We propose a novel formulation of the collision-aware task assignment (CATA) problem and a decentralized auction-based algorithm to solve the problem with optimality bound. Using a collision cone, we predict potential collisions and introduce a binary decision variable into the local reward function for task bidding. We further improve CATA by implementing a receding collision horizon to address the stopping robot scenario, i.e. when robots are confined to their task location and become static obstacles to other moving robots. The auction-based algorithm encourages the robots to bid for tasks with collision mitigation considerations. We validate the improved task assignment solution with both simulation and experimental results, which show significant reduction of overlapping paths as well as deadlocks.