Abstract

argumentation has proven to be a versatile tool to model and analyze various problems in an argumentative setting. The addition of collective attacks syntactically extends Dung’s original argumentation frameworks (AFs), while retaining the most desirable properties—the resulting class of frameworks is called SETAFs. While most reasoning tasks in the realm of abstract argumentation have been shown to be intractable, real-world instances oftentimes are not entirely random but admit a certain structure that allows for efficient computational shortcuts. In certain cases, we can characterize this structure via an integer parameter, and exploit these insights with advanced algorithmic techniques. A thorough analysis of the computational aspects of SETAFs w.r.t. parameterized algorithms has not yet been conducted. We start the investigation of these approaches by applying the backdoor and treewidth approaches to SETAFs. A backdoor is a part of a problem instance, such that removing the backdoor leads to a simple structure. If we can find such a backdoor and guess the solution on this part (respectively, extensions), the rest of the solution follows almost effortlessly. Similarly, the treewidth of a problem instance is a parameter that characterizes the properties of the instance’s graph structure. Intuitively, the lower the treewidth of a graph, the more “tree-like” it is. Since most argumentation problems become easy on trees, one can exploit low treewidth for efficient algorithms. In this paper, we establish that for SETAFs with constant backdoor sizes general argumentation tasks become efficiently solvable—they are fixed-parameter tractable. We generalize the respective techniques that are known for the special case of Dung-style AFs and show that they also apply to the more general case of SETAFs. In addition, we can show an improvement in the asymptotic runtime compared to earlier approaches for AFs via two-valued guesses instead of the state-of-the-art three-valued approach. Along the way, we point out similarities and interesting situations arising from the more general setting. While treewidth is well-studied in the context of AFs with their graph structure, it cannot be directly applied to the (directed) hypergraphs representing SETAFs. We thus introduce two generalizations of treewidth based on different graphs that can be associated with SETAFs, that is, the primal graph and the incidence graph. We show that while some of these notions allow for parameterized tractability results, reasoning remains intractable for other notions, even if we fix the parameter to a small constant. We present parameterized algorithms for efficient reasoning on SETAFs via tree decompositions by characterizing extensions not only by labeling the arguments, but also by assigning (temporary) labels to the attacks.