We develop an algorithm to solve the bottleneck assignment problem (BAP) that is amenable to having computation distributed over a network of agents. This consists of exploring how each component of the algorithm can be distributed, with a focus on one component in particular, i.e., the function to search for an augmenting path. An augmenting path is a common tool used in most BAP algorithms and poses a particular challenge for this distributed approach. Given this significance, we compare the properties of two different methods to search for an augmenting path in a bipartite graph. We evaluate the derived approaches with a simulation-based complexity investigation.