Solving Projected Model Counting by Utilizing Treewidth and its Limits

Abstract

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 projection variables, where multiple solutions that are identical when restricted to the projection variables count as only one solution. Inspired by the observation that the so-called “treewidth” is one of the most prominent structural parameters, our algorithm utilizes small treewidth of the primal graph of the input instance. More precisely, it runs in time O(2²^(k+4)n²) 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. While the algorithm above serves as a first theoretical upper bound and although it might be quite appealing for small values of k, unsurprisingly a naive implementation adhering to this runtime bound suffers already from instances of relatively small width. Therefore, we turn our attention to several measures in order to resolve this issue towards exploiting treewidth in practice: We present a technique called nested dynamic programming, where different levels of abstractions of the primal graph are used to (recursively) compute and refine tree decompositions of a given instance. Further, we integrate the concept of hybrid solving, where subproblems hidden by the abstraction are solved by classical search-based solvers, which leads to an interleaving of parameterized and classical solving. Finally, we provide a nested dynamic programming algorithm and an implementation that relies on database technology for PMC and a prominent special case of PMC, namely model counting (#SAT). Experiments indicate that the advancements are promising, allowing us to solve instances of treewidth upper bounds beyond 200.