In this talk, we consider counting and projected model counting of extensions in abstract argumentation for various semantics.

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 treewidth of the undirected argumentation graph. In particular, we show a complexity dichotomies for counting credulous extensions (#.coNP vs. #.P) as well as for the projected variant lifting the complexity by one level (#.NP vs. #.\Sigma^P_2).

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 (DP). Our algorithms run in time double or triple exponential in the treewidth depending on the considered semantics.

Finally, we take the exponential time hypothesis (ETH) into account and establish tight lower bounds of bounded treewidth algorithms for counting extensions and projected extension.

This talk present joint work with Johannes Fichte and Markus Hecher that was published at AAAI 2019 and presents also some recent advancements on preferred semantics.