## Abstract

Theoretical study of optimization problems in wireless communication often deals with tasks that concern a single point. For example, the power control problem requires computing a power assignment guaranteeing that each transmitting station s_{i} is successfully received at a single receiver point r_{i}. This paper aims at addressing communication applications that require handling two-dimensional tasks (e.g., guaranteeing successful transmission in entire regions rather than at specific points).The natural approach to two-dimensional optimization tasks is to discretize the optimization domain, e.g., by sampling points within the domain. The straightforward implementation of the discretization approach, however, might incur high time and memory requirements, and moreover, it cannot guarantee exact solutions.The alternative proposed and explored in this paper is based on establishing the minimum principle^{1} for the signal to interference and noise ratio (SINR) function with free space path loss (i.e., when the signal decays in proportion to the square of the distance between the transmitter and receiver). Essentially, the minimum principle allows us to reduce the dimension of the optimization domain without losing anything in the accuracy or quality of the solution. More specifically, when the two-dimensional optimization domain is bounded and free from any interfering station, the minimum principle implies that it is sufficient to optimize the SINR function over the boundary of the domain, as the "hardest"points to be satisfied reside on the boundary and not in the interior.We then utilize the minimum principle as the basis for an improved discretization technique for solving two-dimensional problems in the SINR model. This approach is shown to be useful for handling optimization problems over two dimensions (e.g., power control, energy minimization); in providing tight bounds on the number of null cells in the reception map; and in approximating geometric and topological properties of the wireless reception map (e.g., maximum inscribed sphere).The minimum principle, as well as the interplay between continuous and discrete analysis presented in this paper, are expected to pave the way to future study of algorithmic SINR in higher dimensions.

Original language | English |
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Article number | 1 |

Journal | ACM Transactions on Algorithms |

Volume | 19 |

Issue number | 1 |

DOIs | |

State | Published - 9 Mar 2023 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2023 Association for Computing Machinery.

### Funding

Zvi Lotker is partially supported by Foundation des Sciences Mathematiques de Paris, the Ministry of Science Technology and Space, Israel, French-Israeli project MAIMONIDE 31768XL, and the French-Israeli Laboratory FILOFOCS. Erez Kantor is supported in a part by AFOSR Contract Numbers : FA9550-13-1-0042 and FA9550-14-1-0403, and by NSF Awards 0939370-CCF, CCF-1217506 and CCF-AF-0937274. David Peleg and Merav Parter are supported in part by the Israel Science Foundation (grant 894/09), the United States-Israel Binational Science Foundation (grant 2008348), the Israel Ministry of Science and Technology (infrastructures grant), and the Citi Foundation.

Funders | Funder number |
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National Science Foundation | CCF-1217506, CCF-AF-0937274, 0939370-CCF |

Air Force Office of Scientific Research | FA9550-13-1-0042, FA9550-14-1-0403 |

Citi Foundation | |

Ministry of Science, Technology and Space | 31768XL |

United States-Israel Binational Science Foundation | 2008348 |

Israel Science Foundation | 894/09 |

Ministry of science and technology, Israel | |

Fondation Sciences Mathématiques de Paris |

## Keywords

- Additional Key Words and PhrasesSINR
- communication
- power control