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Enumeration of Preferred Extensions in Almost Oriented Digraphs

Authors Serge Gaspers, Ray Li

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ACM Subject Classification
  • Computing methodologies → Knowledge representation and reasoning
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Enumeration
Keywords
  • abstract argumentation
  • exact algorithms
  • exponential time algorithms
  • parameterized algorithms
  • enumeration algorithms
  • semikernels in digraphs

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Abstract

In this paper, we present enumeration algorithms to list all preferred extensions of an argumentation framework. This task is equivalent to enumerating all maximal semikernels of a directed graph. For directed graphs on n vertices, all preferred extensions can be enumerated in O^*(3^{n/3}) time and there are directed graphs with Omega(3^{n/3}) preferred extensions. We give faster enumeration algorithms for directed graphs with at most 0.8004 * n vertices occurring in 2-cycles. In particular, for oriented graphs (digraphs with no 2-cycles) one of our algorithms runs in time O(1.2321^n), and we show that there are oriented graphs with Omega(3^{n/6}) > Omega(1.2009^n) preferred extensions.
A combination of three algorithms leads to the fastest enumeration times for various proportions of the number of vertices in 2-cycles. The most innovative one is a new 2-stage sampling algorithm, combined with a new parameterized enumeration algorithm, analyzed with a combination of the recent monotone local search technique (STOC 2016) and an extension thereof (ICALP 2017).

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Serge Gaspers and Ray Li. Enumeration of Preferred Extensions in Almost Oriented Digraphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 74:1-74:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.MFCS.2019.74

Author Details

Serge Gaspers
  • UNSW Sydney, Sydney, Australia
  • Data61, CSIRO, Australia
Ray Li
  • UNSW Sydney, Sydney, Australia

Acknowledgements

We thank Oliver Fisher for fruitful discussions and collaboration on preliminary results in the early stages of this work.

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