#
Parameterised Enumeration for Modification Problems^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**.**Let R be a set and ⪯ a binary relation on R. Then ⪯ is a preorder (or quasi-order) if we have for all elements $a,b,c\in R$:

- $a\u2aafa$ and
- if $a\u2aafb$ and $b\u2aafc$ then $a\u2aafc$.

**Definition**

**2.**

- I is the set of instances.
- $\kappa :I\to \mathbb{N}$ is the parameterisation function; κ is required to be polynomial time computable.
- $\mathrm{Sol}$ is a function such that for all $x\in I$, $\mathrm{Sol}\left(x\right)$ is a finite set, the set of solutions of x. Further we write $\mathcal{S}={\bigcup}_{x\in I}\mathrm{Sol}\left(x\right)$.
- ⪯ is a quasi-order on $\mathcal{S}$.

**Definition**

**3**

**.**Let $E=(I,\kappa ,\mathrm{Sol},\u2aaf)$ be a parameterised enumeration problem with ordering. Then an algorithm $\mathcal{A}$ is an enumeration algorithm for E if the following holds:

- For every $x\in I$, $\mathcal{A}\left(x\right)$ terminates after a finite number of steps.
- For every $x\in I$, $\mathcal{A}\left(x\right)$ outputs exactly the elements of $\mathrm{Sol}\left(x\right)$ without duplicates.
- For every $x\in I$ and $y,z\in \mathrm{Sol}\left(x\right)$, if $y\u2aafz$ and $z\neg \u2aafy$ then $\mathcal{A}\left(x\right)$ outputs solution y before solution z.

**Definition**

**4**

**.**Let $E=(I,\kappa ,\mathrm{Sol},\u2aaf)$ be a parameterised enumeration problem with ordering and $\mathcal{A}$ be an enumeration algorithm for E. Let $x\in I$ be an instance. The ith delay of $\mathcal{A}$ is the elapsed runtime with respect to $\left|x\right|$ of $\mathcal{A}$ between outputting the ith and $(i+1)$th solution in $\mathrm{Sol}\left(x\right)$. The 0th delay is the precomputation time which is the elapsed runtime with respect to $\left|x\right|$ of $\mathcal{A}$ from the start of the computation to the first output statement. Analogously, the nth delay, for $n=\left|\mathrm{Sol}\right(x\left)\right|$, is the postcomputation time, which is the elapsed runtime with respect to $\left|x\right|$ of $\mathcal{A}$ after the last output statement until $\mathcal{A}$ terminates. Then, the delay of $\mathcal{A}$ is the maximum over all $0\le i\le n$ of the ith delay of $\mathcal{A}$.

**Definition**

**5.**

#### 2.1. Graph Modification Problems

**Definition**

**6**

**.**A graph operation for G is either of the following:

- removing a vertex: A function ${\mathrm{rem}}_{v}:\mathcal{G}\to \mathcal{G}$ such that ${\mathrm{rem}}_{v}\left(G\right)$ is the graph obtained by removing the vertex v from G (if v is present; otherwise ${\mathrm{rem}}_{v}$ is the identity) and deleting all incident edges to v,
- adding/removing an edge: A function ${\mathrm{add}}_{\{u,v\}},{\mathrm{rem}}_{\{u,v\}}:\mathcal{G}\to \mathcal{G}$ such that ${\mathrm{add}}_{\{u,v\}}\left(G\right)$, ${\mathrm{rem}}_{\{u,v\}}\left(G\right)$ is the graph obtained by adding/removing the edge $\{u,v\}$ to G if u and v are present in G; otherwise both functions are the identity

- $o={\mathrm{rem}}_{v}$ and ${o}^{\prime}={\mathrm{rem}}_{\{u,v\}}$ (o removes the vertex v and ${o}^{\prime}$ removes an edge incident to v) or
- $o={\mathrm{rem}}_{\{u,v\}}$ and ${o}^{\prime}={\mathrm{add}}_{\{u,v\}}$ (o removes the edge $\{u,v\}$ and ${o}^{\prime}$ adds the same edge $\{u,v\}$ again).

**Definition**

**7**

**.**Given a graph property $\mathcal{P}$, a graph G, $k\in \mathbb{N}$ and a set of operations O, we say that S is a solution for $(G,k,O)$ with respect to $\mathcal{P}$ if the following three properties hold:

- 1.
- $S\subseteq O$ is a consistent set of operations,
- 2.
- $\left|S\right|\le k$ and
- 3.
- $S\left(G\right)\vDash \mathcal{P}$.

Problem: | ${\mathcal{M}}_{\mathcal{P}}$ |

Input: | $(G,k,O)$, G undirected graph, $k\in \mathbb{N}$, O set of operations on G. |

Parameter: | The integer k. |

Question: | Does there exist a solution for $(G,k,O)$ with respect to $\mathcal{P}$? |

Problem: | Enum-${\mathcal{M}}_{\mathcal{P}}^{\mathrm{LEX}}$ |

Input: | $(G,k,O)$, G undirected graph, $k\in \mathbb{N}$, O ordered set of operations on G. |

Parameter: | The integer k. |

Output: | All solutions of $(G,k,O)$ with respect to $\mathcal{P}$ in lexicographic order. |

Problem: | Enum-${\mathcal{M}}_{\mathcal{P}}^{\mathrm{SIZE}}$ |

Input: | $(G,k,O)$, G undirected graph, $k\in \mathbb{N}$, O set of operations on G. |

Parameter: | The integer k. |

Output: | All solutions of $(G,k,O)$ with respect to $\mathcal{P}$ in non-decreasing size. |

## 3. Enumeration of Graph Modification Problems with Ordering

#### 3.1. Lexicographic Ordering

**Lemma**

**1.**

**Proof.**

`ExistsSol`$(G,k,O)$ that tests if the instance $(G,k,O)$ of the modification problem ${\mathcal{M}}_{\mathcal{P}}$ has a solution. By the assumption of the lemma, ${\mathcal{M}}_{\mathcal{P}}\in \mathrm{FPT}$ so this function runs in FPT time. We use calls to this function to avoid exploration of branches of the recursion tree that do not lead to any output. Moreover, we ensure that the solutions using ${o}_{p}$ have to be consistent. This consistency check runs in polynomial time for graph operations. The rest yields a search tree of depth at most k. From this it follows that, for any instance of length n, the time beween the output of any two solutions is bounded by $f\left(k\right)\xb7p\left(n\right)$ for some polynomial p and a computable function f. □

Algorithm 1: Enumerate all solutions of ${\mathcal{M}}_{\mathcal{P}}$ in lexicographic order |

**Corollary**

**1.**

^{lex}∈ DelayFPT with polynomial space.

**Proof.**

**Definition**

**8.**

**Proposition**

**1**

**.**If a property $\mathcal{P}$ has a finite forbidden set characterisation then ${\mathcal{M}}_{\mathcal{P}}$ is in $\mathrm{FPT}$.

**Corollary**

**2.**

**Proof.**

#### 3.2. Size Ordering

**Definition**

**9.**

- 1.
- For all $x=(G,k,O)\in {I}_{\mathcal{M}}$ and $S\in {\mathrm{Sol}}_{\mathcal{M}}\left(x\right)\cup \left\{\mathbb{O}\right\}$, ${\mathcal{N}}_{\mathcal{M}}(x,S)$ is defined.
- 2.
- For all $x\in {I}_{\mathcal{M}}$, ${\mathcal{N}}_{\mathcal{M}}(x,\mathbb{O})=\varnothing $ if ${\mathrm{Sol}}_{\mathcal{M}}\left(x\right)=\varnothing $, and ${\mathcal{N}}_{\mathcal{M}}(x,\mathbb{O})$ is an arbitrary set of solutions otherwise.
- 3.
- For all $x\in {I}_{\mathcal{M}}$ and $S\in {\mathrm{Sol}}_{\mathcal{M}}\left(x\right)$, if ${S}^{\prime}\in {\mathcal{N}}_{\mathcal{M}}(x,S)$ then $\left|S\right|<|{S}^{\prime}|$.
- 4.
- For all $x\in {I}_{\mathcal{M}}$ and all $S\in {\mathrm{Sol}}_{\mathcal{M}}\left(x\right)$, there exists $p>0$ and ${S}_{1},\cdots ,{S}_{p}\in {\mathrm{Sol}}_{\mathcal{M}}\left(x\right)$ such that (i) ${S}_{1}\in {\mathcal{N}}_{\mathcal{M}}(x,\mathbb{O})$, (ii) ${S}_{i+1}\in {\mathcal{N}}_{\mathcal{M}}(x,{S}_{i})$ for $1\le i<p$ and (iii) ${S}_{p}=S$.

**Theorem**

**1.**

**Proof.**

- $i=1$:
- This proceeds from line 2 of the algorithm.
- $i>1$:
- The solution ${S}_{i-1}$ is inserted in Q by induction hypothesis and hence all elements of ${\mathcal{N}}_{\mathcal{M}}((G,k,O),{S}_{i-1})$, including ${S}_{i}$, are inserted in Q (line 5 of Algorithm 2). Consequently, each ${S}_{i}$ is inserted in Q and then output during the run. In particular, this is true for $S={S}_{p}$.

Algorithm 2:$\mathrm{Delay}\mathrm{FPT}$ algorithm for Enum-$\mathcal{M}$ |

**Theorem**

**2.**

**Proof.**

Algorithm 3: Procedure for computing ${\mathcal{N}}_{\mathcal{M}}((G,k,O),S)$ |

**Corollary**

**3.**

^{size}$\in \mathrm{Delay}\mathrm{FPT}$.

**Proof.**

**Corollary**

**4.**

**Proof.**

**Corollary**

**5.**

^{size}andEnum-Triangulation-Deletion

^{size}are in $\mathrm{Delay}\mathrm{FPT}$.

**Proof.**

^{size}, the forbidden patterns are triangles. The result then follows from Corollary 4. □

## 4. Generalisation to Modification Problems

**Definition**

**10**

**.**Let $Q\subseteq {\Sigma}^{*}$ be a language defined over an alphabet and let $x\in {\Sigma}^{*}$ be an input. A set of operations $\Omega \left(Q\right)=\{\phantom{\rule{0.166667em}{0ex}}{\omega}_{n}:{\Sigma}^{*}\to {\Sigma}^{*}\mid n\in \mathbb{N}\phantom{\rule{0.166667em}{0ex}}\}$ is an infinite set of operations on instances of Q. We say that an operation ω is valid with respect to an instance $x\in Q$, if $\omega \left(x\right)\in Q$. We write $\Omega /x$ for the set of possible (valid) operations on an instance x.

- $\omega \left({\omega}^{\prime}\left(x\right)\right)=x$ or (intuitively, ω and ${\omega}^{\prime}$ cancel out)
- $\omega \left({\omega}^{\prime}\left(x\right)\right)={\omega}^{\prime}\left(x\right)$ or $\omega \left({\omega}^{\prime}\left(x\right)\right)=\omega \left(x\right)$. (intuitively, the order of ω and ${\omega}^{\prime}$ does not matter)

**Example**

**1.**

**Definition**

**11**

**.**Let $Q\subseteq {\Sigma}^{*}$ be a language defined over an alphabet, $S\subseteq \Omega /x$ be a finite set of operations on $x\in Q$ and $\mathcal{P}\subseteq Q$ be a property. We say that S is a solution (of x) if S is a consistent set of operations and $S\left(x\right)\in \mathcal{P}$. Furthermore, we denote by ${\mathcal{S}}_{Q}:={\bigcup}_{x\in Q}\{\phantom{\rule{0.166667em}{0ex}}S\mid S\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{a}\phantom{\rule{4.pt}{0ex}}\mathrm{solution}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}x\phantom{\rule{0.166667em}{0ex}}\}$ the set of all solutions for every instance $x\in Q$. In addition, $\mathrm{Sol}\left(x\right)$ is the set of solutions for every instance $x\in Q$.

**Example**

**2.**

- there is no edge $(i,j)$ added or deleted such that vertex i or j is removed,
- there is no edge $(i,j)$ added and removed and
- $S\left(G\right)\vDash \mathcal{P}$.

Parameter: | ${\Pi}_{\mathcal{P}}$—parameterised modification problem $\Pi $ w.r.t. a property $\mathcal{P}$ over $\Sigma $ |

Input: | $x\in {\Sigma}^{*}$, $k\in \mathbb{N}$, $\Omega /x$ set of operations. |

Parameter: | The integer k. |

Question: | Is there a solution $S\subseteq \Omega /x$ and $\left|S\right|\le k$? |

Parameter: | Enum-Min-${\Pi}_{\mathcal{P}}$—parameterised minimum enumeration modification problem w.r.t. a property $\mathcal{P}$ over $\Sigma $ |

Input: | $x\in {\Sigma}^{*}$, $k\in \mathbb{N}$, $\Omega /x$ set of operations. |

Parameter: | The integer k. |

Output: | All minimal (w.r.t. some order) solutions $S\subseteq \Omega /x$ with $\left|S\right|\le k$. |

**Definition**

**12.**

- 1.
- For all $x\in {\Sigma}^{*}$ and $S\in {\mathrm{Sol}}_{{\Pi}_{\mathcal{P}}}\left(x\right)\cup \left\{\mathbb{O}\right\}$, ${\mathcal{N}}_{{\Pi}_{\mathcal{P}}}(x,S)$ is defined.
- 2.
- For all $x\in {\Sigma}^{*}$, ${\mathcal{N}}_{{\Pi}_{\mathcal{P}}}(x,\mathbb{O})=\varnothing $ if ${\mathrm{Sol}}_{{\Pi}_{\mathcal{P}}}\left(x\right)=\varnothing $, and ${\mathcal{N}}_{{\Pi}_{\mathcal{P}}}(x,\mathbb{O})$ is an arbitrary set of solutions otherwise.
- 3.
- For all $x\in {\Sigma}^{*}$ and $S\in {\mathrm{Sol}}_{{\Pi}_{\mathcal{P}}}\left(x\right)$, if ${S}^{\prime}\in {\mathcal{N}}_{{\Pi}_{\mathcal{P}}}(x,S)$ then $\left|S\right|<|{S}^{\prime}|$.
- 4.
- For all $x\in {\Sigma}^{*}$ and all $S\in {\mathrm{Sol}}_{{\Pi}_{\mathcal{P}}}\left(x\right)$, there exists $p>0$ and ${S}_{1},\cdots ,{S}_{p}\in {\mathrm{Sol}}_{{\Pi}_{\mathcal{P}}}\left(x\right)$ such that (i) ${S}_{1}\in {\mathcal{N}}_{{\Pi}_{\mathcal{P}}}(x,\mathbb{O})$, (ii) ${S}_{i+1}\in {\mathcal{N}}_{{\Pi}_{\mathcal{P}}}(x,{S}_{i})$ for $1\le i<p$ and (iii) ${S}_{p}=S$.

**Corollary**

**6.**

- polynomial space for lexicographic order and
- exponential space for size order.

**Corollary**

**7.**

- polynomial space for lexicographic order and
- exponential space for size order.

#### 4.1. Closest String

**Definition**

**13**

**.**Given a string $w={w}_{1}\cdots {w}_{n}$ with ${w}_{i}\in \{0,1\},n\in \mathbb{N}$ and a set $S\subseteq \{1,\cdots ,n\}$, $S\left(w\right)$ denotes the string obtained from w by flipping the bits indicated by S, more formally $S\left(w\right):=S\left({w}_{1}\right)\cdots S\left({w}_{n}\right)$, where $S\left({w}_{i}\right)=1-{w}_{i}$ if $i\in S$ and $S\left({w}_{i}\right)={w}_{i}$ otherwise.

Problem: | Closest-String |

Input: | A sequence $({s}_{1},{s}_{2},\dots ,{s}_{k})$ of k strings over $\{0,1\}$ each of given length $n\in N$ and an integer $d\in N$. |

Parameter: | The integer d. |

Question: | Does there exist $S\subseteq \{1,\cdots ,n\}$ such that ${d}_{H}(S\left({s}_{1}\right),{s}_{i})\le d$ for all $1\le i\le k$? |

**Theorem**

**3.**

- Enum-Closest-String
^{lex}$\in \mathrm{DelayFPT}$with polynomial space. - Enum-Closest-String
^{lex}$\in \mathrm{DelayFPT}$with exponential space.

**Proof.**

#### 4.2. Backdoors

**Definition**

**14.**

**Definition**

**15**

**.**Let $\mathcal{C}$ be a class of CNF formulas and ϕ be a CNF formula. A set $V\subseteq \mathrm{Vars}\left(\varphi \right)$ of variables of ϕ is a $\mathcal{C}$-deletion backdoor set of ϕ if $\varphi \left[V\right]$ is in $\mathcal{C}$, where $\varphi \left[V\right]$ denotes the formula obtained from ϕ by deleting in ϕ all occurrences of variables from V.

**Definition**

**16**

**.**Let $\mathcal{C}$ be a class of CNF formulas and ϕ be a propositional CNF formula. A set $V\subseteq \mathrm{Vars}\left(\varphi \right)$ of variables from ϕ is a weak $\mathcal{C}$ backdoor set of ϕ if there exists an assignment $\theta \in \Theta \left(V\right)$ such that $\varphi \left[\theta \right]\in \mathcal{C}$ and $\varphi \left[\theta \right]$ is satisfiable.

Problem: | Enum-Weak-BackDoorSet$\left(\mathcal{C}\right)$ |

Input: | A formula $\varphi $ in CNF, $k\in \mathbb{N}$. |

Parameter: | The integer k. |

Output: | The set of all weak $\mathcal{C}$ backdoor sets of $\varphi $ of size at most k. |

**Definition**

**17**

**.**The class $\mathcal{C}$ is a base class if it can be recognised in $\mathrm{P}$ (that is, $\mathcal{C}\in \mathrm{P}$), the satisfiability of its formulas is in $\mathrm{P}$ and the class is closed under isomorphisms w.r.t. variable names. We say that $\mathcal{C}$ is clause defined if for every CNF formula ϕ we have: $\varphi \in \mathcal{C}$ if and only if $\left\{C\right\}\in \mathcal{C}$ for all clauses C from ϕ.

**Proposition**

**3**

**.**For every clause-defined base class $\mathcal{C}$, the detection of weak $\mathcal{C}$ backdoor sets is in $\mathrm{FPT}$ for input formulas in 3-CNF.

**Theorem**

**4.**

- Enum-Weak-$\mathcal{C}$-Backdoors
^{lex}$\in \mathrm{DelayFPT}$with polynomial space and - Enum-Weak-$\mathcal{C}$-Backdoors
^{size}$\in \mathrm{DelayFPT}$with exponential space.

**Proof.**

^{lex}, resp., Enum-Min-Weak-$\mathcal{C}$-Backdoors

^{size}being $\mathrm{FPT}$ enumerable. As the consistency check for solutions is in polynomial time, applying Corollary 7 completes the proof. □

**Theorem**

**5.**

- Enum-Strong-$\mathcal{C}$-Backdoors
^{lex}$\in \mathrm{DelayFPT}$with polynomial space and - Enum-Strong-$\mathcal{C}$-Backdoors
^{size}$\in \mathrm{DelayFPT}$with exponential space.

**Proof.**

#### 4.3. Weighted Satisfiability Problems

**Definition**

**18**

**.**Given a propositional formula ϕ and an assignment θ over the variables in ϕ with $\theta \vDash \varphi $, we say that θ is minimal if there does not exist an assignment ${\theta}^{\prime}\subset \theta $ and ${\theta}^{\prime}\vDash \varphi $. The size $\left|\theta \right|$ of θ is the number of variables it sets to true.

Problem: | Min-MinOnes-SAT^{size}$(\Gamma )$ |

Input: | $(\varphi ,k)$, a $\Gamma $ formula $\varphi $, $k\in \mathbb{N}$. |

Parameter: | The integer k. |

Output: | Generate all inclusion-minimal satisfying assignments $\theta $ of $\varphi $ with $\left|\theta \right|\le k$ by non-decreasing size. |

**Theorem**

**6.**

^{size}$(\Gamma )$ is $\mathrm{FPT}$ enumerable and Enum-MinOnes-SAT

^{size}$(\Gamma )\in \mathrm{Delay}\mathrm{FPT}$ with exponential space.

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Creignou, N.; Ktari, R.; Meier, A.; Müller, J.-S.; Olive, F.; Vollmer, H.
Parameterised Enumeration for Modification Problems. *Algorithms* **2019**, *12*, 189.
https://doi.org/10.3390/a12090189

**AMA Style**

Creignou N, Ktari R, Meier A, Müller J-S, Olive F, Vollmer H.
Parameterised Enumeration for Modification Problems. *Algorithms*. 2019; 12(9):189.
https://doi.org/10.3390/a12090189

**Chicago/Turabian Style**

Creignou, Nadia, Raïda Ktari, Arne Meier, Julian-Steffen Müller, Frédéric Olive, and Heribert Vollmer.
2019. "Parameterised Enumeration for Modification Problems" *Algorithms* 12, no. 9: 189.
https://doi.org/10.3390/a12090189