On the time complexity of achieving optimal throughput in time division multiple access communication networks

The fundamental problem of finding transmission schedules for achieving optimal throughput in time division multiple access (TDMA) communication networks is known to be NP-hard. Let $\mathcal{N}$ be a scheduled $k$-time slot TDMA network with $n$ stations and $m$ links. We showed that an optimal link schedule for $\mathcal{N}$ can be computed recursively with a recursion tree of logarithmic depth $\mathcal{O}(\ln m)$ in expectation. Additionally, we showed that optimal link schedules for those TDMA networks, with recursion trees of depth meeting the expectation, can be found in time $\mathcal{O}(m^{2+\ln k})$. Likewise, we discuss analogous results for computing optimal station schedules of TDMA networks.