We consider the question of determining the scaling of the $n^2$-dimensional balanced unicast and the $n 2^n$-dimensional balanced multicast capacity regions of a wireless network with $n$ nodes placed uniformly at random in a square region of area $n$ and communicating over Gaussian fading channels. We identify this scaling of both the balanced unicast and multicast capacity regions in terms of $Θ(n)$, out of $2^n$ total possible, cuts. These cuts only depend on the geometry of the locations of the source nodes and their destination nodes and the traffic demands between them, and thus can be readily evaluated. Our results are constructive and provide optimal (in the scaling sense) communication schemes.
This paper considers the application of non-orthogonal multiple access (NOMA) to a multi-user network with mixed multicasting and unicasting traffic. The proposed design of beamforming and power allocation ensures that the unicasting performance is improved while maintaining the reception reliability of multicasting. Both analytical and simulation results are provided to demonstrate that the use of the NOMA assisted multicast-unicast scheme yields a significant improvement in spectral efficiency compared to orthogonal multiple access (OMA) schemes which realize multicasting and unicasting serv…
This paper considers a downlink single-cell non-orthogonal multiple access (NOMA) network with uniformly deployed users, while a mixed multicast and unicast traffic scenario is taken into account. By guaranteeing the quality of service (QoS) of the multicast traffic, the multicast outage and secrecy unicast throughput performance is evaluated. In particular, two types of partial channel state information (CSI), namely the imperfect CSI and CSI based on second order statistics (SOS), are investigated in the analytical framework. For both two cases, the closed-form approximations for the multica…
The demand for providing multicast services in cellular networks is continuously and fastly increasing. In this work, we propose a non-orthogonal transmission framework based on layered-division multiplexing (LDM) to support multicast and unicast services concurrently in cooperative multi-cell cellular networks with limited backhaul capacity. We adopt a two-layer LDM structure where the first layer is intended for multicast services, the second layer is for unicast services, and the two layers are superposed with different beamformers. Each user decodes the multicast message first, subtracts i…
Many opportunistic scheduling techniques are impractical because they require accurate channel state information (CSI) at the transmitter. In this paper, we investigate the scheduling of unicast and multicast services in a downlink network with a very limited amount of feedback information. Specifically, unicast users send imperfect (or no) CSI and infrequent acknowledgements (ACKs) to a base station, and multicast users only report infrequent ACKs to avoid feedback implosion. We consider the use of physical-layer rateless codes, which not only combats channel uncertainty, but also reduces the…
In a superimposed unicast and multicast transmission system, one layer of Successive Interference Cancellation (SIC) is required at each receiver to remove the multicast stream before decoding the unicast stream. In this paper, we show that a linearly-precoded Rate-Splitting (RS) strategy at the transmitter can efficiently exploit this existing SIC receiver architecture. By splitting the unicast message into common and private parts and encoding the common parts along with the multicast message into a super-common stream decoded by all users, the SIC is used for the dual purpose of separating …
This paper studies energy-efficient joint transmit and receive beamforming in multi-cell multi-user multiple-input multiple-output systems. We consider conventional network energy efficiency metric where the users can receive unicasting streams in addition to the group-specific common multicasting streams which have certain rate constraints. The goal is to use the transmission resources more efficiently to improve the energy efficiency, when the users are equipped with multiple antennas. Numerical results show the achieved energy efficiency gains by using the additional degrees of freedom of t…
In this paper, we investigate a hybrid multicast/ unicast scheme for a multiple-input single-output cache-aided non-orthogonal multiple access (NOMA) vehicular scenario in the face of rapidly fluctuating vehicular wireless channels. Considering a more practical situation, imperfect channel state information is taking into account. In this paper, we formulate an optimization problem to maximize the unicast sum rate under the constraints of the peak power, the peak backhaul, the minimum unicast rate, and the maximum multicast outage probability. To solve the formulated non-convex problem, a lowe…
In this paper, we consider a joint unicast and multi-group multicast cell-free distributed massive multiple-input multiple-output (MIMO) system, while accounting for co-pilot assignment strategy based channel estimation, pilot contamination and different precoding schemes. Under the co-pilot assignment strategy, we derive the minimum-mean-square error (MMSE) channel state information (CSI) estimation for unicast and multicast users. Given the acquired CSI, the closed-form expressions for downlink achievable rates with maximum ratio transmission (MRT), zero-forcing (ZF) and MMSE beamforming are…
A Non-Orthogonal Unicast and Multicast (NOUM) transmission system allows a multicast stream intended to all receivers to be jointly transmitted with unicast streams in the same time-frequency resource blocks. While the capacity of the two-user multi-antenna NOUM with perfect Channel State Information at the Transmitter (CSIT) is known and achieved by Dirty Paper Coding (DPC)-assisted NOUM with Superposition Coding (SC), the capacity and the capacity-achieving strategy of the multi-antenna NOUM with partial CSIT remain unknown. In this work, we focus on the partial CSIT setting and make two maj…
We study joint unicast and multigroup multicast transmission in single-cell massive multiple-input-multiple-output (MIMO) systems, under maximum ratio transmission. For the unicast transmission, the objective is to maximize the weighted sum spectral efficiency (SE) of the unicast user terminals (UTs) and for the multicast transmission the objective is to maximize the minimum SE of the multicast UTs. These two problems are coupled to each other in a conflicting manner, due to their shared power resource and interference. To address this, we formulate a multiobjective optimization problem (MOOP)…
In a Non-Orthogonal Unicast and Multicast (NOUM) transmission system, a multicast stream intended to all the receivers is superimposed in the power domain on the unicast streams. One layer of Successive Interference Cancellation (SIC) is required at each receiver to remove the multicast stream before decoding its intended unicast stream. In this paper, we first show that a linearly-precoded 1-layer Rate-Splitting (RS) strategy at the transmitter can efficiently exploit this existing SIC receiver architecture. We further propose multi-layer transmission strategies based on the generalized RS an…
We study optimal delivery strategies of one common and $K$ independent messages from a source to multiple users in wireless environments. In particular, two-layered superposition of broadcast/multicast and unicast signals is considered in a downlink multiuser OFDMA system. In the literature and industry, the two-layer superposition is often considered as a pragmatic approach to make a compromise between the simple but suboptimal orthogonal multiplexing (OM) and the optimal but complex fully-layered non-orthogonal multiplexing. In this work, we show that only two-layers are necessary to achieve…
We study the joint unicast and multi-group multicast transmission in massive multiple-input-multiple-output (MIMO) systems. We consider a system model that accounts for channel estimation and pilot contamination, and derive achievable spectral efficiencies (SEs) for unicast and multicast user terminals (UTs), under maximum ratio transmission and zero-forcing precoding. For unicast transmission, our objective is to maximize the weighted sum SE of the unicast UTs, and for the multicast transmission, our objective is to maximize the minimum SE of the multicast UTs. These two objectives are couple…
This work investigates the energy-efficient resource allocation for layered-division multiplexing (LDM) based non-orthogonal multicast and unicast transmission in cell-free massive multiple-input multiple-output (MIMO) systems, where each user equipment (UE) performs wireless information and power transfer simultaneously. To begin with, the achievable data rates for multicast and unicast services are derived in closed form, as well as the received radio frequency (RF) power at each UE. Based on the analytical results, a nonsmooth and nonconvex optimization problem for energy efficiency (EE) ma…
The novel concept of non-orthogonal multiple access (NOMA) aided joint radar and multicast-unicast communication (Rad-MU-Com) is investigated. Employing the same spectrum resource, a multi-input-multi-output (MIMO) dual-functional radar-communication (DFRC) base station detects the radar-centric user (R-user), while transmitting mixed multicast-unicast messages both to the R-user and to the communication-centric user (C-user). In particular, the multicast information is intended for both the R- and C-users, whereas the unicast information is only intended for the C-user. More explicitly, NOMA …
This paper investigates dual-layer non-orthogonally superimposed transmissions for industrial internet of things (IoT) millimeter-wave communications. Essentially, the overlayer is a ubiquitous multicast signal devised to serve all the devices in coverage with a common message, i.e., critical control packet. The underlayer is a composite signal that consists of private unicast messages. Due to safety implications, it is critical that all devices can decode the multicast information. To ensure this requirement, we jointly optimize the hybrid precoder, analog combiners, power allocation, and fai…
This paper investigates rate-splitting (RS) precoding for non-orthogonal unicast and multicast (NOUM) transmissions using fully-digital and hybrid precoders. We study the nonconvex weighted sum-rate (WSR) maximization problem subject to a multicast requirement. We propose FALCON, an approach based on sequential parametric optimization, to solve the aforementioned problem. We show that FALCON converges to a local optimum without requiring judicious selection of an initial feasible point. Besides, we show through simulations that by leveraging RS, hybrid precoders can attain nearly the same perf…
The novel concept of non-orthogonal multiple access (NOMA) aided joint radar and multicast-unicast communication (Rad-MU-Com) is investigated. Employing the same spectrum resource, a multi-input-multi-output (MIMO) dual-functional radar-communication (DFRC) base station detects the radar-centric users (R-user), while transmitting mixed multicast-unicast messages both to the R-user and to the communication-centric user (C-user). In particular, the multicast information is intended for both the R- and C-users, whereas the unicast information is only intended for the C-user. More explicitly, NOMA…
Intelligent reflecting surfaces (IRSs) have received much attention recently and are envisioned to promote 6G communication networks. In this paper, for wireless communications aided by IRS units, we formulate optimization problems for power control under quality of service (QoS) and max-min fair QoS under three kinds of traffic patterns from a base station (BS) to mobile users (MUs): unicast, broadcast, and multicast. The optimizations are achieved by jointly designing the transmit beamforming of the BS and the phase shift matrix of the IRS. For power control under QoS, existing IRS studies i…
In this paper, we investigate the energy-efficient hybrid precoding design for integrated multicast-unicast millimeter wave (mmWave) system, where the simultaneous wireless information and power transform is considered at receivers. We adopt two sparse radio frequency chain antenna structures at the base station (BS), i.e., fully-connected and subarray structures, and design the codebook-based analog precoding according to the different structures. Then, we formulate a joint digital multicast, unicast precoding and power splitting ratio optimization problem to maximize the energy efficiency of…
This paper considers a wireless communication network consisting of multiple interfering multicast sessions. Different from a unicast system where each transmitter has only one receiver, in a multicast system, each transmitter has multiple receivers. It is a well known result for wireless unicast systems that the feasibility of an signal-to-interference-plus-noise power ratio (SINR) without power constraint is decided by the Perron-Frobenius eigenvalue of a nonnegative matrix. We generalize this result and propose necessary and sufficient conditions for the feasibility of an SINR in a wireless…
It is common in the study of secure multicast network coding in the presence of an eavesdropper that has access to $z$ network links, to assume that the source node is the only node that generates random keys. In this setting, the secure multicast rate is well understood. Computing the secure multicast rate, or even the secure unicast rate, in the more general setting in which all network nodes may generate (independent) random keys is known to be as difficult as computing the (non-secure) capacity of multiple-unicast network coding instances --- a well known open problem. This work treats an …
A new coding scheme for general N-node relay networks is presented for unicast, multicast, and broadcast. The proposed distributed decode-forward scheme combines and generalizes Marton coding for single-hop broadcast channels and the Cover-El Gamal partial decode-forward coding scheme for 3-node relay channels. The key idea of the scheme is to precode all the codewords of the entire network at the source by multicoding over multiple blocks. This encoding step allows these codewords to carry partial information of the messages implicitly without complicated rate splitting and routing. This part…
A framework is developed for analyzing capacity gains from user cooperation in slow fading wireless networks when the number of nodes (network size) is large. The framework is illustrated for the case of a simple multipath-rich Rayleigh fading channel model. Both unicasting (one source and one destination) and multicasting (one source and several destinations) scenarios are considered. We introduce a meaningful notion of Shannon capacity for such systems, evaluate this capacity as a function of signal-to-noise ratio (SNR), and develop a simple two-phase cooperative network protocol that achiev…
We propose a new way to construct a multicast coding scheme for linear deterministic relay networks. Our construction can be regarded as a generalization of the well-known multicast network coding scheme of Jaggi et al. to linear deterministic relay networks and is based on the notion of flow for a unicast session that was introduced by the authors in earlier work. We present randomized and deterministic polynomial--time versions of our algorithm and show that for a network with $g$ destinations, our deterministic algorithm can achieve the capacity in $\left\lceil \log(g+1)\right\rceil $ uses …
We consider globally optimal precoder design for rate splitting multiple access in Gaussian multiple-input single-output downlink channels with respect to weighted sum rate and energy efficiency maximization. The proposed algorithm solves an instance of the joint multicast and unicast beamforming problem and includes multicast- and unicast-only beamforming as special cases. Numerical results show that it outperforms state-of-the-art algorithms in terms of numerical stability and converges almost twice as fast.
Packet-switching WDM ring networks with a hotspot transporting unicast, multicast, and broadcast traffic are important components of high-speed metropolitan area networks. For an arbitrary multicast fanout traffic model with uniform, hotspot destination, and hotspot source packet traffic, we analyze the maximum achievable long-run average packet throughput, which we refer to as \textit{multicast capacity}, of bi-directional shortest-path routed WDM rings. We identify three segments that can experience the maximum utilization, and thus, limit the multicast capacity. We characterize the segment …
Multicast transmission has several distinctive traits as opposed to more commonly studied unicast networks. Specially, these include (i) identical packets must be delivered successfully to several nodes, (ii) outage could simultaneously happen at different receivers, and (iii) the multicast rate is dominated by the receiver with the weakest link in order to minimize outage and retransmission. To capture these key traits, we utilize a Poisson cluster process consisting of a distinct Poisson point process (PPP) for the transmitters and receivers, and then define the multicast transmission capaci…
We consider a hybrid delivery scheme for streaming content, combining cache-enabled Orthogonal Multipoint Multicast (OMPMC) and on-demand Single-Point Unicast (SPUC) transmissions for heterogeneous networks. The OMPMC service transmits cached files through the whole network to interested users, and users not being satisfied by this service are assigned to the SPUC service to be individually served. The SPUC fetches the requested files from the core network and unicasts them to UEs using cellular beamforming transmissions. We optimize the delivery scheme to minimize the average resource consump…
This work studies the advantages of coded multicasting for the downlink of a Fog Radio Access Network (F-RAN) system equipped with a multicast fronthaul link. In this system, a control unit (CU) in the baseband processing unit (BBU) pool is connected to distributed edge nodes (ENs) through a multicast fronthaul link of finite capacity, and the ENs have baseband processing and caching capabilities. Each user equipment (UE) requests a file in a content library which is available at the CU, and the requested files are served by the closest ENs based on the cached contents and on the information r…
Immersive video, such as virtual reality (VR) and multi-view videos, is growing in popularity. Its wireless streaming is an instance of general multicast, extending conventional unicast and multicast, whose effective design is still open. This paper investigates the optimization of general rate splitting with linear beamforming for general multicast. Specifically, we consider a multi-carrier single-cell wireless network where a multi-antenna base station (BS) communicates to multiple single-antenna users via general multicast. Linear beamforming is adopted at the BS, and joint decoding is adop…
We consider the multiple unicast problem under network coding over directed acyclic networks when there are two source-terminal pairs, $s_1-t_1$ and $s_2-t_2$. Current characterizations of the multiple unicast capacity region in this setting have a large number of inequalities, which makes them hard to explicitly evaluate. In this work we consider a slightly different problem. We assume that we only know certain minimum cut values for the network, e.g., mincut$(S_i, T_j)$, where $S_i \subseteq \{s_1, s_2\}$ and $T_j \subseteq \{t_1, t_2\}$ for different subsets $S_i$ and $T_j$. Based on these …
Limited bandwidth resources and higher energy efficiency requirements motivate incorporating multicast and broadcast transmission into the next-generation cellular network architectures, particularly for multimedia streaming applications. Layered division multiplexing (LDM), a form of NOMA, can potentially improve unicast throughput and broadcast coverage with respect to traditional orthogonal frequency division multiplexing (FDM) or time division multiplexing (TDM), by simultaneously using the same frequency and time resources for multiple unicast or broadcast transmissions. In this paper, th…
Next generation of cellular networks deploying wireless distributed femtocaching infrastructure proposed by Golrezaei et. al. are studied. By taking advantage of multihop communications in each cell, the number of required femtocaching helpers is significantly reduced. This reduction of femtocaches is achieved by using the underutilized storage and communication capabilities in the User Terminals (UTs), which results in reducing the deployment costs of distributed femtocaches. A multihop index coding technique is proposed to code the cached contents in helpers to achieve order optimal capacity…
Multicast transmission, wherein the same packet must be delivered to multiple receivers, is an important aspect of sensor and tactical networks and has several distinctive traits as opposed to more commonly studied unicast networks. Specially, these include (i) identical packets must be delivered successfully to several nodes, (ii) outage at any receiver requires the packet to be retransmitted at least to that receiver, and (iii) the multicast rate is dominated by the receiver with the weakest link in order to minimize outage and retransmission. A first contribution of this paper is the develo…
Barrage relays networks (BRNs) are ad hoc networks built on a rapid cooperative flooding primitive as opposed to the traditional point-to-point link abstraction. Controlled barrage regions (CBRs) can be used to contain this flooding primitive for unicast and multicast, thereby enabling spatial reuse. In this paper, the behavior of individual CBRs is described as a Markov process that models the potential cooperative relay transmissions. The outage probability for a CBR is found in closed form for a given topology, and the probability takes into account fading and co-channel interference (CCI) …
This paper investigates the problem of secure communication in a wireline noiseless scenario where a source wishes to communicate to a number of destinations in the presence of a passive external adversary. Different from the multicast scenario, where all destinations are interested in receiving the same message, in this setting different destinations are interested in different messages. The main focus of this paper is on characterizing the secure capacity region, when the adversary has unbounded computational capabilities, but limited network presence. First, an outer bound on the secure cap…
We consider the problem of establishing minimum-cost multicast connections over coded packet networks, i.e. packet networks where the contents of outgoing packets are arbitrary, causal functions of the contents of received packets. We consider both wireline and wireless packet networks as well as both static multicast (where membership of the multicast group remains constant for the duration of the connection) and dynamic multicast (where membership of the multicast group changes in time, with nodes joining and leaving the group). For static multicast, we reduce the problem to a polynomial-tim…
In this paper, we introduce novel coding schemes for wireless networks with random transmission delays. These coding schemes obviate the need for synchronicity, reduce the number of transmissions and achieve the optimal rate region in the corresponding wired model for both multiple unicast and multicast cases with up to three users under the equal rate constraint. The coding schemes are presented in two phases; first, coding schemes for line, star and line-star topologies with random transmission delays are provided. Second, any general topology with multiple bidirectional unicast and multicas…
In this paper, we investigate the performance of LTE Multicast-Broadcast Single-Frequency Networks (MBSFN). LTE-MBSFN is viewed as one of the most promising candidates for vehicular communications which can enhance reliability of vehicular application traffic. This is achieved due to the possibility to efficiently support message exchange in-between vehicles by multicasting information to several vehicles in parallel (point-to-multipoint transmission) employing an Multimedia Broadcast/Multicast Service (MBMS). We investigate two metrics to gauge the performance of MBMS/MBSFN transmissions in c…
Not only is network coding essential to achieve the capacity of a single-session multicast network, it can also help to improve the throughput of wireless networks with multiple unicast sessions when overheard information is available. Most previous research aimed at realizing such improvement by using perfectly overheard information, while in practice, especially for wireless networks, overheard information is often imperfect. To date, it is unclear whether network coding should still be used in such situations with imperfect overhearing. In this paper, a simple but ubiquitous wireless networ…
We consider wireless device-to-device (D2D) caching networks with single-hop transmissions. Previous work has demonstrated that caching and coded multicasting can significantly increase per user throughput. However, the state-of-the-art coded caching schemes for D2D networks are generally impractical because content files are partitioned into an exponential number of packets with respect to the number of users if both library and memory sizes are fixed. In this paper, we present two combinatorial approaches of D2D coded caching network design with reduced packetizations and desired throughput …
The problem of serving multicast flows in a crossbar switch is considered. Intra-flow linear network coding is shown to achieve a larger rate region than the case without coding. A traffic pattern is presented which is achievable with coding but requires a switch speedup when coding is not allowed. The rate region with coding can be characterized in a simple graph-theoretic manner, in terms of the stable set polytope of the "enhanced conflict graph". No such graph-theoretic characterization is known for the case of fanout splitting without coding. The minimum speedup needed to achieve 100% thr…
The capacity of multiuser networks has been a long-standing problem in information theory. Recently, Avestimehr et al. have proposed a deterministic network model to approximate multiuser wireless networks. This model, known as the ADT network model, takes into account the broadcast nature of wireless medium and interference. We show that the ADT network model can be described within the algebraic network coding framework introduced by Koetter and Medard. We prove that the ADT network problem can be captured by a single matrix, and show that the min-cut of an ADT network is the rank of this ma…
We present a graph theoretic upper bound on speedup needed to achieve 100% throughput in a multicast switch using network coding. By bounding speedup, we show the equivalence between network coding and speedup in multicast switches - i.e. network coding, which is usually implemented using software, can in many cases substitute speedup, which is often achieved by adding extra switch fabrics. This bound is based on an approach to network coding problems called the "enhanced conflict graph". We show that the "imperfection ratio" of the enhanced conflict graph gives an upper bound on speedup. In p…
We consider arbitrary dense wireless networks, in which $n$ nodes are placed in an arbitrary (deterministic) manner on a square region of unit area and communicate with each other over Gaussian fading channels. We provide inner and outer bounds for the $n\times n$-dimensional unicast and the $n\times 2^n$-dimensional multicast capacity regions of such a wireless network. These inner and outer bounds differ only by a factor $O(\log(n))$, yielding a fairly tight scaling characterization of the entire regions. The communication schemes achieving the inner bounds use interference alignment as a ce…
An intuitive overview of the scalability of a variety of types of wireless networks is presented. Simple heuris- tic arguments are demonstrated here for scaling laws presented in other works, as well as for conditions not previously considered in the literature. Unicast and multicast messages, topology, hierarchy, and effects of reliability protocols are discussed. We show how two key factors, bottlenecks and erasures, can often domi- nate the network scaling behavior. Scaling of through- put or delay with the number of transmitting nodes, the number of receiving nodes, and the file size is de…
The deterministic wireless relay network model, introduced by Avestimehr et al., has been proposed for approximating Gaussian relay networks. This model, known as the ADT network model, takes into account the broadcast nature of wireless medium and interference. Avestimehr et al. showed that the Min-cut Max-flow theorem holds in the ADT network. In this paper, we show that the ADT network model can be described within the algebraic network coding framework introduced by Koetter and Medard. We prove that the ADT network problem can be captured by a single matrix, called the "system matrix". We …
The study of feedback has been mostly limited to single-hop communication settings. In this paper, we consider Gaussian networks where sources and destinations can communicate with the help of intermediate relays over multiple hops. We assume that links in the network can be bidirected providing opportunities for feedback. We ask the following question: can the information transfer in both directions of a link be critical to maximizing the end-to-end communication rates in the network? Equivalently, could one of the directions in each bidirected link (and more generally at least one of the lin…
We consider the problem of energy-efficient transmission in delay constrained cooperative multihop wireless networks. The combinatorial nature of cooperative multihop schemes makes it difficult to design efficient polynomial-time algorithms for deciding which nodes should take part in cooperation, and when and with what power they should transmit. In this work, we tackle this problem in memoryless networks with or without delay constraints, i.e., quality of service guarantee. We analyze a wide class of setups, including unicast, multicast, and broadcast, and two main cooperative approaches, na…
Recently, an evolution of the Cloud Radio Access Network (C-RAN) has been proposed, named as Fog Radio Access Network (F-RAN). Compared to C-RAN, the Radio Units (RUs) in F-CAN are equipped with local caches, which can store some frequently requested files. In the downlink, users requesting the same file form a multicast group, and are cooperatively served by a cluster of RUs. The requested file is either available locally in the cache of this cluster or fetched from the Central Processor (CP) via backhauls. Thus caching some frequently requested files can greatly reduce the burden on backhaul…
Recently, the Degrees-of-Freedom (DoF) region of multiple-input-single-output (MISO) networks with imperfect channel state information at the transmitter (CSIT) has attracted significant attentions. An achievable scheme is known as rate-splitting (RS) that integrates common-message-multicasting and private-message-unicasting. In this paper, focusing on the general $K$-cell MISO IC where the CSIT of each interference link has an arbitrary quality of imperfectness, we firstly identify the DoF region achieved by RS. Secondly, we introduce a novel scheme, so called Topological RS (TRS), whose nove…
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 flo…
