# Correspondence between Multilevel Graph Partitions and Tree Decompositions

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## Abstract

**:**

## 1. Introduction

## 2. Tree Decomposition

- 1.
- For every node $u\in V$, there is a bag $X\in \mathcal{X}$ such that $u\in \mathcal{X}$.
- 2.
- For every edge $\{u,v\}\in E$, there is a bag $X\in \mathcal{X}$ such that $u\in X$ and $v\in X$.
- 3.
- For every pair of bags $X\in \mathcal{X}$ and $Y\in \mathcal{X}$, every bag Z along the unique path in T from X to Y must contain the intersection of X and Y, i.e., $X\cap Y\subseteq Z$.

- 4.
- no leaf bag is a subset of its parent bag; and
- 5.
- for every non-root bag X and node $x\in X$, such that the parent bag but no child bag of X contains x, removing x from X must result in a violation of the edge property.

## 3. Graph Partitioning

#### 3.1. Edge-Based Graph Partition

#### 3.2. Node-Based Graph Partition

#### 3.3. Touching Cells

#### 3.4. Multilevel Graph Partition

- V is a cell, i.e., $V\in \mathcal{C}$;
- no cell is empty;
- all touching cells are totally ordered by set inclusion, i.e., $\forall {C}_{1},{C}_{2}\in \mathcal{C}:{C}_{1}\phantom{\rule{4.pt}{0ex}}\mathrm{touches}\phantom{\rule{4.pt}{0ex}}{C}_{2}\Rightarrow {C}_{1}\subseteq {C}_{2}\vee {C}_{2}\subseteq {C}_{1}$; and
- for all cells ${C}_{1},{C}_{2}\in \mathcal{C}$: if ${C}_{1}\u228a{C}_{2}$, ${C}_{2}$ is an ancestor of ${C}_{1}$, if ${C}_{1}={C}_{2}$, ${C}_{1}$ is an ancestor of ${C}_{2}$ or ${C}_{2}$ is an ancestor of ${C}_{1}$.

## 4. Correspondence between Multilevel Graph Partitioning and Tree Decompositions

**Theorem**

**1.**

#### 4.1. Multilevel Partition to Tree Decomposition

#### 4.2. Tree Decomposition to Multilevel Partition

#### 4.3. Asymptotic Running Times

Algorithm 1: Recursive function that implements MLP2TD. |

Algorithm 2: Recursive function that implements TD2MLP. |

`OutputBags()`in Algorithm 1 recursively calculates the boundary of each cell represented by a separator S. It first calls itself on all children and accumulates their boundaries in B. Then, it marks all nodes in the separator and adds their neighbors to B. Note that marks are global and never cleared. The boundary is then obtained by removing all marked nodes from B. By building the union of S and B, it then outputs the corresponding bag X. To show its correctness, we need to show that

`OutputBags`indeed returns the boundary of the cell represented by S.

**Lemma**

**1.**

`OutputBags`returns the boundary of the cell represented by S.

**Proof.**

`OutputBags`, all separators of children of S are marked and thus at least all nodes in C are marked. Therefore, in line 10, all nodes of C are removed.

**Lemma**

**2.**

**Proof.**

`OutputSeparators()`is called for each bag X recursively, starting with the root bag.

`OutputSeparators()`starts by outputting the nodes in the bag X except nodes that are marked. Then, it marks all nodes in X and calls itself recursively. Algorithm 2 runs in time linear in the input, as it processes the elements of every bag X only a constant number of times. What remains to be shown is that line 4 indeed outputs the separator as defined in TD2MLP.

**Lemma**

**3.**

**Proof.**

#### 4.4. Example Transformation

#### 4.5. Correctness Proof

**Theorem**

**2.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

**Theorem**

**3.**

**Proof.**

#### 4.6. Bijection Proof

**Lemma**

**8.**

**Proof.**

**Lemma**

**9.**

**Proof.**

**Lemma**

**10.**

**Proof.**

**Lemma**

**11.**

**Proof.**

**Lemma**

**12.**

**Proof.**

## 5. Example Applications: Tree Decomposition of Road Network

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Node-based multilevel partition of the running example. The colors correspond to levels. The depicted multilevel partition can be obtained when partitioning the partitions of example Figure 2b.

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Hamann, M.; Strasser, B. Correspondence between Multilevel Graph Partitions and Tree Decompositions. *Algorithms* **2019**, *12*, 198.
https://doi.org/10.3390/a12090198

**AMA Style**

Hamann M, Strasser B. Correspondence between Multilevel Graph Partitions and Tree Decompositions. *Algorithms*. 2019; 12(9):198.
https://doi.org/10.3390/a12090198

**Chicago/Turabian Style**

Hamann, Michael, and Ben Strasser. 2019. "Correspondence between Multilevel Graph Partitions and Tree Decompositions" *Algorithms* 12, no. 9: 198.
https://doi.org/10.3390/a12090198