# Efficiently Finding Simple Schedules in Gaussian Half-Duplex Relay Line Networks

## TL;DR

This paper presents a polynomial-time algorithm for efficiently finding simple schedules in Gaussian Half-Duplex relay line networks, enabling approximate capacity calculation and revealing NP-hardness of HD routing.

## Contribution

It introduces a novel polynomial-time algorithm leveraging graph edge coloring to find simple schedules and derives a closed-form capacity expression for line networks.

## Key findings

- The algorithm achieves the approximate capacity with at most N+1 active states.
- A closed-form expression for the approximate capacity is derived and can be computed distributively.
- The problem of Half-Duplex routing is proven NP-Hard.

## Abstract

The problem of operating a Gaussian Half-Duplex (HD) relay network optimally is challenging due to the exponential number of listen/transmit network states that need to be considered. Recent results have shown that, for the class of Gaussian HD networks with N relays, there always exists a simple schedule, i.e., with at most N +1 active states, that is sufficient for approximate (i.e., up to a constant gap) capacity characterization. This paper investigates how to efficiently find such a simple schedule over line networks. Towards this end, a polynomial-time algorithm is designed and proved to output a simple schedule that achieves the approximate capacity. The key ingredient of the algorithm is to leverage similarities between network states in HD and edge coloring in a graph. It is also shown that the algorithm allows to derive a closed-form expression for the approximate capacity of the Gaussian line network that can be evaluated distributively and in linear time. Additionally, it is shown using this closed-form that the problem of Half-Duplex routing is NP-Hard.