  We consider dynamic equilibria for flows over time under the fluid queuing
model. In this model, queues on the links of a network take care of flow
propagation. Flow enters the network at a single source and leaves at a single
sink. In a dynamic equilibrium, every infinitesimally small flow particle
reaches the sink as early as possible given the pattern of the rest of the
flow. While this model has been examined for many decades, progress has been
relatively recent. In particular, the derivatives of dynamic equilibria have
been characterized as thin flows with resetting, which allowed for more
structural results. Our two main results are based on the formulation of thin
flows with resetting as linear complementarity problem and its analysis. We
present a constructive proof of existence for dynamic equilibria if the inflow
rate is right-monotone. The complexity of computing thin flows with resetting,
which occurs as a subproblem in this method, is still open. We settle it for
the class of two-terminal series-parallel networks by giving a recursive
algorithm that solves the problem for all flow values simultaneously in
polynomial time.
