Exploiting Treewidth for Projected Model Counting and its Limits
This work addresses the computational challenge of counting solutions in Boolean formulas with projected variables, which is incremental as it builds on treewidth-based methods for model counting.
The paper tackles the projected model counting (PMC) problem by introducing a novel algorithm that exploits small treewidth, achieving fixed-parameter tractability with a runtime of O(2^{2^{k+4}} n^2), and establishes asymptotically tight lower bounds based on the exponential time hypothesis.
In this paper, we introduce a novel algorithm to solve projected model counting (PMC). PMC asks to count solutions of a Boolean formula with respect to a given set of projected variables, where multiple solutions that are identical when restricted to the projected variables count as only one solution. Our algorithm exploits small treewidth of the primal graph of the input instance. It runs in time $O({2^{2^{k+4}} n^2})$ where k is the treewidth and n is the input size of the instance. In other words, we obtain that the problem PMC is fixed-parameter tractable when parameterized by treewidth. Further, we take the exponential time hypothesis (ETH) into consideration and establish lower bounds of bounded treewidth algorithms for PMC, yielding asymptotically tight runtime bounds of our algorithm.