Gaussian Half-Duplex Relay Networks: improved constant gap and connections with the assignment problem

TL;DR

This paper improves the understanding of Gaussian half-duplex relay networks by providing tighter capacity bounds, connecting the problem to the assignment problem in graph theory, and analyzing optimal relay scheduling.

Contribution

It introduces a reduced gap to capacity using noisy network coding, links the degrees-of-freedom to linear programs, and characterizes optimal relay schedules for general networks.

Findings

Achieves capacity within 2.021(N+2) bits for N-relay networks.

Shows the degrees-of-freedom problem reduces to a linear program.

Most relay states have zero probability, with at most N+1 active states for N relays.

Abstract

This paper considers a general Gaussian relay network where a source transmits a message to a destination with the help of N half-duplex relays. It proves that the information theoretic cut-set upper bound to the capacity can be achieved to within 2:021(N +2) bits with noisy network coding, thereby reducing the previously known gap. Further improved gap results are presented for more structured networks like diamond networks. It is then shown that the generalized Degrees-of-Freedom of a general Gaussian half-duplex relay network is the solution of a linear program, where the coefficients of the linear inequality constraints are proved to be the solution of several linear programs, known in graph theory as the assignment problem, for which efficient numerical algorithms exist. The optimal schedule, that is, the optimal value of the 2^N possible transmit-receive configurations/states for the relays, is investigated and known results for diamond networks are extended to general relay networks. It is shown, for the case of 2 relays, that only 3 out of the 4 possible states have strictly positive probability. Extensive experimental results show that, for a general N-relay network with N<9, the optimal schedule has at most N +1 states with strictly positive probability. As an extension of a conjecture presented for diamond networks, it is conjectured that this result holds for any HD relay network and any number of relays. Finally, a 2-relay network is studied to determine the channel conditions under which selecting the best relay is not optimal, and to highlight the nature of the rate gain due to multiple relays.