Counting Complexity for Reasoning in Abstract Argumentation

TL;DR

This paper investigates the computational complexity of counting extensions in abstract argumentation, introducing algorithms exploiting treewidth and establishing lower bounds based on ETH.

Contribution

It provides complexity results and novel dynamic programming algorithms for counting extensions, especially for projected counts, parameterized by treewidth.

Findings

Algorithms run in double or triple exponential time in treewidth.

Established lower bounds based on ETH for counting problems.

Provided complexity classifications for various semantics.

Abstract

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.