# A Brief Survey of Fixed-Parameter Parallelism

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

`clique`in G if $G[S]$ is a complete subgraph (i.e., any two elements of S are adjacent in G). A clique on t vertices is denoted by ${K}_{t}$. On the other hand, S is said to be

`independent`or

`stable`in G if no two elements of S are adjacent in G.

`Vertex Cover`which, for a given graph $G=(V,E)$ and a parameter k, asks for a subset C of V of cardinality at most k whose complement is independent. In other words, every edge of G has at least one endpoint in C. Vertex Cover is among the first classical problems shown to be fixed-parameter tractable [3]. A closely related problem is

`Clique`, which for a given graph G and a parameter k, asks whether G has a clique of size at least k. From a parameterized complexity standpoint,

`Clique`is equivalent to

`Independent Set`due a straightforward transformation of the input, without changing the parameter. Both problems are $W[1]$-complete.

`Vertex Cover`is popular, and not as “hard” as other parameterized problems, stems from the fact that it is a special case of many (more general) problems. It is a (i) deletion into maximum-degree d where $d=0$, (ii) deletion into connected components of size p where $p=1$, and (iii) deletion into ${K}_{t}$-free graphs for $t=2$, among other generalizations. A closely related problem is deletion into cycle-free graphs: in the

`Feedback Vertex Set (FVS)`problem we ask for a subset C of V of cardinality at most k such that $V\backslash C$ has no cycles. $FVS$ is fixed-parameter tractable.

- (i)
- Every vertex of G is present in at least one node in T. Moreover $\bigcup {X}_{i}=V$.
- (ii)
- The nodes of T containing some vertex v form a connected subtree. In other words, if ${X}_{i}$ and ${X}_{j}$ both contain a vertex v, then every node ${X}_{k}$ along the unique path between ${X}_{i}$ and ${X}_{j}$ must contain v.
- (iii)
- All edges of G must be represented in the subsets. For every edges $(u,v)$ of G, there is at least one node ${X}_{i}$ in T that contains both u and v.

## 3. Parameterized Parallel Complexity Classes

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

#### 3.1. PNC and NC

#### 3.2. PNC and FPT

#### 3.3. FPP and PNC

#### 3.4. FPP and FPPT

#### 3.5. Parameterized Parallel Intractability

## 4. Key Results and Methods

- -
- Applying $FPP$ branching (i.e., parallel bounded search tree algorithms);
- -
- Applying $FPP$ reductions to kernelize a problem instance before applying a general sequential (possibly branching) algorithm.
- -
- Coloring the input graph (in graph-theoretic problems) with a number of colors that is dependent on the parameter to reduce the problem into one solvable in $FPP$-time.
- -
- Choosing the right auxiliary parameters for which the problem falls in the class $FPP$.

## 5. Implementation Aspects

#### 5.1. Input Representation

#### 5.2. Input Queries

#### 5.3. Input Modification

#### 5.4. Avoiding Excessive Use of Successive Operations

## 6. Common FPT Methods

#### 6.1. Bounded Search Trees

#### 6.2. Kernelization

#### 6.3. Color Coding

#### 6.4. Iterative Compression

## 7. Parallel Algorithms Using FPT Methods

#### 7.1. Bounded Search Trees

#### 7.2. Kernelization

#### 7.3. Color Coding

#### 7.4. Iterative Compression

## 8. Problems not Fixed-Parameter Parallel Tractable

#### 8.1. Problems that Are NC but not FPT (so neither PNC nor FPP)

#### 8.2. Problems that Are FPT but not NC (also neither PNC nor FPP)

#### 8.3. Problems that Are NC and FPT But Not PNC (So Not FPP)

## 9. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Abu-Khzam, F.N.; Al Kontar, K.
A Brief Survey of Fixed-Parameter Parallelism. *Algorithms* **2020**, *13*, 197.
https://doi.org/10.3390/a13080197

**AMA Style**

Abu-Khzam FN, Al Kontar K.
A Brief Survey of Fixed-Parameter Parallelism. *Algorithms*. 2020; 13(8):197.
https://doi.org/10.3390/a13080197

**Chicago/Turabian Style**

Abu-Khzam, Faisal N., and Karam Al Kontar.
2020. "A Brief Survey of Fixed-Parameter Parallelism" *Algorithms* 13, no. 8: 197.
https://doi.org/10.3390/a13080197