# Inapproximability of Rank, Clique, Boolean, and Maximum Induced Matching-Widths under Small Set Expansion Hypothesis

## Abstract

**:**

## 1. Introduction

## 2. Definitions and Known Results

#### 2.1. Graphs, Expansion, Matrices

**Lemma**

**1**

**.**Let A be a matrix over $GF(2)$ such that A has at least p non-zero entries and each row and each column in A has at most q non-zero entries. Then, $\mathit{rank}(A)\ge \frac{p}{{q}^{2}}$ holds.

#### 2.2. Graph Width Parameters

- T is a subcubic tree with $\left|V\right|$ leaves, where a tree is subcubic if every vertex in T has degree 1 or 3;
- ${f}_{T}$ is a bijection from $L(T)$ of T to V.

**Cut-width:**- For a graph $G=(V,E)$, let $\pi :V\to \{1,\dots ,|V\left|\right\}$ be an ordering of V. Let ${S}_{\pi}(i):=$$\{{\pi}^{-1}(1),$${\pi}^{-1}(2),$$\dots ,$${\pi}^{-1}(i)\}$, ${\partial}_{e}(\pi ,i):=E({S}_{\pi}(i),\overline{{S}_{\pi}(i)})$. Then, $cutw(G):=\underset{\pi}{min}\underset{1\le i\le n}{max}\left|{\partial}_{e}(\pi ,i)\right|$.
**Rank-width:**- For a graph $G=(V,E)$, let T be a subcubic tree and ${f}_{T}$ be a bijection from $L(T)$ to V. Then, $rw(G):=\underset{(T,{f}_{T})\in {\mathcal{D}}_{G}}{min}\underset{e\in E(T)}{max}rank\left(M\left[{(V\backslash e)}_{1},{(V\backslash e)}_{2}\right]\right)$.
**Maximum induced matching-width**- For a graph $G=(V,E)$ and a subset A of V, we denote the size of a maximum induced matching in the bipartite graph $(A,\overline{A};E(A,\overline{A}))$ by $mim(A)$. Then, $mimw(G):=\underset{(T,{f}_{T})\in {\mathcal{D}}_{G}}{min}\underset{e\in E(T)}{max}mim({(V\backslash e)}_{1}))$.

## 3. SSE Hypothesis

**SSE hypothesis (strong form)**[Conjecture 2.23 and Remark 2.25 in [9]] There is a constant c such that for every integer $q>1$ and arbitrarily small $\epsilon >0$, the following problem is NP-hard:

**Problem**

**1.**

**Yes**- There exist q disjoint sets ${S}_{1},\dots ,{S}_{q}\subseteq V$ such that ${\Phi}_{G}({S}_{i})\le 2\epsilon $ and $|{S}_{i}|=\frac{\left|V\right|}{q}$ holds for all $1\le i\le q$,
**No**- For every $\frac{\left|V\right|}{10}\le \left|S\right|\le \frac{9\left|V\right|}{10}$, ${\Phi}_{G}(S)\ge c\sqrt{\epsilon}$ holds.

## 4. Method for Showing Inapproximability

- $gw(G)\le {c}_{Y}\epsilon P$ holds if G is a YES instance in Problem 1 (i.e., completeness), and
- $gw(G)\ge {c}_{N}\sqrt{\epsilon}P$ holds if G is a NO instance in Problem 1 (i.e., soundness),

Algorithm 1 DeciInstByAlg$A(G)$ |

Input: a graph GOutput: YES/NO |

## 5. Hardness Results Derived from Inapproximability of Tree-Width

**Theorem**

**1.**

**Proof.**

## 6. Results

#### 6.1. Simpler Proof of the Inapproximability of {Cut, Path, Tree}-Widths

**Lemma**

**2**

**.**Let $q=\frac{1}{\epsilon}$. Let $G=(V,E)$ be d-regular and a YES instance in Problem 1, where d is an universal constant. Then, $cutw(G)\le {c}_{c}\epsilon \left|E\right|$ holds for some universal constant ${c}_{c}$.

**Proof.**

**Lemma**

**3.**

- ${min}_{i}{b}_{e}(i,G)\ge \frac{c}{2}\sqrt{\epsilon}\left|E\right|$, and
- ${min}_{i}{b}_{v}(i,G)\ge \frac{c}{2d}\sqrt{\u03f5}\left|E\right|$

**Proof.**

**Theorem**

**2.**

#### 6.2. Inapproximability of Rank, Clique, Boolean, and Mim-Widths

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 7. Future Research

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Relations among Graph Parameters

- For each $1\le j\le \left|V\right|$,$$\underset{j/2\le i\le j}{min}{b}_{v}(i,G)-1\le tw(G)\le pw(G)\le min\{cutw(G),banw(G)\}.\phantom{\rule{4pt}{0ex}}(\mathrm{see}\mathrm{Lemma}9\mathrm{in}[{27}])$$
- For each $1\le j\le \left|V\right|$,$$\underset{j/2\le i\le j}{min}{b}_{v}(i,G)\le \underset{j/2\le i\le j}{min}{b}_{e}(i,G)\le carw(G)\le cutw(G)\phantom{\rule{4pt}{0ex}}(\mathrm{see}\mathrm{Theorem}\phantom{\rule{3.33333pt}{0ex}}1\mathrm{in}[{28}]),$$
- $$rw(G)\le min\{tw(G)+1,cliw(G)\}\phantom{\rule{4pt}{0ex}}(\mathrm{see}\mathrm{Corollary}\phantom{\rule{3.33333pt}{0ex}}5\mathrm{in}[{29}],\mathrm{Proposition}\phantom{\rule{3.33333pt}{0ex}}6.3\mathrm{in}[{31}]),$$
- $$lclw(G)\le pw(G)+2\phantom{\rule{4pt}{0ex}}(\mathrm{see}\mathrm{Section}\phantom{\rule{3.33333pt}{0ex}}5\mathrm{in}[{30}]),$$
- $$\mathrm{NLC}(G)\le cliw(G)\le 2\mathrm{NLC}(G)\phantom{\rule{4pt}{0ex}}(\mathrm{see}[{36},{37}]),$$
- $$boow(G)\le tw(G)+1\phantom{\rule{4pt}{0ex}}(\mathrm{see}\mathrm{Figure}\phantom{\rule{3.33333pt}{0ex}}1\mathrm{in}[{33}]).$$
- $$mimw(G)\le boow(G)\le mimw(G){log}_{2}(|V(G)|)\phantom{\rule{4pt}{0ex}}(\mathrm{see}\mathrm{Theorem}\phantom{\rule{3.33333pt}{0ex}}\mathrm{4.2.10}\mathrm{in}[{32}]).$$

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**Figure 1.**Scheme showing how to prove the inapproximability. ($g{w}_{1}\to g{w}_{2}$ means that $g{w}_{1}\u2aafg{w}_{2}$).

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Yamazaki, K. Inapproximability of Rank, Clique, Boolean, and Maximum Induced Matching-Widths under Small Set Expansion Hypothesis. *Algorithms* **2018**, *11*, 173.
https://doi.org/10.3390/a11110173

**AMA Style**

Yamazaki K. Inapproximability of Rank, Clique, Boolean, and Maximum Induced Matching-Widths under Small Set Expansion Hypothesis. *Algorithms*. 2018; 11(11):173.
https://doi.org/10.3390/a11110173

**Chicago/Turabian Style**

Yamazaki, Koichi. 2018. "Inapproximability of Rank, Clique, Boolean, and Maximum Induced Matching-Widths under Small Set Expansion Hypothesis" *Algorithms* 11, no. 11: 173.
https://doi.org/10.3390/a11110173