Upper and lower bounds on the capacity of Gaussian multicast relay channels are shown to be quasi-concave in the receiver signal-to-noise ratios and the transmit correlation coefficient. The bounds considered are the cut-set bound, decode-forward (DF) rates, and quantize-forward rates. The DF rates are shown to be quasi-concave in the relay position and this property is used to optimize the relay position for example networks.

The relay positioning problem is addressed for low signal-to-noise ratio Gaussian multicast relay networks where the objective is to maximize the multicast flow. A network coding based decode-and-forward (DF) strategy is presented that achieves strictly larger rates than a previously-studied routing based DF strategy. Properties of the multicast flow function are analyzed and efficient solutions are presented.

This paper shows how to reduce the otherwise hard joint relay positioning and flow optimization problem into a sequence a two simpler decoupled problems. We consider a class of wireless multicast hypergraphs not limited by interference and are mainly characterized by their hyperarc rate functions, that are increasing and convex in power, and decreasing in distance between the transmit node and the farthest end node of the hyperarc. The set-up consists of a single multicast flow session involving a source, multiple destinations and a relay that can be positioned freely. The first problem formulates the relay positioning problem in a purely geometric sense, and once the optimal relay position is obtained the second problem addresses the flow optimization. Furthermore, simple and efficient algorithms are presented that solve these problems.

We consider the broadcast relay channel (BRC), where a single source transmits to multiple destinations with the help of a relay, in the limit of a large bandwidth. We address the problem of optimal relay positioning and power allocations at source and relay, to maximize the multicast rate from source to all destinations. To solve such a network planning problem, we develop a three-faceted approach based on an underlying information theoretic model, computational geometric aspects, and network optimization tools. Firstly, assuming superposition coding and frequency division between the source and the relay, the information theoretic framework yields a hypergraph model of the wideband BRC, which captures the dependency of achievable rate-tuples on the network topology. As the relay position varies, so does the set of hyperarcs constituting the hypergraph, rendering the combinatorial nature of optimization problem. We show that the convex hull C of all nodes in the 2-D plane can be divided into disjoint regions corresponding to distinct hyperarcs sets. These sets are obtained by superimposing all k-th order Voronoi tessellation of C. We propose an easy and efficient algorithm to compute all hyperarc sets, and prove they are polynomially bounded. Then, we circumvent the combinatorial nature of the problem by introducing continuous switch functions, that allows adapting the network hypergraph in a continuous manner. Using this switched hypergraph approach, we model the original problem as a continuous yet non-convex network optimization program. Ultimately, availing on the techniques of geometric programming and p-norm surrogate approximation, we derive a good convex approximation. We provide a detailed characterization of the problem for collinearly located destinations, and then give a generalization for arbitrarily located destinations. Finally, we show strong gains for the optimal relay positioning compared to seemingly interesting positions.

Abstract-We provide a geometric solution to the problem of optimal relay positioning to maximize the multicast rate for low SNR networks. The network we consider consists of a single source, multiple receivers and the only intermediate and locatable node as the relay. We construct network the hypergraph of the system nodes from the underlying information theoretic model of low-SNR regime that operates using superposition coding and FDMA in conjunction (which we call the "achievable hypergraph model"). We make the following contributions. 1) We show that the problem of optimal relay positioning maximizing the multicast rate can be completely decoupled from the flow optimization by noticing and exploiting geometric properties of multicast flow. 2) All the flow maximizing the multicast rate is sent over at most two paths, in succession. The relay position depends on only one path (out of the two), irrespective of the number of receiver nodes in the system. Subsequently, we propose simple and efficient geometric algorithms to compute the optimal relay position. 3) Finally, we show that in our model at the optimal relay position, the difference between the maximized multicast rate and the cut-set bound is minimum. We solve the problem for all (Ps,Pr) pairs of source and relay transmit powers and the path loss exponent α ≥ 2.

The rate optimization for wireless networks with low SNR is investigated. While the capacity in the limit of disappearing SNR is known to be linear for fading and non-fading channels, we study the problem of operating in low SNR wireless network with given node locations that use network coding over flows. The model we develop for low SNR Gaussian broadcast channel and multiple access channel respectively operates in a non-trivial feasible rate region. We show that the problem reduces to the optimization of total network power which can be casted as standard linear multi-commodity min-cost flow program with no inherent combinatorially difficult structure when network coding is used with non integer constraints (which is a reasonable assumption). This is essentially due to the linearity of the capacity with respect to vanishing SNR which helps avoid the effect of interference for the degraded broadcast channel and multiple access environment in consideration, respectively. We propose a fully decentralized Primal-Dual Subgradient Algorithm for achieving optimal rates on each subgraph (i.e. hyperarcs) of the network to support the set of traffic demands (multicast/unicast connections).

Many networks today are composed of numerous subnetworks or domains. The connection points of these domains are important and vulnerable parts of the communication. Therefore, one of the key points of protection schemes today is being able to protect these interconnections. Hence, in this type of networks a protection mechanism is applied, where there is 1+1 protection inside each domain and dual homing at the borders. With this mechanism, the breakdown due to the failure of the interconnection elements is prohibited. This system enables the network to survive any node or link failure in each domain. However, it suffers from the high capacity need on the border links, where data has to be transmitted bidirectionally for each single demand. We are showing that combining network coding with this protection mechanism gives a realization, which outperforms the existing mechanism by providing high resource savings. We present the theoretical basis and the application methodology of the proposed method.

We study the minimum cut between one source and one terminal in a dynamically changing random wireless ad hoc network that is modeled as a random geometric graph. The nature of ad hoc networks is accounted for by letting nodes join and leave. Given the values of all cuts that can be formed in the original network and assuming information about the nodes that join or leave, expressions for the expected value and variance of any particular cut that may arise are derived. However, it is not possible to obtain a closed form expression for the minimum expected cut value in our framework. Nevertheless, we give a simple and reasonable upper bound for the minimum expected cut value which is also extendable to multicast transmissions.