Gaussian half-duplex relay networks: Improved gap and a connection with the assignment problem

Abstract

Abstract-This paper studies a Gaussian relay network, where the relays can either transmit or receive at any given time, but not both. Known upper (cut-set) and lower (noisy network coding) bounds on the capacity of a memoryless full-duplex relay network are specialized to the half-duplex case and shown to be to within a constant gap of one another. For fairly broad range of relay network sizes, the derived gap is smaller than what is known in the literature, and it can be further reduced for more structured networks such as diamond networks. It is shown that the asymptotically optimal duration of the listen and transmit phases for the relays can be obtained by solving a linear program; the coefficients of the linear constraints of this linear program are the solution of certain 'assignment problems' for which efficient numerical routines are available; this gives a general interesting connection between the high SNR approximation of the capacity of a MIMO channel and the 'assignment problem' in graph theory. Finally, some results available for diamond networks are extended to general networks. For a general relay network with 2 relays, it is proved that, out of the 4 possible listen/transmit states, at most 3 have a strictly positive probability. Numerical results for a network with K − 2 < 9 relays show that at most K −1 states have a strictly positive probability, which is conjectured to be true for any number of relays.

The approximate optimality of simple schedules for half-duplex multi-relay networks

“…As mentioned earlier, the result of this paper proves our original conjecture in [7] for Gaussian SISO networks for any number N of relays. Our framework also immediately extends to Gaussian networks with MIMO relays and independent noises since also in this setting independent inputs at all nodes are optimal in the cut-set upper bound to within a constant gap for all choices of the channel matrices.…”

“…In [6], Brahma et al's conjecture was proved for Gaussian HD diamond networks with N ≤ 6 relays; the proof is based on certain properties of submodularity and on linear programming duality; the proof technique does not appear to easily generalize to an arbitrary N . Our numerical experiments in [7] showed that Brahma et al's conjecture on the existence of optimal simple schedules for diamond HD relay networks extends to any Gaussian HD multi-relay network (i.e., not necessarily with a diamond topology) with N ≤ 8; we conjectured that the same holds for any N . Should our more general version of Brahma et al's conjecture be true, then Gaussian HD multi-relay networks have optimal simple schedules irrespectively of their topology.…”

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

“…In a preliminary version of this work [1], by using a bounding technique as in [26, pages 20-5, 20-7] we obtained GAP ∼ = N + 2 2 log (4(N + 2)) bits.…”

“…In a HD relay network with N = 2, we have 4 possible states that may arise with probabilities λ j with j ∈ [1,4]. We let λ 1 = λ 00 , λ 2 = λ 01 , λ 3 = λ 10 and λ 4 = λ 11 , where λ ij = P[S 1 = i, S 2 = j] ≥ 0, (i, j) ∈ {0, 1} 2 , such that λ 00 + λ 01 + λ 10 + λ 11 = 1.…”

On the Gaussian Half-Duplex Relay Channel

“…The goal of this work is to show that this gap of 5 bits is too pessimistic and that it can be reduced to 1 bit with PDF with random switch [5]. In a companion paper we showed that Noisy Network Coding (NNC), proposed in [16] to reduce the gap of wireless FD multi-relay networks from 5(N + 2) [14] to 1.26(N + 2), can be used in HD multi-relay networks with random switch to reduce the gap from 5N [15] to 1.96(N + 2) [17].…”