# Enumeration of Preferred Extensions in Almost Oriented Digraphs

## TL;DR

This paper develops efficient algorithms for enumerating all preferred extensions in argumentation frameworks, especially focusing on almost oriented digraphs, with improved time complexities and novel sampling techniques.

## Contribution

It introduces faster enumeration algorithms for directed graphs with few 2-cycles, including a new 2-stage sampling method and parameterized algorithms using recent local search techniques.

## Key findings

- Enumeration time for all preferred extensions is $O^*(3^{n/3})$.
- Faster algorithms for graphs with limited 2-cycles, e.g., $O(1.2321^n)$ for oriented graphs.
- Existence of graphs with exponentially many preferred extensions, up to $inom{n}{n/2}$.

## 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 $0.8004\cdot n$ 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).