## Abstract

This paper investigates the problem of characterizing the capacity of Half-Duplex (HD) line networks, where a source node communicates to a destination node through a multi-hop path of N relays. If the relays operate in Full-Duplex (FD), it is well known that the capacity of the line network equals the minimum among the point-to-point link capacities in the path. In contrast, this paper considers a different case where the relays operate in HD. In the first part of the paper, it is shown that the approximate capacity (optimal up to a constant additive gap that only depends on the number of nodes in the network) of an HD N-relay line network equals half the minimum of the harmonic means of the point-to-point link capacities of each two consecutive links in the path. It is then proved that the N+1 listen/transmit states (out of the 2 N possible ones) sufficient to characterize the approximate capacity can be found in linear time. In the second part of the paper, it is shown that the problem of finding the path that has the largest HD approximate capacity in a network that can be represented as a graph is NP-hard. However, if the number of cycles in the network is polynomial in the number of nodes, then a polynomial-time algorithm can indeed be designed.

Original language | English (US) |
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Article number | 9103123 |

Pages (from-to) | 4449-4467 |

Number of pages | 19 |

Journal | IEEE Transactions on Information Theory |

Volume | 66 |

Issue number | 7 |

DOIs | |

State | Published - Jul 2020 |

### Bibliographical note

Funding Information:Manuscript received December 22, 2017; revised February 18, 2020; accepted May 11, 2020. Date of publication May 28, 2020; date of current version June 18, 2020. This work was supported in part by the NSF under Award 1514531 and Award 1824568. The work of Martina Cardone was supported in part by the NSF at UCLA under Award 1314937. The work of Daniela Tuninetti was supported in part by the NSF under Award 1527059. This article was presented in part at the 2017 IEEE International Symposium on Information Theory and in part at the 55th Annual Allerton Conference on Communication, Control, and Computing. (Corresponding author: Yahya H. Ezzeldin.) Yahya H. Ezzeldin and Christina Fragouli are with the Electrical and Computer Engineering Department, University of California at Los Angeles, Los Angeles, CA 90095 USA (e-mail: yahya.ezzeldin@ucla.edu; christina.fragouli@ucla.edu).

Publisher Copyright:

© 1963-2012 IEEE.

## Keywords

- Half-duplex (HD) line networks
- NP-hardness
- approximate capacity
- efficient relays scheduling