#
The Inapproximability of k-DominatingSet for Parameterized
AC
0
Circuits †

^{†}

## Abstract

**:**

## 1. Introduction

#### Our Work

**Theorem**

**1.**

- The size of the minimum dominating set is at most k,
- The size of the minimum dominating set is greater than ${\left(\frac{logn}{loglogn}\right)}^{1/\left(\genfrac{}{}{0pt}{}{k}{2}\right)}$.

**Hypothesis**

**1**

**.**There exists $\delta >0$ such that no constant-depth circuits of size ${2}^{\delta n}$ can decide whether the 3-CNF-SAT instance φ is satisfiable, where n is the number of variables of φ.

**Theorem**

**2.**

- The size of the minimum dominating set of G is at most k,
- The size of the minimum dominating set of G is greater than ${\left(\frac{logn}{3loglogn}\right)}^{1/k}$.

## 2. Preliminaries

#### 2.1. Problem Definitions

- In the k-DominatingSet (k-DomSet) problem, our goal is to decide if there is a dominating set of size k in the given graph G.
- In the k-SetCover problem, we are given a bipartite graph $I=(S,U,E)$ and the goal is to decide whether there is a subset X of S with cardinality k such that for each vertex v in U, there exists a vertex u in X that covers v, i.e., $\{u,v\}\in E$.
- In the k-Clique problem, our goal is to determine if there is a clique of size k in the given graph G.
- In the 3-CNF-SAT problem, we are given a propositional formula $\phi $ in which every clause contains at most 3 literals and the goal is to decide whether $\phi $ is satisfiable.

#### 2.2. Circuit Complexity

- ${\mathsf{AC}}^{\mathsf{0}}$ is the class of problems which can be computed by constant-depth circuit families ${({\mathsf{C}}_{n})}_{n\in \mathbb{N}}$ where every ${\mathsf{C}}_{n}$ has size poly(n), and whose gates have unbounded fan-in.
- Para-${\mathsf{AC}}^{\mathsf{0}}$ is the class of parameterized problems which can be computed by a circuit family ${({\mathsf{C}}_{n,k})}_{n,k\in \mathbb{N}}$ satisfying that there exist $d\in \mathbb{N}$ and a computed function f such that for every $n\in \mathbb{N},k\in \mathbb{N}$, ${\mathsf{C}}_{n,k}$ has depth d and size $f(k)\mathrm{poly}(n)$, and whose gates have unbounded fan-in.
- ${\mathsf{NC}}^{\mathsf{1}}$ is the class of problems which can be computed by a circuit family ${({\mathsf{C}}_{n})}_{n\in \mathbb{N}}$ where ${\mathsf{C}}_{n}$ has depth $O(logn)$ and size poly(n), and whose gates have a fan-in of 2.

#### 2.3. Covering Arrays

## 3. Introducing Gap to the k-SetCover Problem

**Definition**

**1.**

- (G1)
- A is partitioned into $({A}_{1},\cdots ,{A}_{m})$ where $|{A}_{i}|=\ell $ for every $i\in [m]$.
- (G2)
- B is partitioned into $({B}_{1},\cdots ,{B}_{k})$ where $|{B}_{j}|=n$ for every $j\in [k]$.
- (G3)
- For each ${b}_{1}\in {B}_{1},\cdots ,{b}_{k}\in {B}_{k}$, there exists ${a}_{1}\in {A}_{1},\cdots ,{a}_{m}\in {A}_{m}$ such that ${a}_{i}$ is adjacent to ${b}_{j}$ for $i\in [m],j\in [k]$.
- (G4)
- For any $X\subseteq B$ and ${a}_{1}\in {A}_{1},\cdots ,{a}_{m}\in {A}_{m}$, if ${a}_{i}$ has $k+1$ neighbors in X for $i\in [m]$, then $|X|>h$.

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**a**and b are adjacent using $O(1)$ gates, for every $\mathit{a}\in A,b\in B$.

**Proof.**

`CA`(n log h; k, n, 2), denoted by $\mathcal{S}$.

**Claim.**

`CA`(n log h; k, n, 2) covering array, there must be a row r such that for each ${i}_{{j}^{\prime}}\in C$, ${M}_{r,{i}_{{j}^{\prime}}}={j}^{\prime}$.

- $A={\cup}_{i\in [m]}{A}_{i}$ with each ${A}_{i}=\{\mathit{a}|\mathit{a}=({a}_{1},\cdots ,{a}_{k}),{a}_{j}\in [h]\mathrm{for}j\in [k]\}$;
- $B={\cup}_{i\in [k]}{B}_{i}$ with ${B}_{i}={S}_{i}$ for $i\in [k]$;
- $E=\{\mathit{a},b\}\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}\mathit{a}\in {A}_{i},b\in {B}_{j}\mathrm{and}{M}_{i,b}=\mathit{a}[j],\mathrm{for}i\in [m],j\in [k]$, that is, for every $i\in [m],j\in [k]$ and every $\mathit{a}\in {A}_{i},b\in {B}_{j}$, if ${M}_{i,b}=\mathit{a}[j]$ then we add an edge between
**a**and b.

**a**and b are connected is determined by

**Lemma**

**3.**

- If there exists ${s}_{1}\in {S}_{1},\cdots ,{s}_{k}\in {S}_{k}$ which could cover U, then the set cover number of ${I}^{\prime}$ is at most k;
- If the set cover number of I is larger than k, then the set cover number of ${I}^{\prime}$ is greater than h.

**Proof.**

- ${S}^{\prime}=S$;
- ${U}^{\prime}={\cup}_{i\in [m]}{U}^{{A}_{i}}$;
- For every $s\in {S}^{\prime}$ and $f\in {U}^{{A}_{i}}$ for each $i\in [m]$, $\{s,f\}\in {E}^{\prime}$ if there is an $a\in {A}_{i}$ such that $\{s,f(a)\}\in E$ and $\{a,s\}\in {E}_{T}$.

**Claim.**

## 4. Inapproximability of k-DominatingSet

#### 4.1. The Unconditional Inapproximability of k-DominatingSet

**Lemma**

**4.**

**Proof.**

**b**(v) the bit representation of v. Note that when i is fixed, every vertex in ${V}_{i}$ could be determined using $logn$ bits.

- $S={E}^{\prime}={\cup}_{1\le i<j\le k}{E}_{i,j}$;
- $U={\cup}_{i\in [k]}{U}_{i}$ with ${U}_{i}=\{({f}^{(i)},l)\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}{f}^{(i)}:\{0,1\}\to [k]\backslash i,l\in [logn]\}$;
- For every $i\in [k]$, we connect every $({f}^{(i)},l)\in {U}_{i}$ to each $\{{v}_{i},{v}_{j}\}\in S$, with ${v}_{i}\in {V}_{i},{v}_{j}\in {V}_{j}$, such that ${f}^{(i)}(\mathit{b}({v}_{i})[l])=j$.

**Lemma**

**5.**

- If G contains a k-clique, then the set cover number of I is at most $\left(\genfrac{}{}{0pt}{}{k}{2}\right)$;
- If G contains no k-clique, then the set cover number of I is greater than ${\left(\frac{logn}{loglogn}\right)}^{1/\left(\genfrac{}{}{0pt}{}{k}{2}\right)}$.

**Proof.**

- If G contains a k-clique, then the set cover number of I is at most $\left(\genfrac{}{}{0pt}{}{k}{2}\right)$;
- If G contains no k-clique, then the set cover number of I is greater than ${(\frac{logm}{loglogm})}^{1/\left(\genfrac{}{}{0pt}{}{k}{2}\right)}\ge {\left(\frac{logn}{loglogn}\right)}^{1/\left(\genfrac{}{}{0pt}{}{k}{2}\right)}$.

**Theorem**

**3.**

- The set cover number of I is at most k, or
- The set cover number of I is greater than ${\left(\frac{logn}{loglogn}\right)}^{1/\left(\genfrac{}{}{0pt}{}{k}{2}\right)}$.

**Proof.**

#### 4.2. The Inapproximability of k-DominatingSet Assuming ${\mathsf{AC}}^{\mathsf{0}}$-ETH

**Lemma**

**6.**

- $|S|+|U|\le N$;
- If φ is satisfiable, then the set cover number of I is at most k;
- If φ is not satisfiable, then the set cover number of I is greater than ${\left(\frac{logN}{3loglogN}\right)}^{1/k}$;

**Proof.**

- If $\phi $ is satisfiable, then the set cover number of I is at most k;
- If $\phi $ is unsatisfiable, then the set cover number of I is greater than h;
- $S={S}^{\prime}$, $|U|=mlogh{|{U}^{\prime}|}^{{h}^{k}}\le \frac{1}{k}m(loglogm-logloglogm){(2{n}^{3})}^{\frac{logm}{loglogm}}\le m\xb7loglogm\xb7({2}^{\frac{logm}{loglogm}}+{n}^{\frac{3n}{k(logn-logk)}})\le m\xb7loglogm\xb7({2}^{\frac{logm}{loglogm}}+{n}^{\frac{3n}{k}})={m}^{4+o(1)}$.

**Theorem**

**4.**

- The set cover number of I is at most k, or
- The set cover number of I is greater than ${\left(\frac{logn}{3loglogn}\right)}^{1/k}$.

**Proof.**

#### 4.3. The Difficulty of Proving ${\mathsf{AC}}^{\mathsf{0}}$-ETH

**Lemma**

**7.**

- $s\in L$ if and only if ${\mathsf{C}}_{|s|}^{\prime}$ outputs 1;
- ${\mathsf{C}}_{n}^{\prime}$ has depth d and size at most ${n}^{3c/2}({2}^{{n}^{2c/d}+1}+1)$.

**Proof.**

**Theorem**

**5.**

**Proof.**

## 5. Conclusions and Open Questions

## Funding

## Acknowledgments

## Conflicts of Interest

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Lai, W.
The Inapproximability of *k*-DominatingSet for Parameterized *Algorithms* **2019**, *12*, 230.
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Lai W.
The Inapproximability of *k*-DominatingSet for Parameterized *Algorithms*. 2019; 12(11):230.
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2019. "The Inapproximability of *k*-DominatingSet for Parameterized *Algorithms* 12, no. 11: 230.
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