Towards Fixed-Parameter Tractable Algorithms for Argumentation

Abstract argumentation frameworks have received a lot of interest in recent years. Most computational problems in this area are intractable but several tractable fragments have been identified. In particular, Dunne showed that many problems can be solved in linear time for argumentation frameworks of bounded tree-width. However, these tractability results, which were obtained via Courcelle’s Theorem, do not directly lead to efficient algorithms. The goal of this paper is to turn the theoretical tractability results into efficient algorithms and to explore the potential of directed notions of tree-width for defining larger tractable fragments.argumentation frameworks have received a lot of interest in recent years. Most computational problems in this area are intractable but several tractable fragments have been identified. In particular, Dunne showed that many problems can be solved in linear time for argumentation frameworks of bounded tree-width. However, these tractability results, which were obtained via Courcelle’s Theorem, do not directly lead to efficient algorithms. The goal of this paper is to turn the theoretical tractability results into efficient algorithms and to explore the potential of directed notions of tree-width for defining larger tractable fragments. Introduction Argumentation has evolved as an important field in AI with abstract argumentation frameworks (AFs, for short) as introduced by Dung (1995) being its most popular formalization. Meanwhile, many semantics for AFs have been proposed (for an overview see (Baroni and Giacomin 2009)). Most computational problems in this area are intractable (see e.g. (Dimopoulos and Torres 1996; Dunne and BenchCapon 2002)), but the importance of efficient algorithms for tractable fragments has been clearly recognized (see e.g. (Dix et al. 2009)). Such tractable fragments are, for instance, symmetric argumentation frameworks (CosteMarquis, Devred, and Marquis 2005) or bipartite argumentation frameworks (Dunne 2007). An interesting approach to dealing with intractable problems comes from parameterized complexity theory and is based on the following observation: Many hard problems become tractable if some problem parameter is bounded by a fixed constant. This property is referred to as fixedparameter tractability (FPT). One important parameter of graphs is the tree-width, which measures the “tree-likeness” of a graph. Indeed, Dunne (2007) showed that many problems in the area of argumentation can be solved in linear time for argumentation frameworks of bounded tree-width. This FPT-result was shown via a seminal result by Courcelle (1990). However, as stated in (Dunne 2007), “rather than This work was supported by the Vienna Science and Technology Fund (WWTF) under grant ICT08-028 and by the Austrian Science Fund (FWF) under grant P20704-N18. Copyright c © 2010, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. synthesizing methods indirectly from Courcelle’s Theorem, one could attempt to develop practical direct methods”. The primary goal of this paper is therefore to present new, direct algorithms for (skeptical and credulous) reasoning. Clearly, the quest for FPT-results in argumentation should not stop at the tree-width, and further parameters should be analyzed. This may of course also lead to negative results. For instance, if we consider as parameter the degree of an argument (i.e., the number of incoming and outgoing attacks), Dunne (2007) showed that reasoning remains intractable, even if we restrict ourselves to AFs with at most two incoming and two outgoing attacks. A number of further parameters is however, still unexplored. Hence, the second major goal of this paper is to explore the potential of further parameters for identifying tractable fragments of argumentation. In particular, since AFs are directed graphs, it is natural to consider directed notions of width to obtain larger classes of tractable AFs. To this end, we investigate the effect of bounded cycle-rank (Eggan 1963) (a precise definition will be given below) on reasoning in AFs. We show that reasoning remains intractable even if we only consider AFs of cycle-rank 2. Actually, many further directed notions of width exist in the literature. However, it has been recently shown in (Berwanger et al. 2006; Hunter and Kreutzer 2008; Gruber 2008) that problems which are hard for bounded cycle-rank remain hard when several other directed variants of the tree-width are bounded. Hence, in the current state of research, bounded tree-width seems to be the most general parameter to obtain FPT. Due to lack of space, we have to restrict ourselves here to the preferred semantics. Roughly speaking, the preferred extensions of an AF are maximal admissible sets of arguments, where admissible means that the selected arguments defend themselves against attacks. One reason for choosing the preferred semantics is that it is widely used. Moreover, admissibility and maximality are prototypical properties common in many other semantics, for instance complete and stable (Dung 1995), stage (Verheij 1996), semi-stable (Caminada 2006), and ideal semantics (Dung, Mancarella, and Toni 2007). Hence, we expect that the methods developed here can also be extended to other semantics. Structure of the paper and summary of results. • After recalling some basic notions and results on AFs and width-measures for graphs, we show that reasoning remains intractable in AFs with bounded cycle-rank (Eggan 1963). As has been mentioned above, this negative result carries over to many other directed notions of width. • A dynamic programming approach is developed to characterize admissible sets of AFs. The time complexity of our algorithm is linear in the size of the AFs (as expected by Courcelle’s Theorem) with a multiplicative constant that is single exponential in the tree-width (which is in great contrast to algorithms derived via Courcelle’s Theorem). • In case of credulous reasoning, the algorithm for admissible sets also applies to the preferred semantics. For skeptical reasoning, we have to extend this algorithm so as to cover also the preferred semantics. Finally, we outline some directions of future research – notably the further extension of our algorithms to other semantics. Argumentation Frameworks In this section we introduce (abstract) argumentation frameworks (Dung 1995), recall the preferred semantics for such frameworks, and highlight some known complexity results. Definition 1. An argumentation framework (AF) is a pair F = (A,R) where A is a set of arguments and R ⊆ A×A is the attack relation. We sometimes use the notation a ֌ b instead of (a, b) ∈ R, in case no ambiguity arises. Further, for S ⊆ A and a ∈ A, we write S ֌ a (resp. a ֌ S) iff there exists b ∈ S, such that b ֌ a (resp. a ֌ b). An argument a ∈ A is defended by a set S ⊆ A iff for each b ∈ A, such that b ֌ a, also S ֌ b holds. An AF can naturally be represented as a directed graph. Example 1. Let F = (A,R) with A = {a, b, c, d, e, f, g} and R = {(a, b), (c, b), (c, d), (d, c), (d, e), (e, g), (f, e), (g, f)}. The graph representation of F is given as follows.