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.

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).

Equations

$$O^{\ast }\left({\left(\min \left({\phi }^{2r}\cdot {\phi }^{1-r},{\left({\left(1+2^{\frac{1}{4}}-\frac{1}{\sqrt{2}}\right)}^r\cdot {\left(2-\frac{1}{\sqrt{2}}\right)}^{1-r}\right)}^{1+\epsilon },3^{1/3}\right)\right)}^n\right)$$

$$\le$$

$$\left|(S\bigsqcup T)\setminus U\right|=\left|S\setminus (U\cap B)\right|+\left|T\setminus (U\setminus B)\right|=(b^{\prime }-b)+(d^{\prime }-d)$$

$$x^n-x^{n-t_1}-x^{n-t_2}-\dots -x^{n-t_r}=0$$

$${\mathcal{F}}_{I,X}^k=\{S\subseteq U_I\setminus X:|S|=k\text{ and }S\cup X\in {\mathcal{F}}_I\}$$

$$\psi (S)=\{v_i:v_i\in S\}\cup \{w_i:v_i\in S\}$$

$$\psi (S)=\{v_i:v_i\in S\}\cup \{w_i:v_i\in S\wedge w_i\text{ exists}\}$$

$$\{\phi \}\cup T\cup W$$

$$\mu (I)$$

$$\ge \frac{1}{2}k+c_{2}b+\frac{c_3}{2}b$$

$$\ge \frac{1}{2}k+\frac{1}{4}b$$

$$\text{∣E(G[V2])∣+ (number of edges from V3 to V2)}=\sum _{v\in V_2}\mathrm{indegree}(v)=2\left|V_2\right|$$

$$\text{∣E(G[V2])∣+ (number of edges from V2 to V3)}=\sum _{v\in V_2}\mathrm{outdegree}(v)=2\left|V_2\right|$$

Full text

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).

1 Introduction

In Dung’s theory of abstract argumentation [17], an argumentation framework (AF) is a digraph $G=(V,E)$, where each vertex represents an argument, and an arc $(u,v)\in E$ denotes that argument $u$ attacks argument $v$. There are various semantics that express what properties a set of arguments should have for a rational agent to stand by that set of arguments. One of the most central semantics is the preferred semantics that was already proposed by Dung in his foundational paper [17]. Let $S\subseteq V$ be a subset of vertices (also called extension) of a digraph $G=(V,E)$. The set $S$ is conflict-free if no arc has both endpoints in $S$. A vertex $v\in V$ is acceptable with respect to $S$ if for each arc $(u,v)\in E$ there is an arc $(w,u)\in E$ with $w\in S$. In other words, for each argument $u$ that attacks $v$, there is an argument $w$ in $S$ that attacks $u$. We say in this case that $w$ defends $v$ against $u$. The set $S$ is admissible if it is conflict-free and each argument in $S$ is acceptable with respect to $S$. The set $S$ is preferred if it is an inclusion-wise maximal admissible set.

While we will use the language of abstract argumentation, we remark that such vertex sets have also been studied in graph theory. Neumann-Lara [33] (see also [25]) defined the notion of semikernels. Maximal semikernels are equal to the preferred extensions in the directed graph where all arcs are reversed. The related notion of kernels [38] has the same correspondence with stable extensions in abstract argumentation, and was introduced as an abstract solution concept in cooperative game theory, but has been extensively studied in the theory of directed graphs. In particular, various issues around the enumeration of kernels and semikernels have been considered in previous work [2, 6, 24, 35].

Motivation A central problem in abstract argumentation is the enumeration of extensions prescribed by a given semantics. In part, this is because exploring what sets of arguments may go together is an inherent issue of AFs. The enumeration of preferred extensions is of particular interest, firstly for its own sake, but also in the study of other semantics as it forms the basis of several other semantics refining this set. A number of existing algorithms and implementations enumerate all preferred extensions of a digraph (see, e.g., [7, 8, 10, 11, 12, 13, 16, 30, 34, 36, 37]). The enumeration of preferred extensions is also a part of the biennial International Competition on Computational Models of Argumentation (ICCMA). Computational problems where the enumeration of extensions are used involve answering the questions: is a given argument in some / all preferred extensions and what is the number of preferred extensions containing a given argument / in total. Upper bounds on the number of extensions under various semantics have also been proposed as fundamental characteristics to compare various semantics in abstract argumentation [4, 18].

Dunne et al. [18], building on the work of Baumann and Strass [3], showed that the number of preferred extensions is $O(3^{\left\lvert V\right\rvert/3})$ (this result also holds for many other semantics [3, 18]). This bound is realized by a disjoint union of triangles, where every edge is replaced by an arc in both directions. The proof is based on the well-known Moon and Moser result [32] for upper bounding the number of maximal cliques in graphs.

We study the enumeration of preferred extensions in digraphs with no, or relatively few, $2$-cycles (i.e., bidirectional arcs). Our aim is to determine how much the presence of $2$-cycles affects the number of preferred extensions of an AF. Mutually attacking arguments play a special role in abstract argumentation [29], but this conflict is often resolved rather easily if the strength of the two attacks can be evaluated [5], or the user’s preference between the two arguments can be elicited [31, 1]. These methods of resolving conflicts motivate the study of problems, and in particular enumeration problems, for AFs with no or few 2-cycles.

Our results We study enumeration algorithms and combinatorial upper bounds on the number of preferred extensions in oriented graphs, which are digraphs without 2-cycles, and generalizations of oriented graphs. Our main concern is the enumeration of all preferred extensions in time moderately exponential in the number of vertices $n$, and we mainly focus on digraphs that are either oriented or have small resolution order. The resolution order of a digraph $G=(V,E)$, denoted $r(G)$, is the number of vertices that belong to a 2-cycle in $G$.

Our main result is an algorithm that, for any $\varepsilon>0$, enumerates all preferred extensions of a digraph $G$ on $n$ vertices in time

[TABLE]

where $r=r(G)/n$, and $\varphi\approx 1.2321$ is the positive root of $1-x^{-1}-x^{-8}$. The $O^{*}$ notation hides factors that are polynomial in the input size. See Figure 1, which plots the base $\alpha$ of the running time expressed as $O^{*}(\alpha^{n})$ for $r$ varying from 0 to 1. For $r=1$, this is best possible and follows from the work in [18, 31].

At the other end of the spectrum, i.e., for oriented graphs where $r=0$, the upper bound is $O^{*}(\varphi^{n})\leq O(1.2321^{n})$ and is obtained via a carefully constructed branching algorithm and running time analysis. We also give a lower bound on the largest number of preferred extensions an oriented graph on $n$ vertices may have of $\Omega(3^{n/6})\geq\Omega(1.2009^{n})$. A construction, which we call the Oriented Translation, reducing an arbitrary digraph $G=(V,E)$ to an oriented graph with $|V|+r(G)$ vertices, such that there is a bijection between their preferred extensions, allows us to generalize these upper and lower bounds to $O^{*}\left(\varphi^{2\cdot r(G)}\cdot\varphi^{n-r(G)}\right)\leq O(1.5180^{r(G)}\cdot 1.2321^{n-r(G)})$ and $\Omega(3^{r(G)/3}\cdot 3^{(n-r(G))/6})\geq\Omega(1.4422^{r(G)}\cdot 1.2009^{n-r(G)})$, respectively.

Our main technical contribution is the third algorithm. It relies on a parameterized enumeration algorithm and extensions of the recent monotone local search framework [21]. The parameterized enumeration algorithm has as input a digraph $G=(V,E)$, a set of arguments $S$, and a non-negative integer $k$, and it enumerates all maximal admissible extensions $T\subseteq S$ of $G$ within distance $k$ of $S$. Its running time can be upper bounded by $O^{*}(2^{k/2+r(G[S])/4})$. This is optimal, since there are instances for which the solution consists of $\Omega(2^{k/2+r(G[S])/4})$ preferred extensions at distance at most $k$ from $S$. Furthermore, under the Strong Exponential Time Hypothesis, the corresponding decision problem has no $O^{*}(2^{(1-\varepsilon)(k/2+r(G[S])/4)})$ time solution for any $\varepsilon>0$. We use this parameterized enumeration algorithm in a new 2-phase monotone local search procedure, where we separately sample vertices from $B$, the set of vertices in at least one 2-cycle, and $V\setminus B$ and then apply the parameterized enumeration algorithm. The running time analysis is a new combination of the results in [21] for the first sampling phase and [27] for the second sampling phase, combined with the parameterized subroutine. From a technical point of view, this is the most innovative part of this paper. (From a conceptual point of view, the most innovative contribution is probably the synergy between modern enumeration algorithmics and the theory of abstract argumentation.) This results in an algorithm enumerating all preferred extensions of a given digraph $G$ in time $O^{*}\left(\left(1+2^{1/4}-2^{-1/2}\right)^{(1+\varepsilon)\cdot r(G)}\cdot\left(2-2^{-1/2}\right)^{(1+\varepsilon)\cdot(n-r(G))}\right)$.

Interpretation of results Figure 1 illustrates the running time of the various algorithms. We have improved algorithms and combinatorial upper bounds whenever $r\leq 0.8004$. The result for $r=0$ shows that for oriented graphs, our new algorithm allows to handle instances with $75\%$ more arguments, compared with the previous best $O(3^{n/3})$ upper bound. (We have that $\log_{1.2321}(3^{1/3})\approx 1.7545$.) The figure also shows that we have significantly narrowed the gap between the best known lower bound and the best known upper bound for oriented graphs, and digraphs with a fraction $r$ of vertices belonging to $2$-cycles, for a wide range of $r$.

Outline Sections 2 and 3 describe our monotone local search algorithm. Section 4 describes our branching algorithm for oriented graphs.

First we introduce the parameterized enumeration problem that will form the subroutine of our Monotone Local Search.

Maximal Admissible Subset Enumeration (MASE)

Input:

Graph $G$, set $S\subseteq V(G)$, integer $k$

Parameter:

$k$

Output:

Enumerate all maximal admissible sets $T\subseteq S$ such that $\left\lvert S\setminus T\right\rvert\leq k$.

There is a subtlety here with how we define maximal. We say $T$ is a maximal admissible subset of $S$ if there does not exist an admissible set $U$ such that $T\subsetneq U\subseteq S$. Notably $T$ is not necessarily a preferred extension (though $T$ is if $S=V(G)$).

In Section 2, we present an algorithm for MASE, parameterized by $k$ and $r(G[S])$, the resolution order of the subgraph of $G$ induced by $S$.

Theorem 1**.**

For an instance $I=(G,S,k)$ of MASE, let $\mu(I):=\frac{k}{2}+\frac{r(G[S])}{4}$. Then MASE can be solved in $O^{*}(2^{\mu(I)})$ time. Furthermore, there are at most $2^{\mu(I)}$ maximal admissible subsets of $S$ within distance $k$ of $S$. Hence, there are at most $2^{\mu(I)}$ preferred extensions that are subsets of $S$ with size $\geq\left\lvert S\right\rvert-k$.

Our algorithm is a standard parameterized branching algorithm. Compared to the enumeration of independent sets, the primary additional tool we have is a powerful simplification rule, (Undefendable), for vertices with in-degree [math]. (Undefendable) also allows our base case to be any conflict-free set.

In Section 3, we extend our parameterized algorithm into a general enumeration algorithm through a novel 2-phase application of monotone local search. Since 2-cycles increase the run time of our MASE subroutine, we modify the classical Monotone Local Search to sample separately between a set of ”bad vertices” (ones contained in a 2-cycle) and ”good vertices” (ones not contained in any 2-cycle). This presents a speed up compared to a more direct application of the Monotone Local Search framework. We believe this may be useful for other problems.

Separately, Section 4 presents a branching algorithm for oriented graphs. Again, (Undefendable) plays a critical role. This time around, we allow our base case to be any induced DAG which further provides a simplification rule for vertices with out-degree 0. We tailor our branching rules to take full advantage of these two simplification rules. This is combined with a lot of careful case analysis and ad-hoc methods (including a graph classification theorem in Subsection H.7).

Each of our algorithms also provide a corresponding combinatorial upper bound on the number of preferred extensions. The various enumeration algorithms and bounds are collected in Section 5.

In the appendix we have a mix of background, further detail and extra results. Some extra results that may be of independent interest:

Appendix Section C describes the Oriented Translation, a parsimonious reduction from a general directed graph to an oriented graph.

Theorem 2**.**

There is a linear time algorithm that transforms any AF $G$ into an oriented AF $G^{\prime}$ with $\left\lvert V(G^{\prime})\right\rvert=\left\lvert V(G)\right\rvert+r(G)+3$ such that there is a bijection between the preferred extensions of $G$ and the preferred extensions of $G^{\prime}$ that can be applied in linear time.

The basic idea of the construction (see Appendix Section C) is carefully converting 2-cycles into 4-cycles by doubling up the vertices that are contained in at least one 2-cycle.

Our primary interest in Theorem 2 is as a tool for extending oriented graph algorithms to algorithms on general graphs, parameterized by resolution order. In Section 5 we apply this to our branching algorithm for oriented graphs.

Theorem 2 is also useful for deriving complexity results on oriented graphs by extending constructions for directed graphs. In Appendix Section F.2 we obtain the following result:

Theorem 3**.**

Unless P=NP, no algorithm enumerates the admissible or preferred extensions of an AF in output-polynomial time, even when the AF is an oriented graph.

In Appendix Section F we also show our algorithm for the decision variant of MASE is optimal, assuming the Strong Exponential Time Hypothesis, and that the enumeration bound in Theorem 1 is tight (i.e: there exist instances with $2^{\mu(I)}$ maximal admissible subsets within distance $k$).

We briefly justify the choice of the measure $\mu$ we use for our MASE algorithm in Appendix Section G.

Notation An oriented graph is a digraph with no 2-cycles.

We will use the notation $N(v)$ to denote the set of vertices adjacent to $v$ and $N[v]$ to denote $N(v)\cup\{v\}$.

We will assume throughout that AFs have no self loops. We can safely make this assumption due to the Loopless Translation (Definition C.1) which, with a (small) constant overhead removes all self loops from an AF while preserving the admissible extensions.

2 Parameterized Enumeration Problems

Our algorithm for MASE will follow a standard template for parameterized branching algorithms. A summary of the necessary concepts can be found in Appendix Section A.

We will consider the measure $\mu(I)=\frac{k}{2}+\frac{b}{4}$ where $k$ is the number of vertices we are allowed to remove and $b:=r(G[S])$ is the resolution order of $G[S]$. We will design a branching algorithm with run time $O^{*}(2^{\mu(I)})$ which recurses into subinstances and collates their results to obtain the maximal admissible subsets of $S$.

However, there is a technical difficulty in the collation step arising from the fact that a maximal admissible subset of $S^{\prime}\subsetneq S$ may not be a maximal admissible subset of $S$. This gives rise to the following subproblem:

Maximal Subset Collation

Input:

Graph $G$, $c$ pairs $(S_{i},C_{i})$ where for each $i$, $S_{i}\subseteq V(G)$ and $C_{i}$ is a set containing only maximal admissible subsets of $S_{i}$

Output:

A set containing all maximal elements of $\bigcup_{i=1}^{c}C_{i}$

The main result we need is:

Lemma 1**.**

Maximal Subset Collation can be solved in $O(c\sum_{i=1}^{c}\left\lvert C_{i}\right\rvert\cdot\text{{poly}}(\left\lvert V\right\rvert))$ time.

An algorithm for Maximal Subset Collation is provided in Appendix Section D.

We are now ready to solve MASE.

2.1 An $O^{*}(2^{\mu(I)})$ algorithm for Maximal Admissible Subset Enumeration

The overall structure of our algorithm is described in Algorithm 1.

The following table lists the branching rules in application order with the first row being the base case. The second column describes the instances each rule is applicable to. After applying simplification rules, our branching algorithm will always apply the first rule that is applicable to the input instance.

[TABLE]

We note that these requirements are exhaustive, hence there will always be at least one applicable rule in any instance.

We include the base case and short remarks on each of the branching cases. Full proofs of each case can be found in Appendix Section E.

The key ingredient for solving the base case and optimizing our branching cases is the following simplification rule. It is essentially a rephrasing of a well known result that has been applied before in the context of enumeration algorithms (e.g., [16][9]).

Simplification Rule 1** (Undefendable).**

Suppose we are trying to enumerate all maximal admissible subextensions of $S$.

Let $u\in S$ be a vertex such that there exists a vertex $a\in V(G)$, $a$ attacks $u$ and no vertex in $S$ attacks $a$. Then there is no admissible subset of $S$ that contains $u$ so we can safely set $S\leftarrow S\setminus\{u\}$.

Lemma 2**.**

(Undefendable) is sound.

Proof.

Vertex $u$ can never be defended from $a$ in any subextension of $S$ and hence is in no admissible subextension of $S$. ∎

Instead of solving the base case directly we will instead solve the more general case where $G[S]$ is a DAG. The increased generality will be needed in Section 4.

The base case, that $S$ is conflict-free, follows trivially (since $G[S]$ is a DAG if $S$ is conflict-free).

2.2 MASE when $G[S]$ is a DAG

The results in this section are all well-known and follow directly from (Undefendable).

Lemma 3**.**

If $G[S]$ is a DAG, the rule (Undefendable) is not applicable if and only if $S$ is admissible.

Proof.

Clearly (Undefendable) is not applicable if and only if every $v\in S$ is acceptable with respect to $S$ (recall this means that for all arcs $(u,v)\in E(G)$, there is an arc $(w,u)\in E(G)$ with $w\in S$). By the definition of admissibility, it remains to show that if (Undefendable) is not applicable, then $S$ is conflict-free. This follows from noting (Undefendable) is applicable to any vertex attacked by a maximal vertex in $G[S]$. Such a vertex exists in any weakly connected DAG with more than one vertex. Hence all weakly connected components of $G[S]$ have size one. ∎

(Undefendable) takes polynomial time to apply. The resulting subset from applying (Undefendable) until it is not applicable is the unique maximal admissible subset of $S$. It is admissible by Lemma 3. It is the unique maximal admissible subset as (Undefendable) only removes vertices that are in no admissible subsets of $S$. Hence:

Lemma 4**.**

Suppose $S\subseteq V(G)$ induces a DAG in $G$. Then $S$ has exactly one maximal admissible subset and it can be found in polynomial time.

2.3 Remarks on branching cases

Full proofs can be found in Appendix Section E.

Case 1 follows from noting $G[S]$ must be a family of cycles.

Let $v$ be the vertex in the 2-cycle with higher total degree. Case 2 follows from a 2-way branch, in one branch enumerating all maximal admissible subsets that include $v$, in the other branch enumerating all that do not include $v$.

Case 3 is another 2-way branch, except applied to any vertex with total degree at least 4.

Case 4 is probably the most instructive. It best showcases the power of (Undefendable). First we show a vertex $v$ exists with in-degree 1, out-degree 2. Then a 2-way branch is applied to the vertex attacking $v$, using (Undefendable) to improve the branch where the vertex is excluded and $v$ is left with in-degree 0.

2.4 Running Time Analysis

The base case is solvable in polynomial time. Each of our branching rules has branching number $\leq 2$. Hence, if we ignore the calls to Maximal Subset Collation, by the Combine Analysis Lemma (Lemma 6) our algorithm has running time $O^{*}(2^{\mu(I)})$.

Maximal Subset Collation is applied to each admissible subset encountered by the MASE algorithm (i.e: each leaf in the search tree) at most $d$ times, where $d$ is the maximum depth of the search tree. By Lemma 1 each application incurs a $O(\text{{poly}}(\left\lvert V\right\rvert))$ cost ($c\leq 2$ for our branching rules). Hence the overall cost incurred by the Maximal Subset Collation step is

$O(A\cdot d\cdot\text{{poly}}(\left\lvert V\right\rvert))$

where $A$ is the total number of admissible subsets encountered by the MASE algorithm (equivalently, the number of leaves in the search tree). By the Combine Analysis Lemma for Enumeration (Lemma 7), $A$ is $O(2^{\mu(I)})$. The maximum depth $d$ is $O(\mu(I))$ which we may take to be $O(\left\lvert V\right\rvert)$. Hence Maximal Subset Collation incurs an overhead cost of $O^{*}(2^{\mu(I)})$.

As noted, our algorithm also satisfies the requirements for applying the Combine Analysis Lemma for Enumeration (Lemma 7). We summarize all the above results in the following theorem.

Theorem 4**.**

Let $\mu(I)=\frac{k}{2}+\frac{b}{4}$, where $b$ is the resolution order of $G[S]$. Then MASE can be solved in $O^{*}(2^{\mu(I)})$ time. Furthermore, there are at most $2^{\mu(I)}$ maximal admissible subsets of $S$ within distance $k$ of $S$. Hence, there are at most $2^{\mu(I)}$ preferred extensions that are subsets of $S$ with size $\geq\left\lvert S\right\rvert-k$.

3 Monotone Local Search

In this section, we apply Monotone Local Search to our $O^{*}(2^{\mu(I)})$ algorithm for Maximal Admissible Subset Enumeration. A basic exposition of Monotone Local Search will be provided here, additional background can be found in Appendix Section B.

The framework normally applies to extension problems. However we can just as easily apply it to removal problems (formally by focusing on the complement of each set, we can turn any removal problem into an extension problem). Hence, we will freely use the framework with removal problems instead.

For our application, the instance $I$ is the graph $G$ and the family we are looking to enumerate, $\mathcal{F}_{I}$, is the set of all preferred extensions of $G$. We will apply Monotone Local Search using MASE as our subroutine.

For a MASE instance $I^{\prime}=(G,X,k)$, let $\mathcal{F}_{I,X}^{k}$ denote the set of all maximal admissible sets $T\subseteq X$ such that $\left\lvert X\setminus T\right\rvert\leq k$. Hence $\mathcal{F}_{I,X}^{k}$ contains all preferred extensions that are subsets of $X$ within distance $k$, but may also contain admissible extensions that are not preferred extensions.

Because of this, our Monotone Local Search will enumerate $F_{I}$, however, it may also enumerate some non-maximal admissible extensions. Our result, Theorem 5, will account for this, however, for simplicity we will just speak of enumerating $F_{I}$ throughout this section.

A naive application of the Monotone Local Search framework with our $O^{*}\left(2^{\frac{k}{2}+\frac{b}{4}}\right)$ algorithm for enumerating $\mathcal{F}_{I,X}^{k}$ yields an $O^{*}\left(2^{\frac{b}{4}}\left(2-\frac{1}{\sqrt{2}}\right)^{n+o(n)}\right)\approx O^{*}(1.1893^{b}\cdot 1.2929^{n})$ time algorithm that enumerates all preferred extensions of $G$. To improve this we need a basic understanding of how Monotone Local Search works.

3.1 Basic Overview of Monotone Local Search

We need the following definition from [21] (slightly modified to account for our preference for removal problems):

Definition 3.1** ([21]).**

Let $U$ be a universe of size $n$ and let $0\leq p\leq q\leq n$. A family $C\subseteq\binom{U}{q}$ is an $(n,p,q)$-set-containing-family if for every set $S\in\binom{U}{p}$, there exists a $Y\in C$ such that $S\subseteq Y$.

For any fixed $s$, a value $t\geq s$ will somehow be chosen. Then, a $(n,s,t)$-set-containing-family $C_{s}$ is constructed. For each set in $C_{s}$, its subsets from $\mathcal{F}_{I}$ obtained by removing at most $t-s$ elements are then enumerated using an enumeration subroutine. This enumerates all elements of $\mathcal{F}_{I}$ with size $s$. Supposing the subroutine has run time $O^{*}(\alpha^{k})$ (where $k$ is the parameter), this step of Monotone Local Search has run time $O^{*}(\left\lvert C_{s}\right\rvert\cdot\alpha^{t-s})$.

Repeating this for all $s$, Monotone Local Search has running time $O^{*}(\max\limits_{1\leq s\leq n}\left\lvert C_{s}\right\rvert\cdot\alpha^{t-s})$. With the right choices of $t$, [21] shows the running time $O^{*}((2-\frac{1}{\alpha})^{n+o(n)})$.

3.2 Improving our Monotone Local Search

We start with some notation. For any digraph $G=(V,E)$, let $B$ be the set of vertices in $V$ in at least one 2-cycle and let $D:=V\setminus B$. The key idea is we will sample vertices from $B$ and $D$ separately. The overall structure is described in Algorithm 2. Except for the separate sampling, it is identical to a standard application of Monotone Local Search.

First, we argue correctness, i.e: that every preferred extension is enumerated at least once. Fix a preferred extension, say $U$, and suppose $U$ contains $b$ vertices in $B$ and $d$ vertices in $D$. Then, by the definition of set-containing-families, there exists a $S\in C_{b}$ such that $U\cap B\subseteq S$ and a $T\in C_{d}$ such that $U\setminus B\subseteq T$. Now we note that $S\subseteq B,T\subseteq V\setminus B$ to get:

[TABLE]

Hence $U$ is enumerated in the call to $\mathrm{MASE}(S\cup T,(b^{\prime}-b)+(d^{\prime}-d))$ as required.

Now we argue the runtime. For a fixed $b,d$ the calls to MASE have total run time $O^{*}(\left\lvert C_{b}\right\rvert\cdot\left\lvert C_{d}\right\rvert\cdot 2^{\frac{(b^{\prime}-b)+(d^{\prime}-d)}{2}+\frac{b^{\prime}}{4}})$ for some choice of $b^{\prime},d^{\prime}$. Our overall running time is

$O^{*}\left(\max\limits_{1\leq b\leq\left\lvert B\right\rvert}\max\limits_{1\leq d\leq\left\lvert D\right\rvert}\left\lvert C_{b}\right\rvert\cdot\left\lvert C_{d}\right\rvert\cdot 2^{\frac{(b^{\prime}-b)+(d^{\prime}-d)}{2}+\frac{b^{\prime}}{4}}\right)$

We can split this into two terms to get a complexity of $O^{*}\left(\max\limits_{1\leq d\leq\left\lvert D\right\rvert}\left\lvert C_{d}\right\rvert\cdot 2^{\frac{(d^{\prime}-d)}{2}}\right)$ multiplied by $O^{*}\left(\max\limits_{1\leq b\leq\left\lvert B\right\rvert}\left\lvert C_{b}\right\rvert\cdot 2^{\frac{(b^{\prime}-b)}{2}+\frac{b^{\prime}}{4}}\right)$.

We will now analyze the running time with the right choices of $b^{\prime}$ and $d^{\prime}$. Theorem 7 summarizes the results we use to analyze the complexity.

The first of these two terms can be analyzed using the analysis in [21]. This term is the complexity one attains for Monotone Local Search with a $O^{*}(2^{\frac{k}{2}})$ subroutine. Hence the analysis in [21] gives a complexity of $O^{*}((2-\frac{1}{\sqrt{2}})^{\left\lvert D\right\rvert+o(\left\lvert D\right\rvert)})$.

We need the extended analysis presented in [27] to analyze the second term. This term is the complexity one attains for Monotone Local Search with a $O^{*}(2^{\frac{k}{2}+\frac{n-\left\lvert X\right\rvert}{4}})$ subroutine (where $k$ is the parameter, $n=\left\lvert U\right\rvert$ the size of the underlying set and $X$ is the set we are extending). Hence the analysis in [27] gives a complexity of $O^{*}((1+2^{\frac{1}{4}}-\frac{1}{\sqrt{2}})^{\left\lvert B\right\rvert+o(\left\lvert B\right\rvert)})$.

The papers we have cited also give a corresponding combinatorial upper bound on the number of preferred extensions. We summarize the above results as follows.

Theorem 5**.**

Let $G=(V,E)$ be a digraph. Let $r$ be the proportion of vertices in $V$ that are in at least one 2-cycle.

Then there exists a $O^{*}(((1+2^{\frac{1}{4}}-\frac{1}{\sqrt{2}})^{r}(2-\frac{1}{\sqrt{2}})^{1-r})^{\left\lvert V\right\rvert+o(\left\lvert V\right\rvert)})$ $\approx$ $O^{*}((1.4822^{r}1.2929^{1-r})^{\left\lvert V\right\rvert})$ time algorithm that enumerates all preferred extensions of $G$; however it may also enumerate some non-maximal admissible extensions. Furthermore, there are at most $O^{*}(((1+2^{\frac{1}{4}}-\frac{1}{\sqrt{2}})^{r}(2-\frac{1}{\sqrt{2}})^{1-r})^{\left\lvert V\right\rvert})$ $\approx$ $O^{*}((1.4822^{r}1.2929^{1-r})^{\left\lvert V\right\rvert})$ preferred extensions in $G$.

4 Improved Enumeration Algorithm for Oriented Graphs

Finally, we outline a branching algorithm with a finer analysis for oriented graphs. As with MASE, we will follow a standard template for branching algorithms. A summary of the necessary concepts can be found in Appendix Section A. Our algorithm creates a search tree with at most $\varphi^{n}$ leaves where $\varphi\approx 1.2321$ is the branching number for branching vector $(8,1)$.

In Section 5 we will use the Oriented Translation (Theorem 2) to obtain a general enumeration algorithm parameterized by the number of vertices in at least one 2-cycle.

4.1 Overview

The overall structure of our algorithm is described in Algorithm 3.

The state of the algorithm consists of the subset $\operatorname{Und}\subseteq V(G)$ and a queue of vertices $\operatorname{Def}$ ($\operatorname{Und}$ for undecided and $\operatorname{Def}$ for deferred). $\operatorname{Und}$ is the set of vertices we have yet to make a decision on whether we should include them. $\operatorname{Def}$ is a queue of vertices which have no outgoing arcs to $\operatorname{Und}$. These vertices will be handled in the base case. While branching we can essentially assume the vertices in $\operatorname{Def}$ do not exist.

We maintain the following invariants. We outline why they are invariant, it is straight-forward to verify that each case of our branching algorithm maintains these invariants.

$\operatorname{Und}$ and $\operatorname{Def}$ are disjoint. This holds as vertices are only ever deleted from or moved between $\operatorname{Und}$ and $\operatorname{Def}$, never copied. 2. 2.

$G[\operatorname{Def}]$ is a DAG where each vertex $v\in\operatorname{Def}$ only attacks vertices that were added to $\operatorname{Def}$ before $v$. This is crucial for the base case since a DAG has 1 maximal admissible subset. This holds as only vertices with out-degree [math] in $G[\operatorname{Und}]$ are ever moved to $\operatorname{Def}$. 3. 3.

There is no attack from any vertex in $\operatorname{Def}$ to any vertex in $\operatorname{Und}$. This holds for the same reason as Invariant 2.

Our measure is $\mu=\left\lvert\operatorname{Und}\right\rvert$. For any instance, our algorithm creates a search tree with at most $\varphi^{\mu}$ leaves. Hence, calling $\mathrm{OrientedEnumeration}(V(G),[])$ will return all preferred extensions of $G$ by traversing a search tree with at most $\varphi^{|V(G)|}$ leaves.

4.2 Extra Notation

We will say a vertex has degree $(a,-)$ if it has in-degree $a$, a vertex has degree $(-,b)$ if it has out-degree $b$ and a vertex has degree $(a,b)$ if it has in-degree $a$ and out-degree $b$.

4.3 Simplification Rules

Both of these are applicable in polynomial time and decrease $\mu=\left\lvert\operatorname{Und}\right\rvert$.

Simplification Rule 2** (Out-degree 0).**

Let $v$ be a vertex in $G[\operatorname{Und}]$ with out-degree [math]. Move $v$ from $\operatorname{Und}$ to the end of the queue $\operatorname{Def}$.

Simplification Rule 3** (In-degree 0).**

Let $v$ be any vertex in $G[\operatorname{Und}]$ with in-degree [math]. Then by Invariant 3, $v$ has in-degree [math] in $G[\operatorname{Und}\cup\operatorname{Def}]$. Applying Simplification Rule (Undefendable) we can set $\operatorname{Und}\leftarrow\operatorname{Und}\setminus N(v)$. After that, $v$ has out-degree 0 in $G[\operatorname{Und}]$ and hence we move $v$ from $\operatorname{Und}$ to $\operatorname{Def}$.

Our new instance $I^{\prime}=(\operatorname{Und}^{\prime},\operatorname{Def}^{\prime})$ has:

• •

$\operatorname{Und}^{\prime}=\operatorname{Und}\setminus N[v]$.

• •

$\operatorname{Def}^{\prime}=\operatorname{Def}\cup\{v\}$.

Due to these rules, henceforth we may assume each vertex in $G[\operatorname{Und}]$ has in-degree $\geq 1$, out-degree $\geq 1$ and (total) degree $\geq 2$.

4.4 Branching Rules

Our algorithm will apply the first rule applicable to the instance:

[TABLE]

We note the base case and cases 3 and 6 cover all possible inputs. Hence there will always be at least one applicable rule.

Our primary strategy is the following branching rule applied to a specifically chosen $v$.

4.5 2-way branch on whether to include $v$

We will often pick a specific vertex $v$ and do a 2-way branch, in one branch enumerating all maximal admissible subsets that include $v$, in the other branch enumerating all that do not include $v$.

• •

In the branch where we decide to include $v$ we can not include any of $v$’s neighbors. Hence we create a new instance $I^{\prime}=(\operatorname{Und}^{\prime},\operatorname{Def}^{\prime})$ where $\operatorname{Und}^{\prime}=\operatorname{Und}\setminus N(v)$.

Now $v$ is isolated in $G[\operatorname{Und}^{\prime}]$ so by Simplification Rule 0 we may move $v$ from $\operatorname{Und}^{\prime}$ to $\operatorname{Def}^{\prime}$. Hence in our new instance we finally have:

• –

$\operatorname{Und}^{\prime}=\operatorname{Und}\setminus N[v]$.

• –

$\operatorname{Def}^{\prime}=\operatorname{Def}\cup\{v\}$.

and hence $\mu(I^{\prime})=\mu(I)-\left\lvert N[v]\right\rvert=\mu(I)-\deg(v)-1$.

As a technical note, not every subset enumerated in this branch contains $v$, however, every maximal admissible subset that contains $v$ will be enumerated in this branch.

• •

In the other branch we decide to exclude $v$. Hence our new instance $I^{\prime}=(\operatorname{Und}^{\prime},\operatorname{Def}^{\prime})$ has:

• –

$\operatorname{Und}^{\prime}=\operatorname{Und}\setminus\{v\}$.

• –

$\operatorname{Def}^{\prime}=\operatorname{Def}$.

and hence $\mu(I^{\prime})=\mu(I)-1$.

Hence branching on whether we should include $v$ leads to a branching rule with branching vector $(\deg(v)+1,1)$.

Definition 4.1** (2-way branch on whether to include $v$).**

The phrase 2-way branch on whether to include $v$ will be used as shorthand for the above branching rule.

4.6 Remarks on cases

Full proofs can be found in Appendix Section H.

In the base case, by Invariant 2, $G[\operatorname{Und}\cup\operatorname{Def}]$ is a DAG. The base case then follows trivially from Lemma 4.

Case 1 follows from a 2-way branch on whether to include $v$(Definition 4.1), where $v$ is any vertex with total degree at least 7.

Case 2 is a good example of the strategy of picking a specific $v$ to apply a 2-way branch on. We will pick a $v$ that allows us to apply our simplification rules in the branch where $v$ is excluded. However, the choice of $v$ will depend on some case analysis.

Case 3 is similar to Case 2, a specific $v$ is chosen on which we apply a 2-way branch.

Case 4 is done through a graph classification theorem(Theorem 10).

Case 5 is just a 4-way branch on any vertex $v$ and its $3$ in-neighbors.

For Case 6, we first show there is a vertex with degree $(3,3)$ attacking a vertex with degree $(2,2)$, say $b$. Then we apply a 3-way branch on $b$ and its $2$ in-neighbors.

4.7 Summary of results

In our base case we enumerate 1 extension in polynomial time.

Otherwise, we apply a branching rule with branching number $\leq\varphi$.

As in MASE, the Maximal Subset Collation subroutine (Algorithm 4) does not incur additional overhead as the search tree’s depth is bounded by $\left\lvert V(G)\right\rvert$ (see Subsection 2.4 for the argument which we can apply verbatim).

Applying the Combine Analysis Lemma(Lemma 6) and Combine Analysis Lemma for Enumeration (Lemma 7) we obtain:

Theorem 6**.**

Let $G=(V,E)$ be an oriented graph. Then there is an algorithm that enumerates all preferred extensions of $G$ with running time $O^{*}(\varphi^{\left\lvert V\right\rvert})$ where $\varphi$ is the unique positive root of $1-x^{-1}-x^{-8}=0$, $\varphi\approx 1.23205<1.2321$.

Furthermore, $G$ has at most $\varphi^{\left\lvert V\right\rvert}$ preferred extensions.

5 Bounds on number of preferred extensions

In this section, we collect our results bounding the number of preferred extensions.

5.1 Bounds on general directed graphs

A tight upper bound is $O(3^{\frac{\left\lvert V\right\rvert}{3}})$ [19]. This bound is easily attainable since preferred extensions coincide with maximal independent sets(MIS) in graphs where every edge is in a 2-cycle.

We can also enumerate them in $O^{*}(3^{\frac{\left\lvert V\right\rvert}{3}})$. We note each MIS has a single maximal admissible subset. We then take a branching algorithm for MIS[26], allowing us to apply the Maximal Subset Collation subroutine (Algorithm 4) to remove the non preferred extensions without additional overhead. Hence, all preferred extensions can be enumerated in $O^{*}(3^{\frac{\left\lvert V\right\rvert}{3}})$.

5.2 Parameterizing by Resolution Order

We will give bounds based on $r$, the proportion of vertices that are in at least one 2-cycle.

5.2.1 Lower Bound

An undirected 3-cycle has 3 preferred extensions. Hence, applying the Oriented Translation to it, we obtain an oriented structure with 6 vertices and 3 preferred extensions.

Our construction for lower bounding will be to include as many undirected 3-cycles as possible and then include as many oriented translations of 3-cycles as possible. We can include $\frac{r\left\lvert V\right\rvert}{3}$ undirected 3-cycles and $\frac{(1-r)\left\lvert V\right\rvert}{6}$ oriented translations, obtaining an AF with $\Omega((3^{\frac{r}{3}}3^{\frac{1-r}{6}})^{\left\lvert V\right\rvert})\approx((1.44^{r}1.2^{1-r})^{\left\lvert V\right\rvert})$ preferred extensions.

5.2.2 Upper Bound

There are 3 different upper bounds that are all optimal in a different range. See also Figure 1.

• •

$(\varphi^{2r}\varphi^{1-r})^{\left\lvert V\right\rvert}\approx O((1.5180^{r}1.2321^{1-r})^{\left\lvert V\right\rvert})$ where $\varphi$ is the unique positive root of $1-x^{-1}-x^{-8}$. This bound is obtained from using the Oriented Translation along with Theorem 6. This is best for $r$ up to around 0.6684.

• •

$O^{*}(((1+2^{\frac{1}{4}}-\frac{1}{\sqrt{2}})^{r}(2-\frac{1}{\sqrt{2}})^{1-r})^{\left\lvert V\right\rvert})$ $\approx$ $O^{*}((1.4822^{r}1.2929^{1-r})^{\left\lvert V\right\rvert})$, the bound from our 2-phase Monotone Local Search. This is best for a small range where $0.6685\leq r\leq 0.8004$.

• •

$3^{\frac{\left\lvert V\right\rvert}{3}}$. This is best for $r\geq 0.8005$.

6 Conclusion

We again note that the concept of an admissible (resp. preferred) extension has also been studied as a semikernel[33] (resp. maximal semikernel) in graph theory. Hence our result may be interpreted as a combinatorial upper bound on the number of maximal semikernels, parameterized by the proportion of vertices in at least one 2-cycle.

Acknowledgment

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

Appendix A Background on Analysis of Branching Algorithms

This section provides the required background for the analysis of branching algorithms used in this paper. All results here can be found in standard textbooks. We mostly follow Fomin & Kratsch [23]:

For the analysis of branching algorithms, we use Measure & Conquer [22] with relatively simple measures. A measure $\mu$ is a function assigning a non-negative number to each instance $I$. For each instance, either the instance is directly solvable in polynomial time or our algorithm will apply a polynomial time branching rule to generate multiple sub-instances with smaller measure that are then recursed on.

The following notation (which is used in standard textbooks on exact exponential algorithms [23] and parameterized algorithms [14]) will simplify the discussion of these algorithms.

Definition A.1** (Branching Vector [23]).**

Let $\mu$ be a measure. Let $b$ be a branching rule that for any input instance, say $I$, branches into $r$ instances with measures $\mu(I)-t_{1},\mu(I)-t_{2},\ldots,\mu(I)-t_{r}$ such that for all $i$, $t_{i}>0$. Then we call $\mathbf{b}=(t_{1},t_{2},\ldots,t_{r})$ the branching vector of branching rule $b$.

The following lemma from [23] forms the basis for the analysis of our branching algorithms.

Lemma 5** ([23]).**

Let $b$ be a branching rule with branching vector $(t_{1},t_{2},\ldots t_{r})$. Then the running time of the branching algorithm using only branching rule $b$ is $O^{*}(\alpha^{\mu(I)})$ where $\alpha$ is the unique positive real root of:

[TABLE]

We call $\alpha$ the branching number of the branching vector $\mathbf{b}$.

In [23], it is further noted that if an algorithm has multiple branching rules, then its running times is $O^{*}(\alpha^{n})$ where $\alpha=\max_{i}\alpha_{i}$. We will actually need the following simple extension, which has been simplified from [26] and restated to use our notation.

Lemma 6** (Combine Analysis Lemma [26]).**

Let $A$ be an algorithm for a problem which for each instance, say $I$, either directly solves the instance in $O^{*}(\alpha_{0}^{\mu(I)})$ time or after polynomial time, applies one of $r$ branching rules $b_{i}$, the $i$-th of which has branching number $\alpha_{i}$. Then $A$ has running time $O^{*}(\alpha^{\mu(I)})$ where $\alpha=\max_{0\leq i\leq r}(\alpha_{i})$.

For our enumeration bounds we will need the following variant of the above lemma.

First, we define the search tree of a branching algorithm to be the tree formed by the recursive calls, with the leaves being cases that can be solved directly. Generally, and in all our applications, for enumeration algorithms the number of leaves is an upper bound on the number of objects being enumerated (equality may not hold as the same object may appear as a leaf multiple times).

Lemma 7** (Combine Analysis Lemma for Enumeration).**

Let $A$ be an algorithm for a problem which for each instance, say $I$, either directly solves the instance with a branching algorithm that generates at most $\alpha_{0}^{\mu(I)}$ leaves or applies one of $r$ branching rules $b_{i}$, the $i$-th of which has branching number $\alpha_{i}$. Then the search tree for applying $A$ to $I$ has at most $\alpha^{\mu(I)}$ leaves where $\alpha=\max_{i}(\alpha_{i})$.

Appendix B Background on Monotone Local Search

Monotone Local Search [21] is a technique to design exponential-time algorithms for subset problems based on single-exponential parameterized algorithms. We give an exposition of the corresponding enumeration algorithms from [21], and the extension in [27].

An implicit set system is a function $\Phi$ that takes as input a string $I\in\{0,1\}^{*}$ and outputs a set system $(U_{I},\mathcal{F}_{I})$, where $U_{I}$ is a universe and $\mathcal{F}_{I}$ is a collection of subsets of $U_{I}$. The string $I$ is referred to as an instance and we denote by $|U_{I}|=n$ the size of the universe and by $|I|=N$ the size of the instance. We assume that $N\geq n$. An implicit set system $\Phi$ is polynomial time computable if: (a) there exists a polynomial time algorithm that given $I$ produces $U_{I}$, and (b) there exists a polynomial time algorithm that given $I$, $U_{I}$ and a subset $S$ of $U_{I}$ determines whether $S\in\mathcal{F}_{I}$.

Let $c,b\geq 1$ be real valued constants and $\Phi$ be an implicit set system. Then $\Phi$ is ($b,c$)-uniform if for every instance $I$, set $X\subseteq U_{I}$, and integer $k\leq n-|X|$, the cardinality of the collection

[TABLE]

is at most $b^{n-|X|}c^{k}n^{O(1)}$. If there is an algorithm that given any instance $I$, set $X\subseteq U_{I}$, and integer $k\leq n-|X|$, enumerates all elements of $\mathcal{F}_{I,X}^{k}$ in time $b^{n-|X|}c^{k}N^{O(1)}$, then $\Phi$ is said to be efficiently ($b,c$)-uniform. In this case, the algorithm from [27] enumerates $\mathcal{F}_{I}$ in $O^{*}\left(\left(1+b-\frac{1}{c}\right)^{n+o(n)}\right)$ time.

Theorem 7**.**

Let $c,b\geq 1$ and $\Phi$ be a polynomial time computable implicit set system.

If $\Phi$ is $(b,c)$-uniform, then $|\mathcal{F}_{I}|\leq\left(1+b-\frac{1}{c}\right)^{n}n^{O(1)}$ for every instance $I$.

If $\Phi$ is efficiently $(b,c)$-uniform, then there is an algorithm that, given $I$ as input, enumerates $\mathcal{F}_{I}$ in time $\left(1+b-\frac{1}{c}\right)^{n+o(n)}N^{O(1)}$.

The original paper on Monotone Local Search [21] proved the theorem for $b=1$.

Appendix C Oriented Translation

In this section we present the construction we use to transform directed graphs into oriented graphs (see Theorem 2) while preserving the preferred extensions.

This will be useful for lifting complexity results from directed graphs to oriented graphs. It is also used for automatically converting oriented graph algorithms into algorithms for general graphs parameterized by resolution order.

To remove special cases, we first present a translation that eliminates self loops from an AF. The only purpose of self loops is to prevent vertices from being included in admissible sets. Hence, we can replace self loops with attacks from an undefendable vertex.

Loopless Translation

Definition C.1** (Loopless Translation).**

Let $G=(V,E)$ be an arbitrary AF. The loopless translation of $G$ is the graph $G^{\prime}=(V^{\prime},E^{\prime})$ where:

• •

$V^{\prime}=V\cup\{l_{1},l_{2},l_{3}\}$.

• •

$E^{\prime}=(E\setminus\{(v,v):v\in V\})\cup\{(l_{1},v):v\in V,(v,v)\in E\}\cup\{(l_{1},l_{2}),(l_{2},l_{3}),(l_{3},l_{1})\}$**

An example translation of $V=\{v_{1},v_{2}\},E=\{(v_{1},v_{1}),(v_{1},v_{2})\}$ can be found in Figure 2.

Lemma 8**.**

The admissible extensions of $G$ are exactly the admissible extensions of the Loopless Translation $G^{\prime}$ of $G$.

Proof.

The odd cycle among $\{l_{1},l_{2},l_{3}\}$ ensures none of $l_{1},l_{2},l_{3}$ are in any admissible extension of $G^{\prime}$. Hence $l_{1}$ can not be attacked in any admissible extension of $G^{\prime}$. Hence, no vertex with a self loop in $G$ is in any admissible extension of $G^{\prime}$.

Now it is straight forward to check $S\subseteq V(G^{\prime})\setminus\{l_{1},l_{2},l_{3}\}$ is an admissible extension of $G^{\prime}$ if and only if it is an admissible extension of $G$. ∎

We summarize the result as follows.

Lemma 9**.**

Let $G$ be an AF. Then there is a linear time algorithm that transforms $G$ into an AF $G^{\prime}$, with $\left\lvert V(G^{\prime})\right\rvert=\left\lvert V(G)\right\rvert+3$, that has no self loops.

Furthermore there is an inclusion preserving bijection between the admissible extensions of $G$ and the admissible extensions of $G^{\prime}$ that can be applied in linear time.

This is why we assume in all other results that graphs have no self loops.

Simple Oriented Translation

For pedagogical purposes we first present an oriented translation that doubles the number of vertices. The full oriented translation algorithm follows from noting that only vertices in a 2-cycle need to be duplicated.

Our strategy is to replace each 2-cycle with a 4-cycle. However, a bit more care is necessary to ensure a bijection between preferred extensions in both graphs.

Definition C.2** (Simple Oriented Translation).**

Let $G=(V,E)$ be an arbitrary AF. First apply the Loopless Translation (Definition C.1) to $G$. Arbitrarily order the vertices in $V$ with a total order $<$. For simplicity, relabel $V$ such that $V=\{v_{1},v_{2},\ldots v_{n}\}$ with $v_{1}<v_{2}<\ldots<v_{n}$. Let $F$ be the set of edges in $G$ that are in a 2-cycle. Then the simple oriented translation of $G$ is the graph $G^{\prime}=(V^{\prime},E^{\prime})$ where:

• •

$V^{\prime}=\{v_{i}:1\leq i\leq n\}\cup\{w_{i}:1\leq i\leq n\}$**

• •

$E^{\prime}$* is the union of the following sets:*

• –

4-cycle edges: $\{(v_{i},v_{j}),(v_{j},w_{i}),(w_{i},w_{j}),(w_{j},v_{i}):(v_{i},v_{j})\in F\land i<j\}$.

• –

Unidirectional edges: $\{(v_{i},v_{j}),(v_{i},w_{j}),(w_{i},v_{j}),(w_{i},w_{j}):(v_{i},v_{j})\in E,(v_{i},v_{j})\notin F\}$.

An example translation of $V=\{v_{1},v_{2},v_{3}\},E=\{(v_{1},v_{2}),(v_{2},v_{1}),(v_{2},v_{3})\}$ can be found in Figure 3.

We need to show that $G^{\prime}$ has a few nice properties. First:

Lemma 10**.**

$G^{\prime}$* is oriented.*

Proof.

This is clear from a case analysis on the edges of $G^{\prime}$. ∎

Next, we want to show this translation nicely preserves preferred extensions. We first need a lemma controlling the structure of preferred extensions in $G^{\prime}$.

Lemma 11**.**

Let $D$ be a preferred extension of $G^{\prime}=(V^{\prime},E^{\prime})$. For any $i$, if $v_{i}$ or $w_{i}$ are in $D$, then both $v_{i}$ and $w_{i}$ are in $D$.

Proof.

First we note $N(v_{i})=N(w_{i})$ so if $v_{i}$ is conflict-free with respect to $D$, then $w_{i}$ is conflict-free with respect to $D\cup\{v_{i}\}$.

We then note that every vertex attacking $w_{i}$ either also attacks $v_{i}$ or is attacked by $v_{i}$. Hence if $v_{i}$ is acceptable with respect to $D$, then so is $w_{i}$.

Therefore if $v_{i}\in D$ is admissible, then so is $w_{i}$ with respect to $D$. Hence, by maximality, $w_{i}\in D$. This clearly also applies if $w_{i}\in D$. ∎

Now the main result:

Lemma 12**.**

Let $G$ be an AF. Let $G^{\prime}$ be the oriented AF obtained from applying the simple oriented translation on $G$. Let $\psi$ be the following function from subsets of $V(G)$ to subsets of $V(G^{\prime})$:

[TABLE]

Then $\psi$ is a bijection between the preferred extensions of $G$ and the preferred extensions of $G^{\prime}$.

Proof.

It is clear that $(v_{i},v_{j})\in G\iff\{v_{i},w_{i}\}$ attacks both $v_{j}$ and $w_{j}$. Hence $\psi$ is a bijection between the preferred extensions of $G$ and the preferred extensions of $G^{\prime}$ where $(v_{i}\in S\lor w_{i}\in S)\implies(v_{i}\in S\land w_{i}\in S)$. However Lemma 11 states that every preferred extension of $G^{\prime}$ satisfies this property. ∎

Full Oriented Translation

We have shown that the simple oriented translation does biject preferred extensions. We now optimize the translation which is necessary for our parameterized results.

Namely, if $v_{i}\in V$ is not in any 2-cycles, then it is easy to see that $v_{i}$ and $w_{i}$ have the exact same in-neighbors and out-neighbors in $G^{\prime}$. Hence, $v_{i}$ and $w_{i}$ are essentially identical and merging $v_{i}$ and $w_{i}$ into a single vertex still preserves all the above lemmas. Formally, an explicit construction of the oriented translation is:

Definition C.3** (Oriented Translation).**

Let $G=(V,E)$ be an arbitrary AF. First apply the Loopless Translation (Definition C.1) to $G$. Arbitrarily order the vertices in $V$ with a total order $<$. For simplicity, relabel $V$ such that $V=\{v_{1},v_{2},\ldots v_{n}\}$ with $v_{1}<v_{2}<\ldots<v_{n}$. Let $S$ be the vertices in $G$ that are in at least one 2-cycle. Let $F$ be the set of edges in $G$ that are in a 2-cycle.

Then the oriented translation of $G$ is the graph $G^{\prime}=(V^{\prime},E^{\prime})$ where:

• •

$V^{\prime}=\{v_{i}:1\leq i\leq n\}\cup\{w_{i}:v_{i}\in S\}$**

• •

$E^{\prime}$* is the union of the following sets:*

• –

4-cycle edges: $\{(v_{i},v_{j}),(v_{j},w_{i}),(w_{i},w_{j}),(w_{j},v_{i}):(v_{i},v_{j})\in F\land i<j\}$. We note that all these vertices exist as $v_{i},v_{j}\in S$ since there is a 2-cycle between $v_{i}$ and $v_{j}$.

• –

Unidirectional edges: $\{(v_{i},v_{j}),(v_{i},w_{j}),(w_{i},v_{j}),(w_{i},w_{j}):(v_{i},v_{j})\in E,(v_{i},v_{j})\notin F\}$. Leave out of $E^{\prime}$ all edges in this category that have a non-existent endpoint (this may happen as $w_{i}$ does not necessarily exist for all $i$).

From this construction, it is clear the lemmas for the Simple Oriented Translation still hold in slightly modified form:

Lemma 13**.**

$G^{\prime}$* is oriented.*

Lemma 14**.**

Let $G$ be an AF. Let $G^{\prime}$ be the oriented AF obtained from applying the oriented translation to $G$. Let $\psi$ be the following function from subsets of $V(G)$ to subsets of $V(G^{\prime})$:

[TABLE]

Then $\psi$ is a bijection between the preferred extensions of $G$ and the preferred extensions of $G^{\prime}$.

The following theorem summaries the main results of this section:

Theorem 8**.**

Let $G$ be an AF. Let $r(G)$ be the number of vertices in $G$ that are in at least one 2-cycle.

There is a linear time algorithm that transforms any AF $G$ into an oriented AF $G^{\prime}$ with $\left\lvert V(G^{\prime})\right\rvert=\left\lvert V(G)\right\rvert+r(G)+3$ such that there is a bijection between the preferred extensions of $G$ and the preferred extensions of $G^{\prime}$ that can be applied in linear time.

The 3 in $\left\lvert V(G^{\prime})\right\rvert=\left\lvert V(G)\right\rvert+r(G)+3$ comes from the Loopless Translation (see Lemma 9).

Appendix D Maximal Subset Collation

Recall we define the problem as:

Maximal Subset Collation

Input:

Graph $G$, $c$ pairs $(S_{i},C_{i})$ where for each $i$, $S_{i}\subseteq V(G)$ and $C_{i}$ is a set containing only maximal admissible subsets of $S_{i}$

Output:

A set containing all maximal elements of $\bigcup_{i=1}^{c}C_{i}$

The key to our algorithm for Maximal Subset Collation is the following lemma:

Lemma 15**.**

Suppose $T\in C_{i}$ but $T\subsetneq U\in\bigcup_{i=1}^{c}C_{i}$. Then $T$ is the unique maximal admissible subset of $U\cap S_{i}$.

Proof.

First, we note that $U\cap S_{i}$ has a unique maximal admissible subset due to Lemma 4 ($U$ is admissible, hence conflict-free, hence $U\cap S_{i}$ induces a DAG).

It is clear $T\subseteq U\cap S_{i}$. Then, since $T$ is a maximal admissible subset of $S_{i}$, $T$ must be the unique maximal admissible subset of $U\cap S_{i}$ as required. ∎

Now, our algorithm for Maximal Subset Collation is straight-forward with pseudocode in Algorithm 4. Lemma 15 guarantees all non-maximal subsets in $\bigcup_{i=1}^{c}C_{i}$ are removed. By Lemma 4, finding the Unique Maximal Admissible Subset of $U\cap S_{i}$ can be done in polynomial time. Using a hashmap or balanced binary search tree to store $R$ (in Algorithm 4) yields the following result:

Lemma 16**.**

Maximal Subset Collation can be solved in $O(c\sum_{i=1}^{c}\left\lvert C_{i}\right\rvert\cdot\text{{poly}}(\left\lvert V\right\rvert))$ time.

Appendix E Detailed Analysis of our $O^{*}(2^{\mu(I)})$ algorithm for Maximal Admissible Subset Enumeration

Recall we have the following cases and apply them in the listed order:

[TABLE]

These requirements are exhaustive, hence there will always be at least one applicable rule in any instance.

E.1 Base Case - $S$ is conflict-free

By Lemma 4, $S$ has a unique maximal admissible subset, $T$, which can be found in polynomial time. Return this subset if $\left\lvert S\setminus T\right\rvert\leq k$.

E.2 Case 1 - Oriented, degree $\leq 2$ graph

As the maximum total degree in $G[S]$ is $\leq 2$, ignoring arc directions, $G[S]$ decomposes into disjoint paths and cycles. Since (Undefendable) is not applicable, each non-isolated vertex in $G[S]$ must have an in-neighbor. Hence, we can further deduce that $G[S]$ decomposes into disjoint directed cycles and isolated vertices.

Let $v\in S$ be any non-isolated vertex in $G[S]$ (such a vertex exists as $S$ is not conflict-free since the base case is not applicable). We do a 2-way branch on whether $v$ is in the admissible extension.

In one branch, we assume $v$ is in the admissible extension. Then neither of $v$’s neighbors are in the admissible extension. Hence our new instance $I^{\prime}=(S^{\prime},k^{\prime},b^{\prime})$ has:

• •

$S^{\prime}=S\setminus N(v)$.

• •

$k^{\prime}=k-2$. The two neighbors of $v$ are distinct as $G[S]$ is oriented.

• •

$b^{\prime}\leq b$.

and hence $\mu(I^{\prime})\leq\mu(I)-\frac{2}{2}=\mu(I)-1$.

In the other branch, we remove $v$ from $S$. After this, the directed cycle that $v$ was a part of is now a directed path. This path has at least two vertices as, by assumption, $G[S]$ is oriented. Hence, after removing $v$, we can apply the rule (Undefendable) to prune at least one more vertex, say $w$, from $S$. Hence our new instance $I^{\prime}=(S^{\prime},k^{\prime},b^{\prime})$ has:

• •

$S^{\prime}=S\setminus\{v,w\}$.

• •

$k^{\prime}=k-2$.

• •

$b^{\prime}\leq b$.

and hence $\mu(I^{\prime})\leq\mu(I)-\frac{2}{2}=\mu(I)-1$.

Hence we get a branching vector of $(1,1)$ with branching number $2$.

E.3 Case 2 - 2-cycle exists

Let $v,w\in S$ be two vertices such that $(v,w),(w,v)\in E$. There are two cases depending on whether $\{v,w\}$ is a maximal weakly connected component in $G[S]$.

Case 1: If $\{v,w\}$ is a maximal weakly connected component, then we branch on whether $v$ is in the admissible extension.

In the branch where we include $v$ we must remove $w$. Hence our new instance has $S^{\prime}=S\setminus\{w\},k^{\prime}=k-1$.

In the other branch we remove $v$. Hence our new instance has $S^{\prime}=S\setminus\{v\},k^{\prime}=k-1$.

In both cases $v$ and $w$ are no longer in any 2-cycles and hence $b^{\prime}=b-2$. Therefore, in both cases our new instance has $\mu(I^{\prime})=\mu(I)-\frac{1}{2}-\frac{2}{4}=\mu(I)-1$ and we get a branching vector of $(1,1)$ with branching number $2$.

Case 2: If $\{v,w\}$ is not a maximal weakly connected component, then one of these vertices has another neighbor. Without loss of generality, we assume $v$ has at least two distinct neighbors. Again, we branch on whether we include $v$ in the admissible extension.

In the branch where we include $v$ we must remove $N(v)$. Furthermore, since $v$ is no longer in any 2-cycles and $w$ has been removed, $b$ decreases by at least 2. Hence our new instance $I^{\prime}=(S^{\prime},k^{\prime},b^{\prime})$ has:

• •

$S^{\prime}=S\setminus N(v)$.

• •

$k^{\prime}=k-\left\lvert N(v)\right\rvert\leq k-2$.

• •

$b^{\prime}\leq b-2$.

and hence $\mu(I^{\prime})\leq\mu(I)-\frac{2}{2}-\frac{2}{4}=\mu(I)-\frac{3}{2}$.

In the branch where we exclude $v$ we remove $v$ from $S$. Hence in our new instance $I^{\prime}=(S^{\prime},k^{\prime},b^{\prime})$:

• •

$S\leftarrow S\setminus\{v\}$.

• •

$k^{\prime}=k-1$.

• •

$b^{\prime}\leq b-1$ as $v$ is in a 2-cycle.

and hence $\mu(I^{\prime})\leq\mu(I)-\frac{1}{2}-\frac{1}{4}=\mu(I)-\frac{3}{4}$.

Hence we get a branching vector of $(\frac{3}{2},\frac{3}{4})$ with branching number $\approx 1.9<2$.

E.4 Case 3 - Total Degree $\geq 4$ exists

Let $v$ be any vertex in $G[S]$ with total degree $\geq 4$. So $\left\lvert N(v)\right\rvert\geq 4$ as we are assuming $G$ has no self-loops and after case 2, we may also assume $G[S]$ is oriented.

We branch on whether to include $v$ in our admissible subset. If we include $v$, we must remove its $\geq 4$ neighbors. Otherwise we remove $v$. In the first case the new instance has $k^{\prime}\leq k-4$. In the second case the new instance has $k^{\prime}=k-1$.

Hence we get a branching vector of $(2,\frac{1}{2})$ with branching number $\approx 1.91<2$.

E.5 Case 4 - Degree 3 exists

We need the following lemma on the graph structure at this point. In this branching rule the conditions are satisfied as otherwise an earlier branching rule would have been applicable.

Lemma 17**.**

Suppose $G[S]$ is oriented, (Undefendable) is not applicable to $S$ and the maximum total degree in $S$ is 3. Then there exists a vertex in $G[S]$ with in-degree 1 and out-degree 2.

Proof.

Let $u$ be any vertex with total degree 3 in $G[S]$. First, $u$ cannot have in-degree 0, otherwise (Undefendable) would be applicable. Hence if $u$ has higher out-degree than in-degree, then $u$ must have in-degree 1 and out-degree 2.

Otherwise, as the sum of in-degrees in $G[S]$ is equal to the sum of out-degrees, there must be a vertex, $v$, in $G[S]$ with higher out-degree than in-degree. Since there are no vertices with degree $\geq 4$ and no vertices with in-degree 0 and out-degree $\neq 0$, necessarily $v$ must have in-degree 1 and out-degree 2.

Hence, in either case there exists a vertex in $G[S]$ with in-degree 1 and out-degree 2, as required. ∎

Let $v$ be any vertex in $G[S]$ with in-degree 1 and out-degree 2. Let $u$ be the unique vertex in $G[S]$ attacking $v$. We branch on whether to include $u$.

If we include $u$, then in our new instance $I^{\prime}=(S^{\prime},k^{\prime},b^{\prime})$ we have:

• •

$S^{\prime}=S\setminus N(u)$.

• •

$k^{\prime}=k-\left\lvert N(u)\right\rvert\leq k-2$. We note $u$ has at least $1$ in-neighbor (as (Undefendable) is not applicable) and this in-neighbor is distinct from $v$ as $G[S]$ is oriented.

• •

$b^{\prime}=b$.

and hence $\mu(I^{\prime})\leq\mu(I)-\frac{2}{2}=\mu(I)-1$.

If we do not include $u$, then after removing $u$ from $S$, $v$ has in-degree 0. Hence we can apply (Undefendable) to the two vertices that $v$ attacks, removing those as well. These 3 vertices are all distinct as $G[S]$ is oriented. Hence in our new instance $I^{\prime}=(S^{\prime},k^{\prime},b^{\prime})$ we have:

• •

$S^{\prime}=S\setminus N(v)$.

• •

$k^{\prime}=k-3$.

• •

$b^{\prime}=b$.

and hence $\mu(I^{\prime})\leq\mu(I)-\frac{3}{2}$.

Hence we get a branching vector of $(1,\frac{3}{2})$ with branching number $\approx 1.76<2$.

Appendix F Complexity Lower Bounds

In this section we will first show, assuming P != NP, that there is no output-polynomial time algorithm (an algorithm where the running time is upper bounded by a polynomial in input size plus output size) for preferred or admissible enumeration.

We also show our parameterized algorithm is tight, assuming the Strong Exponential Time Hypothesis (SETH). For this, we instead focus on the decision version of the problem:

Admissible Removal (AR)

Input:

Graph $G$, set $S\subseteq V(G)$, integer $k$

Parameter:

$k$

Question:

Does there exist an admissible set $T\subseteq S$ such that $\left\lvert S\setminus T\right\rvert\leq k$

We show, assuming SETH, there is no $O^{*}(2^{(1-\varepsilon)\mu(I)})$ algorithm for AR. Separately, we will also show there is no $O^{*}(2^{(1-\varepsilon)k})$ algorithm for AR. We briefly note a straight-forward 2-way branching $O^{*}(2^{k})$ algorithm exists for AR and MASE.

F.1 Required Background

We will be using the following form of the Strong Exponential Time Hypothesis (SETH):

Hypothesis 1** (Strong Exponential Time Hypothesis).**

There is no $O(2^{(1-\varepsilon)n})$ algorithm for SAT for any $\varepsilon>0$. Here, $n$ denotes the number of variables in the input instance.

The primary tool used to relate known complexity results to problems for Argumentation Frameworks is a translation from Boolean formulae in Conjunctive Normal Form (CNF) to AFs that has been called the standard translation (see, e.g., [20]). We will need a slight variation of the standard translation that was used in [15].

Definition F.1** (Extended translation [15]).**

Given $\varphi(z_{1},\ldots z_{n})$, a CNF with clauses $\{C_{1},\ldots C_{m}\}$, first let $C^{\prime}=\{C_{1},\ldots C_{m}\}\cup\{(z_{i}\lor\neg z_{i}):1\leq i\leq n\}$. The AF $(V,E)$ constituting the extended translation from $\varphi$ has

• •

$V=\{\varphi\}\cup\{C^{\prime}_{1},\ldots C^{\prime}_{m+n}\}\cup\{z_{1},\ldots z_{n}\}\cup\{\neg z_{1},\ldots,\neg z_{n}\}\cup\{A_{i}:0\leq i\leq 2\}$**

• •

$\begin{aligned} E&=\{(C^{\prime}_{j},\varphi):1\leq j\leq\left\lvert C^{\prime}\right\rvert\}\\ &\cup\{(z_{i},\neg z_{i}),(\neg z_{i},z_{i}):1\leq i\leq n\}\\ &\cup\{(y_{i},C^{\prime}_{j}):y_{i}\in C^{\prime}_{j}\}\\ &\cup\{(\varphi,A_{0})\}\\ &\cup\{(A_{0},A_{1}),(A_{1},A_{2}),(A_{2},A_{0})\}\\ &\cup\{(A_{0},z_{i}),(A_{0},\neg z_{i}):1\leq i\leq n\}\end{aligned}$**

An example translation can be found in Figure 4. The idea is to convert the notion of $y_{i}$ satisfying $C^{\prime}_{j}$ in the CNF into the AF notion of $y_{i}$ defending $\varphi$ from $C^{\prime}_{j}$.

In particular we have:

Lemma 18**.**

Let $G$ be an AF obtained from applying the Extended Translation to a CNF formula $\varphi(z_{1},\ldots z_{n})$. The nontrivial admissible extensions of $G$ are exactly the sets

[TABLE]

where

• •

$T$* is a subset of $\{z_{i}:1\leq i\leq n\}\cup\{\neg z_{i}:1\leq i\leq n\}$ that satisfies:*

• –

for all $1\leq i\leq n$, $T$ contains either $z_{i}$ or $\neg z_{i}$, and

• –

the literals in $T$ form a satisfying assignment of $\varphi(z_{1},\ldots z_{n})$.

• •

$W$* is either $\emptyset$ or $\{A_{1}\}$.*

Proof.

Let $S$ be a nontrivial admissible extension of $G$. In our proof, we argue which vertices may belong to $S$.

$\mathbf{\{A_{0},A_{1},A_{2},\varphi\}}$.

As $A_{1}$ is both adjacent to $A_{0}$ and the only vertex that defends $A_{0}$ from $A_{2}$, we deduce that $A_{0}\notin S$. For similar reasons, we deduce that $A_{2}\notin S$.

Through a simple chain of deductions, it can be verified that if $\varphi\notin S$ and $A_{0}\notin S$, then $S$ must be empty. Hence $\varphi\in S$.

Now $A_{1}$ is defended by $S$ and only attacks an irrelevant vertex $A_{2}$. Hence $A_{1}$ may or may not be in $S$ and its presence or absence does not affect the presence or absence of any other vertices in $S$.

$\mathbf{\{C^{\prime}_{i}\}}$.

As each $C^{\prime}_{i}$ is adjacent to $\varphi$, $C^{\prime}_{i}\notin S$ for all $i$.

$\mathbf{\{z_{i},\neg z_{i}\}}$.

By adjacency it is impossible that both $z_{i}$ and $\neg z_{i}$ are in $S$ for any $i$.

To defend $\varphi\in S$, each vertex corresponding to a clause must be attacked. By construction of the graph, to attack the vertex corresponding to a clause $C^{\prime}_{i}$, there must be a vertex in $S$ that represents a literal in $C^{\prime}_{i}$. Hence, the vertices in $S$ that correspond to literals form a satisfying assignment of $\varphi(z_{1},\ldots z_{n})$.

Finally, for all $i$ the clause $(z_{i}\lor\neg z_{i})$ is in $C^{\prime}$. Hence $S$ must contain either $z_{i}$ or $\neg z_{i}$.

The above shows that the conditions given are necessary. It is easy to see that the conditions given are sufficient. Hence the nontrivial admissible extensions of $G$ have exactly the prescribed form. ∎

From this we obtain the required relation:

Lemma 19**.**

Let $\varphi(z_{1},\ldots z_{n})$ be a CNF formula and $G$ be the AF obtained by applying the Extended Translation. Then there is a bijection between the non-trivial admissible extensions of $G$ with $W=\emptyset$ and the satisfying assignments of $\varphi$.

Proof.

This is clear from the description in Lemma 18. ∎

F.2 No Output-Polynomial Enumeration Algorithm

As a first result, we partly justify our decision to only focus on worst case running times by showing, assuming P != NP, that no output-polynomial time algorithm (an algorithm where the running time upper bounded by a polynomial in the input size plus the output size) exists for the enumeration of admissible or preferred extensions.

In [15], using the Extended Translation, it is proven that it is NP-complete to determine if an AF has a nontrivial admissible extension or a nontrivial preferred extension. Hence assuming P $\neq$ NP, there is no output-polynomial time algorithm that enumerates the admissible extensions or the preferred extensions of a digraph. In [28], it was shown that the preferred extensions cannot be enumerated in output-polynomial time, unless P=NP, even for bipartite AFs.

By applying the oriented translation algorithm to the AF obtained from the extended translation, it is straight forward to show that these impossibility results also hold on oriented graphs. Hence:

Theorem 9**.**

There is no algorithm that enumerates the admissible or preferred extensions of an AF in output-polynomial time. These results further hold when restricted to the class of oriented graphs.

F.3 Parameterized Complexity Lower Bounds

In this section we show, assuming SETH, that our parameterized algorithm for MASE is tight for the measure we use. In particular we show there is no $O^{*}(2^{(1-\varepsilon)\mu(I)})$ algorithm for AR. Separately, we will also show there is no $O^{*}(2^{(1-\varepsilon)k})$ algorithm for AR and describe a straight forward $O^{*}(2^{k})$ algorithm for MASE (and hence AR).

Lemma 20**.**

Assuming SETH, there is no $O^{*}(2^{(1-\varepsilon)\mu(I)})$ algorithm for AR.

Proof.

Consider the AR problem where

• •

$G$ is the Extended Translation of $\varphi(z_{1},\ldots z_{n})$,

• •

$S=\{\varphi\}\cup\{z_{i}\}\cup\{\neg z_{i}\}$, using the same notation as in the definition of the Extended Translation, and

• •

$k=n$.

By Lemma 18, any non-trivial admissible extension of $G$ contains $\varphi$ and exactly one vertex from each of the $n$ pairs $(z_{i},\neg z_{i})$ and corresponds to a satisfying assignment for the original CNF. As $k=n$, which is just enough to remove one vertex from each literal pair, this instance of admissible removal is equivalent to finding if there exists a satisfying assignment to $\varphi(z_{1},\ldots z_{n})$.

We now note $b=2n$ (there are $n$ disjoint 2-cycles, one for each variable). Hence $\mu(I)=n$ and an AR algorithm with complexity $O^{*}(2^{(1-\varepsilon)\mu(I)})$ would imply an $O^{*}(2^{(1-\varepsilon)n})$ algorithm for SAT, contradicting SETH. ∎

Incidentally, the $2^{\mu(I)}$ bound is also tight for the number of preferred extensions.

Lemma 21**.**

There are instances $I=(G,S,k)$ with $2^{\mu(I)}$ preferred subsets of $S$ that are at distance at most $k$ from $S$.

Proof.

Consider an instance $I=(G,S,k)$, where $G=(V,E)$ is a disjoint union of $n/2$ $2$-cycles, $S=V$, and $k=n/2$. It suffices to note that $\mu(I)=\frac{k}{2}+\frac{b}{4}=\frac{n}{2}$, and that $G$ has $2^{n/2}=2^{\mu(I)}$ preferred subsets, all of which are at distance $k$ from $S$. ∎

In the same manner we also obtain:

Lemma 22**.**

Assuming SETH, there is no $O^{*}(2^{(1-\varepsilon)k})$ algorithm for AR. Furthermore, a simple $O^{*}(2^{k})$ algorithm exists for MASE (hence AR). In addition, there are at most $2^{k}$ preferred extensions that are subsets of $S$ with size $\geq\left\lvert S\right\rvert-k$.

Proof.

Consider the same instance of AR as in the above proof. Since $k=n$, we again have that an $O^{*}(2^{(1-\varepsilon)k})$ algorithm for AR would imply an $O^{*}(2^{(1-\varepsilon)n})$ algorithm for SAT, contradicting SETH.

For existence, we just consider a simple 2-way branching algorithm. While there is an edge, say $u\to v$ in $G[S]$, do a 2-way branch, in 1 branch removing $u$ and in another branch removing $v$. As before, we glue the solutions of our branches together by applying our Maximal Subset Collation algorithm. The base case is when $G[S]$ is conflict-free which can be solved in polynomial time by Lemma 4.

The enumeration upper bound immediately follows as our branching algorithm enumerates every maximal admissible subset of $S$ (which contains all preferred extensions that are subsets of $S$). ∎

Appendix G Discussion of measure

We have parameterized by the number of vertices in 2-cycles. It is natural to ask whether we should instead be parameterizing by the number of 2-cycles or some mixture of the two. In this section, we provide some partial results that somewhat justify our choice of measure.

For any instance $I$ of AR, let $b$ be the number of vertices in at least one 2-cycle in $G[S]$ and let $d$ be the number of 2-cycles in $G[S]$. We restrict our consideration to measures of the form $c_{1}k+c_{2}b+c_{3}d$ where $c_{1},c_{2},c_{3}$ are real constants. We will prove some results regarding such measures.

Throughout, we will say a measure $\mu$ is valid if the existence of a $O^{*}(2^{\mu(I)})$ algorithm for AR does not contradict SETH.

Lemma 23**.**

For any valid measure $\mu$, $c_{1}\geq\frac{1}{2}$ assuming SETH.

Proof.

Let $\varphi(z_{1},\ldots z_{n})$ be an arbitrary CNF formula. Now we consider the AR problem where

• •

$G$ is the Oriented Translation applied to the Extended Translation of $\varphi(z_{1},\ldots z_{n})$. Explicitly this is the same construction as the Extended Translation except each literal pair $(z_{i},\neg z_{i})$ now maps to a 4-cycle $z_{i,1}\to\neg z_{i,1}\to z_{i,2}\to\neg z_{i,2}\to z_{i,1}$,

• •

$S=\{\varphi\}\cup\{z_{i,c}:c\in\{1,2\}\}\cup\{\neg z_{i,c}:c\in\{1,2\}\}$, using the same notation as in our description of $G$, and

• •

$k=2n$.

Theorem 8 states that the preferred extensions of $G$ are in a bijection with the preferred extensions of the Extended Translation. The preferred extensions of $G$ hence correspond directly to satisfying assignments of $\varphi(z_{1},\ldots z_{n})$ by Lemma 18. As $k=2n$ which is just enough to remove two vertices from each literal 4-cycle, this instance of AR reduces to checking if there exists a satisfying assignment to $\varphi(z_{1},\ldots z_{n})$.

This instance has $b=d=0$. Hence, if $c_{1}k<\frac{1}{2}k$, this would imply an $O^{*}(2^{(1-\varepsilon)n})$ algorithm for SAT, contradicting SETH. ∎

Lemma 24**.**

For any valid measure $\mu$, $c_{1}+2c_{2}+c_{3}\geq 1$, assuming SETH.

Proof.

Consider the translation from SAT used for our complexity lower bound for MASE/AR on general directed graphs (Lemma 20). This instance has $k=n$, $b=2n$, $d=n$. Assuming SETH, we must have $c_{1}n+2c_{2}n+c_{3}n\geq n$ or, equivalently: $c_{1}+2c_{2}+c_{3}\geq 1$. ∎

Lemma 25**.**

Consider the measures with $c_{1}=\frac{1}{2}$. Assuming SETH, among these measures, $\mu(I)=\frac{k}{2}+\frac{b}{4}$ is optimal in the sense that, for any other valid measure $\mu^{\prime}$ with $c_{1}=\frac{1}{2}$, $\mu^{\prime}(I)\geq\mu(I)$ for all instances $I$.

Proof.

Let $I$ be an instance of AR. Let $\mu$ be any valid measure with $c_{1}=\frac{1}{2}$. Then,

[TABLE]

This concludes the proof of the lemma. ∎

Hence our earlier measure is the optimal valid measure with $c_{1}=\frac{1}{2}$, assuming SETH.

We note this says nothing about measures with $c_{1}>\frac{1}{2}$. Indeed, the measure $\mu(I)=k$ is better than our new measure on graphs with many vertices in 2-cycles. It also says nothing about other classes of measures. For instance, we believe $\mu(I)=\frac{k}{2}+\min(\frac{k}{2},\frac{b}{4})$ is a valid measure that is worth exploring.

Appendix H Detailed Case Analysis of our Improved Enumeration Algorithm

As a reminder, our cases are:

[TABLE]

And we always apply the first applicable case.

We also recall our simplification rules:

H.1 Simplification Rules

Both of these are applicable in polynomial time and decrease $\mu=\left\lvert\operatorname{Und}\right\rvert$.

Simplification Rule 0** (Out-degree 0).**

Let $v$ be a vertex in $G[\operatorname{Und}]$ with out-degree [math]. Move $v$ from $\operatorname{Und}$ to the end of the queue $\operatorname{Def}$.

Simplification Rule 0** (In-degree 0).**

Let $v$ be any vertex in $G[\operatorname{Und}]$ with in-degree [math]. Then by Invariant 3, $v$ has in-degree [math] in $G[\operatorname{Und}\cup\operatorname{Def}]$. Applying Simplification Rule (Undefendable) we can set $\operatorname{Und}\leftarrow\operatorname{Und}\setminus N(v)$. After that, $v$ has out-degree 0 in $G[\operatorname{Und}]$ and hence we move $v$ from $\operatorname{Und}$ to $\operatorname{Def}$. Our new instance $I^{\prime}=(\operatorname{Und}^{\prime},\operatorname{Def}^{\prime})$ has:

• •

$\operatorname{Und}^{\prime}=\operatorname{Und}\setminus N[v]$.

• •

$\operatorname{Def}^{\prime}=\operatorname{Def}\cup\{v\}$.

Due to these rules, henceforth we may assume each vertex in $G[\operatorname{Und}]$ has in-degree $\geq 1$, out-degree $\geq 1$ and (total) degree $\geq 2$.

H.2 Notation

As in Subsection 4.2 we will say a vertex has degree $(a,-)$ if it has in-degree $a$, a vertex has degree $(-,b)$ if it has out-degree $b$ and a vertex has degree $(a,b)$ if it has in-degree $a$ and out-degree $b$.

We will again use the shorthand 2-way branch on whether to include $v$ (see Subsection 4.5).

Similarly, we will use the phrase include $v$ as shorthand for the first part of the branching rule described in Subsection 4.5. This terminology will be useful for describing our 3-way and 4-way branches (these will be similar to our 2-way branches except there will be a choice on which vertex to include from a specified set).

H.3 Base Case - $\operatorname{Und}=\emptyset$

We need to enumerate all maximal admissible subsets of $\operatorname{Und}\cup\operatorname{Def}=\operatorname{Def}$. By Invariant 2, $G[\operatorname{Def}]$ is a DAG. Hence by Lemma 4 there is exactly one maximal admissible subset of $\operatorname{Def}$ and we can find it in polynomial time.

H.4 Case 1 - Total Degree $\geq 7$

Let $v$ be any vertex in $G[\operatorname{Und}]$ with total degree $\geq 7$. We do a 2-way branch on whether to include $v$(Definition 4.1).

This gives a branching vector at least as good as $(8,1)$ with branching number at most $\varphi$ (recall that $\varphi$ was defined to be the branching number for $(8,1)$).

H.5 Case 2 - Degree $(1,-)$ exists

Let $v$ be any vertex in $G[\operatorname{Und}]$ with degree $(1,-)$. Let $a$ be the vertex attacking $v$ and let $b$ be any vertex $v$ attacks (due to Simplification Rule 0, $v$ has out-degree $>0$). We note that $a\neq b$ as $G$ is oriented. See Figure 5 for a depiction.

We consider a few cases depending on the degrees of $a$, $b$, and $v$.

Case 2.1 - $\deg(a)\geq 3$ or $\deg(v)\geq 3$

If either $a$ or $v$ has degree at least $3$, then we do a 2-way branch on whether to include $a$(Definition 4.1). This gives a branching vector of $(\deg(a)+1,1)$.

Furthermore, in the branch where we exclude $a$, $v$ is left with in-degree [math]. Hence, we can apply Simplification Rule 0 to $v$. Therefore, the branch that excludes $a$ also removes all of $N[v]$ from $\operatorname{Und}$.

This is a branching rule with branching vector $(\deg(a)+1,\deg(v)+1)$. After our simplification rules, all vertices in $G[\operatorname{Und}]$ have total degree at least $2$. By assumption at least one of $a$ or $v$ has degree at least 3. Hence, our branching vector is at least as good as $(4,3)$.

Case 2.2 - $\deg(b)\geq 3$

Vertices $a$ and $v$ do not have in-degree or out-degree [math] due to our simplification rules. Hence, they must have degree $(1,1)$ (else the previous case is applicable).

If $\deg(b)\geq 3$, we do a 2-way branch on whether to include $b$(Definition 4.1).

In the branch where $b$ is excluded $v$ will have out-degree 0. Hence we may apply Simplification Rule 0 to $v$. After this, $a$ will also have out-degree 0 and we may apply Simplification Rule 0 again to $a$. Hence our new instance $I^{\prime}=(\operatorname{Und}^{\prime},\operatorname{Def}^{\prime})$ has:

• •

$\operatorname{Und}^{\prime}=\operatorname{Und}\setminus\{a,v,b\}$.

• •

$\operatorname{Def}^{\prime}=\operatorname{Def}\cup\{v,a\}$.

and hence $\mu(I^{\prime})=\mu(I)-3$.

Hence we have branching vector $(\deg(b)+1,3)$, which is at least as good as $(4,3)$.

Case 2.3 - $\deg(a)=\deg(v)=\deg(b)=2$

Due to our simplification rules, this is the only case remaining. If these three vertices are connected to no other vertices, then no preferred extension can contain any vertex in $\{a,v,b\}$ as $\{a,v,b\}$ is an isolated odd length cycle. Hence, we can remove all three vertices from $\operatorname{Und}$.

Otherwise, the vertex that $b$ attacks is distinct from $\{a,v,b\}$. Let this vertex be $w$. We have a chain of attacks $a\to v\to b\to w$. Do a 2-way branch on whether to include $a$(Definition 4.1).

In the branch where we include $a$, since $v\in N[a]$, $v$ is removed from $\operatorname{Und}$. This leaves $b$ with in-degree 0. Hence we may apply Simplification Rule 0 to $b$. Our new instance $I^{\prime}=(\operatorname{Und}^{\prime},\operatorname{Def}^{\prime})$ has:

• •

$\operatorname{Und}^{\prime}=\operatorname{Und}\setminus\{a,v,b,w\}$.

• •

$\operatorname{Def}^{\prime}=\operatorname{Def}\cup\{a,b\}$.

and hence $\mu(I^{\prime})=\mu(I)-4$.

In the branch where we exclude $a$, $v$ has in-degree 0. We may apply Simplification Rule 0 to $v$ removing $b$ from $\operatorname{Und}$. Hence our new instance $I^{\prime}=(\operatorname{Und}^{\prime},\operatorname{Def}^{\prime})$ has:

• •

$\operatorname{Und}^{\prime}=\operatorname{Und}\setminus\{a,v,b\}$.

• •

$\operatorname{Def}^{\prime}=\operatorname{Def}\cup\{v\}$.

with $\mu(I^{\prime})=\mu(I)-3$.

Hence we have branching vector $(4,3)$.

Case 2 - Summary

Hence there is always a branching rule with branching vector $(4,3)$ which has branching number $\approx 1.221<\varphi$.

After this case, all vertices can be assumed to have in-degree $\geq 2$, out-degree $\geq 1$ and total degree $\geq 3$.

H.6 Case 3 - Vertex with in-degree $\neq$ out-degree

As the sum of all in-degrees is equal to the sum of all out-degrees in $G[S]$, there exists a vertex $v$ in $G[S]$ with larger out-degree than in-degree.

Due to the above cases, $v$ has in-degree $\geq 2$ and total degree $\leq 6$. Hence, $v$ has degree $(2,3)$ or $(2,4)$. There are two cases depending on the vertices attacking $v$.

Case 3.1 - A degree-3 vertex attacks $v$

If either of the vertices attacking $v$ has total degree $3$, then it must have degree $(2,1)$ due to the previous cases. Let $u$ be such a vertex.

We do a 2-way branch on whether to include $v$(Definition 4.1). This gives a branching vector of $(\deg(v)+1,1)$.

Furthermore, in the branch where we exclude $v$, $u$ will have out-degree 0. Hence we may apply Simplification Rule 0 to move $u$ from $\operatorname{Und}$ to $\operatorname{Def}$.

Hence our branching rule has a branching vector at least as good as $(6,2)$, with branching number $\approx 1.211$.

Case 3.2 - Both in-neighbors of $v$ have degree at least $4$

Otherwise, both vertices attacking $v$ have total degree at least $4$. Let the 2 vertices be $a$ and $b$.

We do a 3-way branch, in the first branch we include $a$, in the second branch we include $b$ and in the third branch we exclude both $a$ and $b$. The branches including $a$ and $b$ remove at least 5 vertices from $\operatorname{Und}$ as $\deg(a),\deg(b)\geq 4$.

The branch excluding both $a$ and $b$ results in $v$ having in-degree 0. Hence we may apply Simplification Rule 0 to $v$. Our new instance $I^{\prime}=(\operatorname{Und}^{\prime},\operatorname{Def}^{\prime})$ then has:

• •

$\operatorname{Und}^{\prime}=\operatorname{Und}\setminus N[v]$.

• •

$\operatorname{Def}^{\prime}=\operatorname{Def}\cup\{v\}$.

with $\mu(I^{\prime})\leq\mu(I)-6$.

This is a $(6,5,5)$ branching with branching number $\approx 1.2298$.

H.7 Case 4 - $G[\operatorname{Und}]$ contains a weakly connected component of degree $(2,2)$ vertices

Let $C$ be a weakly connected component in $G[\operatorname{Und}]$ such that each $v\in C$ has degree $(2,2)$. We consider two cases.

Case 4.1 - There exist non-adjacent attackers

In this subcase we assume there exists a vertex $v\in C$ such that its two in-neighbors $a,b\in C$ are not adjacent. Then we do a 3-way branch: include $a$, exclude $a$ but include $b$, exclude both.

In the branch where we include $a$ our new instance $I^{\prime}=(\operatorname{Und}^{\prime},\operatorname{Def}^{\prime})$ has:

• •

$\operatorname{Und}^{\prime}=\operatorname{Und}\setminus N[a]$.

• •

$\operatorname{Def}^{\prime}=\operatorname{Def}\cup\{a\}$.

with $\mu(I^{\prime})=\mu(I)-5$.

In the branch where we include $b$ and exclude $a$ our new instance $I^{\prime}=(\operatorname{Und}^{\prime},\operatorname{Def}^{\prime})$ has:

• •

$\operatorname{Und}^{\prime}=\operatorname{Und}\setminus(N[b]\sqcup\{a\})$.

• •

$\operatorname{Def}^{\prime}=\operatorname{Def}\cup\{b\}$.

with $\mu(I^{\prime})=\mu(I)-6$.

In the branch where we exclude $a$ and $b$, $v$ has in-degree 0.

Hence we may apply Simplification Rule 0 to $v$. Our new instance $I^{\prime}=(\operatorname{Und}^{\prime},\operatorname{Def}^{\prime})$ then has:

• •

$\operatorname{Und}^{\prime}=\operatorname{Und}\setminus N[v]$.

• •

$\operatorname{Def}^{\prime}=\operatorname{Def}\cup\{v\}$.

with $\mu(I^{\prime})=\mu(I)-5$.

This is a $(6,5,5)$ branching with branching number $\approx 1.2298$.

Case 4.2 - Attackers of each vertex are adjacent

In this subcase, each $v\in C$ has degree $(2,2)$ with adjacent in-neighbors. We handle this case by showing that the graph $G[C]$ has a very restricted structure. In particular, we show that $G[C]$ must look like Figure 6.

Theorem 10**.**

Let $G=(V,E)$ be a weakly connected, oriented graph that satisfies the following properties:

1. P1

every vertex has in-degree 2 and out-degree 2, and 2. P2

the in-neighbors of every vertex are adjacent.

Then $\left\lvert V\right\rvert\geq 5$ and $G$ is isomorphic to $F_{\left\lvert V\right\rvert}$ where we define $F_{n}$ for $n\in\mathbb{N}$ as:

• •

$V(F_{n})=\{0,1,\ldots,n-1\}$**

• •

$E(F_{n})=\bigcup\limits_{i=0}^{n-1}(i,(i+1)\mod n)\cup\bigcup\limits_{i=0}^{n-1}(i,(i+2)\mod n)$**

Proof.

Consider any arbitrary vertices $v_{0},v_{1},v_{2}$ with $(v_{1},v_{0}),(v_{2},v_{1}),(v_{2},v_{0})\in E$. Such a triplet exists as we can arbitrarily pick any $v_{0}$ and then pick $v_{1}$ and $v_{2}$ to be its in-neighbors such that the edge between $v_{1}$ and $v_{2}$ is $(v_{2},v_{1})$. Let $V_{2}=[v_{0},v_{1},v_{2}]$. Then, while $v_{k-1}$ has only one in-neighbor in $G[V_{k}]$, we construct $V_{k+1}$ to be $V_{k}$ appended with $v_{k+1}$ where $v_{k+1}$ is the other in-neighbor of $v_{k-1}$.

Lemma 26**.**

Each $V_{k}$ satisfies the following property:

Local Attack Property:* For all $i\geq 1$, $v_{i}$ attacks $v_{i-1}$ and for all $i\geq 2$, $v_{i}$ attacks $v_{i-2}$.*

Proof.

This is obvious for the base case $V_{2}$. We can then confirm it for $V_{k},k\geq 3$ by induction:

The vertex $v_{k}$ attacks $v_{k-2}$ by construction.

Hence, by (P2), either $(v_{k},v_{k-1})\in E$ or $(v_{k-1},v_{k})\in E$. However, since $V_{k-1}$ satisfies the Local Attack Property and $k-1\geq 2$, $v_{k-1}$ attacks both $v_{k-2}$ and $v_{k-3}$. Hence, it already has two out-neighbors distinct from $v_{k}$. Therefore, since $v_{k-1}$ has out-degree 2 by (P1), it must be the case that $(v_{k},v_{k-1})\in E$ as required. ∎

Now, consider the largest such constructed sequence, $V_{s}$. Then, since $V_{s}$ could not be extended, there must be a vertex $u\in V_{s},u\neq v_{s}$ that attacks $v_{s-1}$. Since $V_{s}$ satisfies the Local Attack Property, every vertex other than $v_{s}$, $v_{1}$ and $v_{0}$ already has 2 out-neighbors that are not $v_{s-1}$. Hence $u=v_{0}$ or $u=v_{1}$. We now note that since $G$ is oriented, for $v_{0}$ or $v_{1}$ to have an edge to $v_{s-1}$, $s$ must be at least 4.

Now, as $s\geq 4$, $v_{s}$ attacks two vertices that are not $u$. Hence, by (P2), $u$ must also attack $v_{s}$. Therefore, by out-degree considerations, $u=v_{0}$. Finally, by (P1), the other in-neighbor of $v_{s}$ must attack $v_{0}$. By simple degree considerations, we conclude that the other in-neighbor of $v_{s}$ is $v_{1}$. Hence, $(v_{0},v_{s-1}),(v_{0},v_{s}),(v_{1},v_{s})\in E$ and $V_{s}$ actually satisfies the following stronger condition:

For all $i\leq s$, $(v_{i},v_{(i-1)\mod s}),(v_{i},v_{(i-2)\mod s})\in E$.

Therefore, $G[V_{s}]$ contains a subgraph isomorphic to $F_{s+1}$ with an obvious isomorphism. Furthermore, by degree considerations and (P1), this subgraph is $G[V_{s}]$ itself and $G[V_{s}]$ is isomorphic to $F_{s+1}$. Finally, since $G$ is weakly connected and $G[V_{s}]$ satisfies (P1), we conclude that $V_{s}$ contains all of $V$ and $s+1=\left\lvert V\right\rvert$. Hence, since $s\geq 4$, we further conclude that $\left\lvert V\right\rvert\geq 5$. ∎

Theorem 10 shows that our weakly connected component $C$ is isomorphic to $F_{n}$ for some $n\geq 5$.

We do a 3-way branch on including $v_{0}$, including $v_{1}$, and excluding both. In each of these cases $v_{0}$ and $v_{1}$ are both removed from $\operatorname{Und}$ (as $v_{0}$ is adjacent to $v_{1}$) and it is easy to confirm that the resulting graph, $C^{\prime}=C\setminus\{v_{0},v_{1}\}$, is a DAG. We can repeatedly apply Simplification Rule 0 to vertices with out-degree [math] in the DAG $C^{\prime}$ until all vertices in $C^{\prime}$ have been moved to $\operatorname{Def}$. Hence in each of these branches, once we finish applying our simplification rules, the size of $\operatorname{Und}$ decreases by $\left\lvert C\right\rvert$.

We now note that $F_{5}$ has no non-trivial admissible extensions. This also follows from a more general argument: if $v_{i}$ is in an admissible extension that is a subset of $\operatorname{Und}\cup\operatorname{Def}$, then so is $v_{(i+3k)\mod n}$ for all $k$. Hence if $3\nmid n$, then $F_{n}$ has only the trivial admissible extension.

Hence we may assume $n\geq 6$. Hence our 3-way branching rule has branching vector at least as good as $(6,6,6)$ with branching number $3^{\frac{1}{6}}<\varphi$.

H.8 Case 5 - $G[\operatorname{Und}]$ contains a weakly connected component of degree $(3,3)$ vertices

Let $C$ be a weakly connected component in $G[\operatorname{Und}]$ such that each $v\in C$ has degree $(3,3)$. Pick any vertex $v\in C$ and let its 3 attackers be $\{a,b,c\}$. Then we do a 4-way branch: include $a$, include $b$, include $c$, and exclude all of $\{a,b,c\}$.

By assumption, $a,b,c$ all have degree $(3,3)$. Hence the first 3 cases each remove at least 7 vertices from $\operatorname{Und}$. In the last case, $v$ has in-degree 0 and hence by Simplification Rule 0, $N[v]$ is removed from $\operatorname{Und}$.

Hence, this is a $(7,7,7,7)$ branching with branching number $4^{\frac{1}{7}}\approx 1.219$.

H.9 Case 6 - Every vertex has the same in-degree as out-degree

In this case, every vertex has the same in-degree as out-degree. From case 2 no vertex has in-degree $\leq 1$. From case 1 no vertex has total degree $\geq 7$. Hence, each vertex has degree $(2,2)$ or $(3,3)$.

Furthermore, due to the previous two cases, each weakly connected component $C$ of $G[\operatorname{Und}]$ contains both a vertex with degree $(2,2)$ and a vertex with degree $(3,3)$.

Hence we may apply the following lemma:

Lemma 27**.**

Let $G=(V,E)$ be a weakly connected, oriented graph where every vertex $v\in G$ has degree $(2,2)$ or $(3,3)$. Further suppose $G$ contains at least one vertex with degree $(2,2)$ and at least one vertex with degree $(3,3)$.

Then there exists a $v\in G$ with degree $(2,2)$ that is attacked by a vertex $a\in G$ with degree $(3,3)$.

Proof.

Let $V_{2}\subseteq V$ be the vertices with degree $(2,2)$. Let $V_{3}=V\setminus V_{2}$ be the vertices with degree $(3,3)$.

Accounting for the edges in $G[V_{2}]$ and the edges between $V_{2}$ and $V_{3}$ separately, we have:

[TABLE]

Similarly, we have:

[TABLE]

Hence:

number of edges from $V_{3}$ to $V_{2}$ = number of edges from $V_{2}$ to $V_{3}$

As $G$ is weakly connected and neither $V_{2}$ nor $V_{3}$ are empty, there exists at least one edge between $V_{2}$ and $V_{3}$.

Hence there exists an edge from $V_{3}$ to $V_{2}$ as required. ∎

Let $v$ and $a$ be as in the above lemma and let $b$ be $v$’s other attacker.

Now we do a 3-way branch.

• •

Include $a$. Then we remove $\left\lvert N[a]\right\rvert=7$ vertices from $\operatorname{Und}$.

• •

Include $b$. Then we remove $\left\lvert N[b]\right\rvert\geq 5$ vertices from $\operatorname{Und}$.

• •

Exclude $a$ and $b$. Then $v$ has in-degree [math] and applying Simplification Rule 0 removes $\left\lvert N[v]\right\rvert=5$ vertices from $\operatorname{Und}$.

This is a $(7,5,5)$ branching rule with branching number $\approx 1.218$.