Network Simplification in Half-Duplex: Building on Submodularity

Abstract

This paper explores the network simplification problem in the context of Gaussian Half-Duplex (HD) diamond networks. Specifically, given an N -relay diamond network, this problem seeks to derive fundamental guarantees on the capacity of the best k-relay subnetwork, as a function of the full network capacity. The main focus of this work is on the case when k = N − 1 relays are selected out of the N possible ones. First, a simple algorithm, which removes the relay with the minimum capacity (i.e., the worst relay), is analyzed and it is shown that the remaining (N −1)-relay subnetwork has an approximate (i.e., optimal up to a constant gap) HD capacity that is at least half of the approximate HD capacity of the full network. This fraction guarantee is shown to be tight if only the single relay capacities are known, i.e., there exists a class of Gaussian HD diamond networks with N relays where, by removing the worst relay, the subnetwork of the remaining k = N − 1 relays has an approximate capacity equal to half of the approximate capacity of the full network. Next, this work proves a fundamental guarantee, which improves over the previous fraction: there always exists a subnetwork of k = N − 1 relays that achieves at least a fraction N −1 N of the approximate capacity of the full network. This fraction is proved to be tight and it is shown that any optimal schedule of the full network can be used by at least one of the N subnetworks of N − 1 relays to achieve a worst-case performance guarantee of N −1 N . Additionally, these results are extended to derive lower bounds on the fraction guarantee for general k ∈ [1 : N ]. The key steps in the proofs lie in the derivation of properties of submodular functions, which provide a combinatorial handle on the network simplification problem in Gaussian HD diamond networks. Finally, this work provides comparisons between the simplification problem for HD M.

Wireless Network Simplification: The Performance of Routing

“…The work in [17] extended the result in [10] for some scenarios of the Gaussian FD diamond network with multiple antennas at the nodes. The network simplification problem have also been studied recently in [18] for Gaussian half-duplex diamond relay networks, where the authors showed we can always select N − 1 relays and approximately achieve N −1 N of the Gaussian half-duplex relay network capacity. As a scheme-specific performance guarantee (as opposed to guaranteeing capacity fractions), the work of [19] proved upper bounds on multiplicative and additive gaps for relay selection based on the amplify-and-forward scheme, primarily for diamond full-duplex networks.…”

Half-duplex routing is NP-hard

“…The above example shows that in general it is suboptimal to find the best HD path by using as optimization metric the FD capacity of the path. In fact, one can show that there exist networks for which routing based on the FD capacities yields a route with HD capacity equal to half that of the best HD route [8]. This observation naturally suggests the question: Does there exist an efficient (polynomial-time) algorithm that finds the route in a network with the best HD capacity?…”