Gaussian Half-Duplex Relay Networks: Improved Constant Gap and Connections With the Assignment Problem

• Published in: IEEE transactions on information theory Vol. 60; no. 6; pp. 3559 - 3575
• Main Authors: Cardone, Martina, Tuninetti, Daniela, Knopp, Raymond, Salim, Umer
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.06.2014; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Artificial neural networks; Assignment problem; Constants; Diamonds; Exact sciences and technology; Gaussian; Graph theory; High definition video; Information theory; Information, signal and communications theory; Mathematical models; Networks; Normal distribution; Optimization; Relay; Relay networks; Relays; Signal to noise ratio; Switching and signalling; Systems, networks and services of telecommunications; Telecommunications; Telecommunications and information theory; Transmission and modulation (techniques and equipments); Upper bound; Vectors; inner bound; Assignment problem; outer bound; relay networks; generalized degrees-of-freedom; weighted bipartite matching problem; capacity to within a constant gap; half-duplex; Packet switching; Relay; Relay network; Graph theory; Data broadcast; Information dissemination; Multicast; Duplex process; Upper bound; Algorithm performance; Network coding; Optimal planning; Numerical algorithm; Information theory
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2014.2314636

This paper considers a Gaussian relay network where a source transmits a message to a destination with the help of N half-duplex relays. The information theoretic cut-set upper bound to the capacity is shown to be achieved to within 1.96(N+2) bits by noisy network coding, thereby reducing the previously known gap. This gap is obtained as a special case of a more general constant gap result for Gaussian half-duplex multicast networks. It is then shown that the generalized degrees-of-freedom of this 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 referred as the assignment problem in graph theory, for which efficient numerical algorithms exist. The optimal schedule, that is, the optimal value of the 2 N possible transmit-receive configuration 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 N=2 relays, that only N+1=3 out of the 2 N =4 possible states have a strictly positive probability and suffice to characterize the capacity to within a constant gap. Extensive experimental results show that, for a general N -relay network with N≤8 , the optimal schedule has at most N+1 states with a strictly positive probability. As an extension of a conjecture presented for diamond networks, it is conjectured that this result holds for any half-duplex relay network and any number of relays. Finally, a network with N=2 relays is studied in detail to illustrate 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.