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The aircraft routing problem (ARP) is formulated as a time-dependent network flow problem and proved to be NP-hard, using a reduction from the NP-complete 3-dimensional matching problem (3DM). Flow scheduling and dynamic network theory concepts are used to develop an Integer Program (IP) formulation of the ARP, which can be solved exactly through software such as CPLEX. Linear program (LP) relaxation and rounding techniques are used to solve this formulation in pseudo-polynomial time. A heuristic first-come-first-served (FCFS) is also implemented. Scenarios of routing under congestion and rerouting due to weather show that the FCFS has the fastest run-time but worst performance, while the IP formulation is optimal but has no guarantees on run-time. For a given time horizon, the LP formulation runs in polynomial time and is often optimal, with bounded suboptimality otherwise.