We study a delay-sensitive information flow problem where a source streams
information to a sink over a directed graph G(V,E) at a fixed rate R possibly
using multiple paths to minimize the maximum end-to-end delay, denoted as the
Min-Max-Delay problem. Transmission over an edge incurs a constant delay within
the capacity. We prove that Min-Max-Delay is weakly NP-complete, and
demonstrate that it becomes strongly NP-complete if we require integer flow
solution. We propose an optimal pseudo-polynomial time algorithm for
Min-Max-Delay, with time complexity O(\log (Nd_{\max}) (N^5d_{\max}^{2.5})(\log
R+N^2d_{\max}\log(N^2d_{\max}))), where N = \max\{|V|,|E|\} and d_{\max} is the
maximum edge delay. Besides, we show that the integrality gap, which is defined
as the ratio of the maximum delay of an optimal integer flow to the maximum
delay of an optimal fractional flow, could be arbitrarily large