The multi-state hard core model on a regular tree

The classical hard core model from statistical physics, with activity $\lambda > 0$ and capacity $C=1$, on a graph $G$, concerns a probability measure on the set ${\mathcal I}(G)$ of independent sets of $G$, with the measure of each independent set $I \in {\mathcal I}(G)$ being proportional to $\lambda^{|I|}$. Ramanan et al. proposed a generalization of the hard core model as an idealized model of multicasting in communication networks. In this generalization, the {\em multi-state} hard core model, the capacity $C$ is allowed to be a positive integer, and a configuration in the model is an assignment of states from $\{0,\ldots,C\}$ to $V(G)$ (the set of nodes of $G$) subject to the constraint that the states of adjacent nodes may not sum to more than $C$. The activity associated to state $i$ is $\lambda^{i}$, so that the probability of a configuration $\sigma:V(G)\rightarrow \{0,\ldots, C\}$ is proportional to $\lambda^{\sum_{v \in V(G)} \sigma(v)}$. In this work, we consider this generalization when $G$ is an infinite rooted $b$-ary tree and prove rigorously some of the conjectures made by Ramanan et al. In particular, we show that the $C=2$ model exhibits a (first-order) phase transition at a larger value of $\lambda$ than the $C=1$ model exhibits its (second-order) phase transition. In addition, for large $b$ we identify a short interval of values for $\lambda$ above which the model exhibits phase co-existence and below which there is phase uniqueness. For odd $C$, this transition occurs in the region of \lambda = (e/b)^{1/\ceil{C/2}}, while for even $C$, it occurs around $\lambda=(\log b/b(C+2))^{2/(C+2)}$. In the latter case, the transition is first-order.Comment: Will appear in {\em SIAM Journal on Discrete Mathematics}, Special Issue on Constraint Satisfaction Problems and Message Passing Algorithm

Precoder Design for Physical Layer Multicasting

This paper studies the instantaneous rate maximization and the weighted sum delay minimization problems over a K-user multicast channel, where multiple antennas are available at the transmitter as well as at all the receivers. Motivated by the degree of freedom optimality and the simplicity offered by linear precoding schemes, we consider the design of linear precoders using the aforementioned two criteria. We first consider the scenario wherein the linear precoder can be any complex-valued matrix subject to rank and power constraints. We propose cyclic alternating ascent based precoder design algorithms and establish their convergence to respective stationary points. Simulation results reveal that our proposed algorithms considerably outperform known competing solutions. We then consider a scenario in which the linear precoder can be formed by selecting and concatenating precoders from a given finite codebook of precoding matrices, subject to rank and power constraints. We show that under this scenario, the instantaneous rate maximization problem is equivalent to a robust submodular maximization problem which is strongly NP hard. We propose a deterministic approximation algorithm and show that it yields a bicriteria approximation. For the weighted sum delay minimization problem we propose a simple deterministic greedy algorithm, which at each step entails approximately maximizing a submodular set function subject to multiple knapsack constraints, and establish its performance guarantee.Comment: 37 pages, 8 figures, submitted to IEEE Trans. Signal Pro

Coordinated Multicast Beamforming in Multicell Networks

We study physical layer multicasting in multicell networks where each base station, equipped with multiple antennas, transmits a common message using a single beamformer to multiple users in the same cell. We investigate two coordinated beamforming designs: the quality-of-service (QoS) beamforming and the max-min SINR (signal-to-interference-plus-noise ratio) beamforming. The goal of the QoS beamforming is to minimize the total power consumption while guaranteeing that received SINR at each user is above a predetermined threshold. We present a necessary condition for the optimization problem to be feasible. Then, based on the decomposition theory, we propose a novel decentralized algorithm to implement the coordinated beamforming with limited information sharing among different base stations. The algorithm is guaranteed to converge and in most cases it converges to the optimal solution. The max-min SINR (MMS) beamforming is to maximize the minimum received SINR among all users under per-base station power constraints. We show that the MMS problem and a weighted peak-power minimization (WPPM) problem are inverse problems. Based on this inversion relationship, we then propose an efficient algorithm to solve the MMS problem in an approximate manner. Simulation results demonstrate significant advantages of the proposed multicast beamforming algorithms over conventional multicasting schemes.Comment: 10pages, 9 figure

Intelligent Reflecting Surface Aided Multigroup Multicast MISO Communication Systems

Intelligent reflecting surface (IRS) has recently been envisioned to offer unprecedented massive multiple-input multiple-output (MIMO)-like gains by deploying large-scale and low-cost passive reflection elements. By adjusting the reflection coefficients, the IRS can change the phase shifts on the impinging electromagnetic waves so that it can smartly reconfigure the signal propagation environment and enhance the power of the desired received signal or suppress the interference signal. In this paper, we consider downlink multigroup multicast communication systems assisted by an IRS. We aim for maximizing the sum rate of all the multicasting groups by the joint optimization of the precoding matrix at the base station (BS) and the reflection coefficients at the IRS under both the power and unit-modulus constraint. To tackle this non-convex problem, we propose two efficient algorithms. Specifically, a concave lower bound surrogate objective function has been derived firstly, based on which two sets of variables can be updated alternately by solving two corresponding second-order cone programming (SOCP) problems.Then, in order to reduce the computational complexity, we further adopt the majorization—minimization (MM) method for each set of variables at every iteration, and obtain the closed form solutions under loose surrogate objective functions. Finally, the simulation results demonstrate the benefits of the introduced IRS and the effectiveness of our proposed algorithms

Quantum network communication -- the butterfly and beyond

We study the k-pair communication problem for quantum information in networks of quantum channels. We consider the asymptotic rates of high fidelity quantum communication between specific sender-receiver pairs. Four scenarios of classical communication assistance (none, forward, backward, and two-way) are considered. (i) We obtain outer and inner bounds of the achievable rate regions in the most general directed networks. (ii) For two particular networks (including the butterfly network) routing is proved optimal, and the free assisting classical communication can at best be used to modify the directions of quantum channels in the network. Consequently, the achievable rate regions are given by counting edge avoiding paths, and precise achievable rate regions in all four assisting scenarios can be obtained. (iii) Optimality of routing can also be proved in classes of networks. The first class consists of directed unassisted networks in which (1) the receivers are information sinks, (2) the maximum distance from senders to receivers is small, and (3) a certain type of 4-cycles are absent, but without further constraints (such as on the number of communicating and intermediate parties). The second class consists of arbitrary backward-assisted networks with 2 sender-receiver pairs. (iv) Beyond the k-pair communication problem, observations are made on quantum multicasting and a static version of network communication related to the entanglement of assistance.Comment: 15 pages, 17 figures. Final versio