Counting Complexity for Reasoning in Abstract Argumentation

In this paper, we consider counting and projected model counting of extensions in abstract argumentation for various semantics, including credulous reasoning. When asking for projected counts, we are interested in counting the number of extensions of a given argumentation framework, while multiple extensions that are identical when restricted to the projected arguments count as only one projected extension. We establish classical complexity results and parameterized complexity results when the problems are parameterized by the treewidth of the undirected argumentation graph. To obtain upper bounds for counting projected extensions, we introduce novel algorithms that exploit small treewidth of the undirected argumentation graph of the input instance by dynamic programming. Our algorithms run in double or triple exponential time in the treewidth, depending on the semantics under consideration. Finally, we establish lower bounds of bounded treewidth algorithms for counting extensions and projected extension under the exponential time hypothesis (ETH).