# Online Bottleneck Matching Problem with Two Heterogeneous Sensors in a Metric Space

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**available**if the ${s}_{i}$ is not matched by one request. If ${r}_{j}$ is matched with the sensor ${s}_{i}$, the cost of pair $({r}_{j},{s}_{i})$ is

## 3. The OBM(2) Problem

#### 3.1. The Case $1\le w\le \beta $

**Theorem**

**1.**

**Proof.**

Algorithm 1:A1 |

**Theorem**

**2.**

**Proof.**

**Case 1.**$\sigma \left({r}_{j}\right)={s}_{1}$ and ${\sigma}^{*}\left({r}_{k}\right)={s}_{1}$ ($\sigma \left({r}_{k}\right)={s}_{2}$ and ${\sigma}^{*}\left({r}_{j}\right)={s}_{2})$.

**Case 2.**$\sigma \left({r}_{j}\right)={s}_{1}$ and ${\sigma}^{*}\left({r}_{k}\right)={s}_{2}$.

**Case 3.**$\sigma \left({r}_{j}\right)={s}_{2}$ and ${\sigma}^{*}\left({r}_{k}\right)={s}_{2}$ ($\sigma \left({r}_{k}\right)={s}_{1}$ and ${\sigma}^{*}\left({r}_{j}\right)={s}_{1})$.

**Case 4.**$\sigma \left({r}_{j}\right)={s}_{2}$ and ${\sigma}^{*}\left({r}_{k}\right)={s}_{1}$.

#### 3.2. The Case $w>\beta $

**Theorem**

**3.**

**Proof.**

Algorithm 2:A2 |

**Theorem**

**4.**

**Proof.**

**Case 1.**$\sigma \left({r}_{j}\right)={s}_{1}$ and ${\sigma}^{*}\left({r}_{k}\right)={s}_{1}$ ($\sigma \left({r}_{k}\right)={s}_{2}$ and ${\sigma}^{*}\left({r}_{j}\right)={s}_{2})$.

**Case 2.**$\sigma \left({r}_{j}\right)={s}_{1}$ and ${\sigma}^{*}\left({r}_{k}\right)={s}_{2}$.

**Case 3.**$\sigma \left({r}_{j}\right)={s}_{2}$ and ${\sigma}^{*}\left({r}_{k}\right)={s}_{2}$ ($\sigma \left({r}_{k}\right)={s}_{1}$ and ${\sigma}^{*}\left({r}_{j}\right)={s}_{1})$.

**Case 4.**$\sigma \left({r}_{j}\right)={s}_{2}$ and ${\sigma}^{*}\left({r}_{k}\right)={s}_{1}$.

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Xiao, M.; Yang, Y.; Li, W.
Online Bottleneck Matching Problem with Two Heterogeneous Sensors in a Metric Space. *Computation* **2022**, *10*, 217.
https://doi.org/10.3390/computation10120217

**AMA Style**

Xiao M, Yang Y, Li W.
Online Bottleneck Matching Problem with Two Heterogeneous Sensors in a Metric Space. *Computation*. 2022; 10(12):217.
https://doi.org/10.3390/computation10120217

**Chicago/Turabian Style**

Xiao, Man, Yaru Yang, and Weidong Li.
2022. "Online Bottleneck Matching Problem with Two Heterogeneous Sensors in a Metric Space" *Computation* 10, no. 12: 217.
https://doi.org/10.3390/computation10120217