On the Minimum Cost Range Assignment Problem

TL;DR

This paper addresses the minimum cost range assignment problem for radio networks, providing an optimal algorithm for 1D cases, polynomial solutions for specific constraints, and improved approximation algorithms for higher dimensions.

Contribution

It introduces an optimal $O(n^2)$ algorithm for 1D, polynomial-time solutions for t-spanner constraints, and enhances approximation ratios for higher-dimensional cases.

Findings

Optimal $O(n^2)$ algorithm for 1D range assignment

Polynomial-time algorithm for t-spanner constrained networks in 1D

Improved approximation ratio of 1.5 - ε for higher dimensions

Abstract

We study the problem of assigning transmission ranges to radio stations placed arbitrarily in a $d$-dimensional ($d$-D) Euclidean space in order to achieve a strongly connected communication network with minimum total power consumption. The power required for transmitting in range $r$ is proportional to $r^\alpha$, where $\alpha$ is typically between $1$ and $6$, depending on various environmental factors. While this problem can be solved optimally in $1$D, in higher dimensions it is known to be $NP$-hard for any $\alpha \geq 1$. For the $1$D version of the problem, i.e., radio stations located on a line and $\alpha \geq 1$, we propose an optimal $O(n^2)$-time algorithm. This improves the running time of the best known algorithm by a factor of $n$. Moreover, we show a polynomial-time algorithm for finding the minimum cost range assignment in $1$D whose induced communication graph is a $t$-spanner, for any $t \geq 1$. In higher dimensions, finding the optimal range assignment is $NP$-hard; however, it can be approximated within a constant factor. The best known approximation ratio is for the case $\alpha=1$, where the approximation ratio is $1.5$. We show a new approximation algorithm with improved approximation ratio of $1.5-\epsilon$, where $\epsilon>0$ is a small constant.