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. 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. 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 lower bounds of bounded treewidth algorithms for counting extensions and projected extension.Comment: Extended version of a paper published at AAAI-1

Efficient algorithms for computing the Euler-Poincar\'e characteristic of symmetric semi-algebraic sets

Let $\mathrm{R}$ be a real closed field and $\mathrm{D} \subset \mathrm{R}$ an ordered domain. We consider the algorithmic problem of computing the generalized Euler-Poincar\'e characteristic of real algebraic as well as semi-algebraic subsets of $\mathrm{R}^k$, which are defined by symmetric polynomials with coefficients in $\mathrm{D}$. We give algorithms for computing the generalized Euler-Poincar\'e characteristic of such sets, whose complexities measured by the number the number of arithmetic operations in $\mathrm{D}$, are polynomially bounded in terms of $k$ and the number of polynomials in the input, assuming that the degrees of the input polynomials are bounded by a constant. This is in contrast to the best complexity of the known algorithms for the same problems in the non-symmetric situation, which are singly exponential. This singly exponential complexity for the latter problem is unlikely to be improved because of hardness result ($\#\mathbf{P}$-hardness) coming from discrete complexity theory.Comment: 29 pages, 1 Figure. arXiv admin note: substantial text overlap with arXiv:1312.658