We consider optimal resolution of air traffic (AT) conflicts. Aircraft are assumed to cruise within a given altitude layer and are modeled as a kinematic system with constant velocity and curvature bounds. Aircraft cannot get closer to each other than a predefined safety distance. For such a system of multiple aircraft, we consider the problem of planning optimal paths among given waypoints. Necessary conditions for optimality of solutions are derived and used to devise a parametrization of possible trajectories that turns into efficient numerical solutions to the problem. Simulation results for a realistic aircraft conflict scenario are provided. A decentralized implementation of the optimal conflict resolution scheme is introduced that may allow free-flight coordination in a cooperative airspace management scheme. Impact of decentralization on performance and safety is finally discussed with the help of extensive simulations.