A new combinatorial dynamic traffic assignment (CDTA) algorithm for multi-destination transportation networks is developed. The algorithm, stated on a discrete space-time network, uses the cell transmission model (CTM) to propagate traffic, thereby ensuring that traffic dynamics such as queue evolution, link spill-over, and shockwave propagation are adequately captured. The CDTA algorithm assigns vehicles to optimal time-dependent shortest paths in one shot by finding connectivity between origin-destination pairs in the time-expanded CTM network. The CDTA algorithm runs in polynomial time and is guaranteed to find a user optimal assignment in single-destination networks. However, vehicular trajectories could potentially violate first-in first-out (FIFO) condition in a multi-destination network (thereby yielding an optimal though infeasible solution). FIFO flows are achieved by simulating the vehicular trajectories using a simulation-based DTA model. These flows in turn serve as an initial feasible DTA solution - this method is called "warm starting" a simulation-based DTA model. The algorithm has been tested for the Anaheim and Winnipeg networks for varying demand levels. The warm started DTA models performed better than the non-warm started models in terms of equilibrium convergence metrics. In particular, for solutions involving small path sets, the DTA model warm started using the CDTA algorithm provided better solutions than the purely simulation-based model.