In mobile robot navigation, cost functions are a popular approach to generate feasible, safe paths that avoid obstacles and that allow the robot to get from its starting position to the goal position. Alternative ways to navigate around the obstacles typically correspond to different local minima in the cost function. In this paper we present a highly effective approach to overcome such local minima and to quickly propose a set of alternative, topologically different and optimized paths. We furthermore describe how to maintain a set of optimized trajectory alternatives to reduce optimization efforts when the robot has to adapt to changes in the environment. We demonstrate in experiments that our method outperforms a state-of-the-art approach by an order of magnitude in computation time, which allows a robot to use our method online during navigation. We furthermore demonstrate that the approach of using a set of qualitatively different trajectories is beneficial in shared autonomy settings, where a user operating a wheelchair can quickly switch between topologically different trajectories.