# Approximation Algorithms for the Geometric Firefighter and Budget Fence Problems

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

#### 1.1. Our Contributions

#### 1.2. Related Results

## 2. Barriers

## 3. NP-Hardness

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 4. An Approximation Algorithm for the Geometric Firefighter Problem

**while**there is still any unconsidered job

**do**

**end-while**

**Theorem**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

**Theorem**

**8.**

## 5. A PTAS for the Budget Fence Problem and a Special Case of the Firefighter Problem

**Theorem**

**9.**

**Corollary**

**10.**

**Proof.**

## 6. Generalizations and Refinements

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Klein, R.; Levcopoulos, C.; Lingas, A. Approximation Algorithms for the Geometric Firefighter and Budget Fence Problems. *Algorithms* **2018**, *11*, 45.
https://doi.org/10.3390/a11040045

**AMA Style**

Klein R, Levcopoulos C, Lingas A. Approximation Algorithms for the Geometric Firefighter and Budget Fence Problems. *Algorithms*. 2018; 11(4):45.
https://doi.org/10.3390/a11040045

**Chicago/Turabian Style**

Klein, Rolf, Christos Levcopoulos, and Andrzej Lingas. 2018. "Approximation Algorithms for the Geometric Firefighter and Budget Fence Problems" *Algorithms* 11, no. 4: 45.
https://doi.org/10.3390/a11040045