Cooperative path-finding can be abstracted as computing non-colliding paths for multiple agents between their start and goal locations on a graph. This paper proposes a fast algorithm that can provide completeness guarantees for a general class of problems without any assumptions about the graph's topology. Specifically, the approach can address any solvable instance where there are at most $n$-2 agents in a graph of size $n$. The algorithm employs two primitives: a ``push'' operation where agents move towards their goals up to the point that no progress can be made, and a ``swap'' operation that allows two agents to swap positions without altering the configuration of other agents. Simulated experiments are provided on hard instances of cooperative path-finding, including comparisons against alternative methods. The results are favorable for the proposed algorithm and show that the technique scales to problems that require high levels of coordination, involving hundreds of agents.