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In this paper, we consider counting and projected model counting ofextensions in abstract argumentation for various semantics. When asking forprojected counts we are interested in counting the number of extensions of agiven argumentation framework while multiple extensions that are identical whenrestricted to the projected arguments count as only one projected extension. Weestablish classical complexity results and parameterized complexity resultswhen the problems are parameterized by treewidth of the undirectedargumentation graph. To obtain upper bounds for counting projected extensions,we introduce novel algorithms that exploit small treewidth of the undirectedargumentation graph of the input instance by dynamic programming (DP). Ouralgorithms run in time double or triple exponential in the treewidth dependingon the considered semantics. Finally, we take the exponential time hypothesis(ETH) into account and establish lower bounds of bounded treewidth algorithmsfor counting extensions and projected extension.

Counting Complexity for Reasoning in Abstract Argumentation