Towards fixed-parameter tractable algorithms for abstract argumentation

Abstract

We study the complexity of reasoning in abstracts argumentation frameworks close to a graph classes that allow for efficient reasoning methods, i.e. to one of the classes of acyclic, noeven, biparite and symmetric AFs. In this work we show that certain reasoning problems on the second level of the polynomial hierarchy still maintain their full complexity when restricted to instances of fixed distance to one of the above graph classes. OverviewThis work complements the studies in [9] on augmenting tractable fragments of abstract argumentation, but in contrast solely addresses negative results.That is we consider abstracts argumentation which are close to a graph classes which allows for efficient reasoning methods, i.e. to one of the classes of acyclic [4], noeven [6], biparite [5] and symmetric [2] AFs. We show that certain reasoning problems on the second level of the polynomial hierarchy still maintain their full complexity when restricted to instances of fixed distance to one of the above graph classes. This improves results from [9], showing hardness for the * This work has been funded by the Vienna Science and Technology Fund (WWTF) through project ICT08-028.

Complexity-sensitive decision procedures for abstract argumentation

“…A noteworthy exception is the family of labelling-based algorithms (Verheij 2007;Modgil and Caminada 2009;Podlaszewski, Caminada, and Pigozzi 2011). Other current branches in abstract argumentation are certain preprocessing techniques (Baumann 2011;Baumann, Brewka, and Wong 2011) to divide the problem into small pieces as well as implementations of fixed-parameter tractable algorithms (Dvořák, Pichler, and Woltran 2010;Dvořák, Szeider, and Woltran 2010;Dvořák et al 2011b). While the concept of complexity-sensitivity as used in our approach bears some resemblance to fixedparameter tractability and parameterized complexity, there are fundamental differences.…”

“…For weakly cyclic AFs (these are the AFs where the graph parameter cycle-rank is at most 1 (Dvořák, Pichler, and Woltran 2011)), we can make direct use of the following complexity result. Proposition 1 ( (Dvořák, Pichler, and Woltran 2011)). For weakly cyclic AFs, the problem Skept prf is coNP-complete.…”

“…The reasoning problems for stage and semi-stable semantics still maintain their full complexity when restricted to weak cyclic AFs (Dvořák, Pichler, and Woltran 2011).…”

“…Tractable (i.e., polynomial-time decidable) fragments have been quite thoroughly studied in the literature (see, e.g., (Coste-Marquis, Devred, and Marquis 2005;Dunne 2007;Dvořák, Szeider, and Woltran 2010;Dvořák, Pichler, and Woltran 2011;Ordyniak and Szeider 2011)). However, there is only little work on the identification of fragments which are located on the first level (NP-coNP layer), that is, inbetween tractability and full second-level complexity.…”

Formalizing Cognitive Acceptance of Arguments: Durum Wheat Selection Interdisciplinary Study

“…This notion corresponds to the procedural biases as introduced by [21] and [27] that is left for future work. However, it is interesting to mention that such approaches might pave the way to use any-time approximation algorithms (as already investigated in argumentation by [16]). …”