  Coflow scheduling models communication requests in parallel computing
frameworks where multiple data flows between shared resources need to be
completed before computation can continue. In this paper, we introduce
Path-based Coflow Scheduling, a generalized problem variant that considers
coflows as collections of flows along fixed paths on general network topologies
with node capacity restrictions. For this problem, we minimize the coflows'
total weighted completion time. We show that flows on paths in the original
network can be interpreted as hyperedges in a hypergraph and transform the
path-based scheduling problem into an edge scheduling problem on this
hypergraph. We present a $(2\lambda + 1)$-approximation algorithm when node
capacities are set to one, where $\lambda$ is the maximum number of nodes in a
path. For the special case of simultaneous release times for all flows, our
result improves to a $(2\lambda)$-approximation. Furthermore, we generalize the
result to arbitrary node constraints and obtain a $(2\lambda\Delta + 1)$- and a
$(2\lambda\Delta)$-approximation in the case of general and zero release times,
where $\Delta$ captures the capacity disparity between nodes.
