role as canonical problem for cost-optimal probabilistic reasoning. We study the problem from a complexity-theoretic point of view for unary weights and prove that the decision version is $\mathrm{D}_{2}^{\mathrm{P}}{-}$ complete. We then focus on structural parameters and provide an ETH lower bound with respect to the inputs treewidth, as well as a treewidth-aware reduction from $\text{MAX}\#\exists \text{SAT}$ to $\text{MAX}\#\exists \text{SAT}$.