# NP-Hardness of the Problem of Optimal Box Positioning

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Previous Results

- B does not intersect with ${P}_{-}$, and
- the cardinality of $B\bigcap {P}_{+}$ is maximal over all boxes that satisfy the first condition.

## 3. Formal Definitions

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

## 4. NP-Hardness of the Problem of Optimal Box Positioning

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Lemma 1.**

**Proof.**

**Lemma 2.**

**Proof.**

**Lemma 3.**

**Theorem**

**3.**

**Theorem**

**4.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Galatenko, A.V.; Nersisyan, S.A.; Zhuk, D.N.
NP-Hardness of the Problem of Optimal Box Positioning. *Mathematics* **2019**, *7*, 711.
https://doi.org/10.3390/math7080711

**AMA Style**

Galatenko AV, Nersisyan SA, Zhuk DN.
NP-Hardness of the Problem of Optimal Box Positioning. *Mathematics*. 2019; 7(8):711.
https://doi.org/10.3390/math7080711

**Chicago/Turabian Style**

Galatenko, Alexei V., Stepan A. Nersisyan, and Dmitriy N. Zhuk.
2019. "NP-Hardness of the Problem of Optimal Box Positioning" *Mathematics* 7, no. 8: 711.
https://doi.org/10.3390/math7080711