14 problems

Simple-schedule conjecture. Simple schedules suffice for approximate capacity characterization of Gaussian half-duplex relay networks.

A Gaussian half-duplex diamond network has a source, a destination, no direct source–destination link, and $N$ non-interfering half-duplex relays. A relay scheduling policy is appr…

Private-message extension conjecture. If the capacity of the $K$-user relay network can be found for any $K$, then the capacity of the $K$-user relay network in which the relay exc…

Let the optimization scheme be the coding scheme described above for a multiple-link relay network, and let its achieved rate be compared with the rates achieved by the baseline sc…

General DoF upper-bound conjecture. The total degrees of freedom satisfies

Operational-significance conjecture. As in the corresponding point-to-point MIMO results, link adaptation and HARQ will reduce or eliminate the need to use degrees of freedom for d…

Consider any Gaussian half-duplex multi-relay network, with an arbitrary network topology. A relay scheduling policy is simple if it has at most $N+1$ active states, where an activ…

Consider a half-duplex Gaussian multi-relay network. A relay schedule is simple if at most $N+1$ of the $2^N$ possible relay listen-transmit configurations have strictly positive p…

Consider the $K$-node relay network with relay messages, consisting of $K-1$ users that exchange messages through the $K$-th node, which also transmits private messages to each use…

Active-state conjecture. For a general HD relay network with $N$ relays, there always exists an optimal schedule that maximizes the gDoF with at most $N+1$ active states.

Consider a general half-duplex diamond network with $K-2$ relays. For each relay state $s\in[0:2^{K-2}-1]$, let $\lambda_s^\star$ be its optimal probability in the cut-set upper bo…

Consider a general half-duplex diamond network with $K-2$ relays and state probabilities $\lambda_s$, indexed by $s\in[0:2^{K-2}-1]$. Let $\lambda_s^\star$ be optimal state probabi…

Let $R(d_1,d_2)$ denote the rate-distortion function for lossy broadcasting in a two-way relay network, and let $R_L(d_1,d_2)$ be its cut-set lower bound, with distortion pair…

Let ${\bf D_1}$ and ${\bf D_2}$ denote the relay-gain matrices in the three-hop relay-network optimization algorithm described above. The algorithm alternately optimizes the relay…