Survivable Directed Supports for Synchronous Learning

We study minimum-cost directed communication supports that retain prescribed coordinator–worker connections after bounded arc failures, subject to a limit of $\ell$ provisioned arcs. For requirements of multiplicities $b_1,\dots ,b_q$, an integral-flow dynamic program stores only boundary imbalances and the number of selected arcs. On a supplied width-$\tau$ decomposition, its running time is $2^{O((\tau +1){\sum }_g\log (2b_g+1))}\text{poly}(N,\ell ,L)$. The bounded state is complete because each requirement admits a witness consisting of simple arc-disjoint paths, even though the algorithm permits circulations. Rooted localization then gives a $2^{O({\ell }^2)}\text{poly}(N,L)$ algorithm on planar and fixed apex-minor-free underlays. On a 40-instance benchmark, the flow DP completes 22 instances, the labelled-path DP completes 20, and MILP completes all 40 under common limits. A separate provisioning study shows how shared resilient supports preserve the prescribed synchronous learning updates under an explicit arc-failure model.