The assignment problem constitutes one of the fundamental problems in the context of linear programming. Besides its theoretical significance, its frequent appearance in the areas of distributed control and facility allocation, where the problemspsila size and the cost for global computation and information can be highly prohibitive, gives rise to the need for local solutions that dynamically assign distinct agents to distinct tasks, while maximizing the total assignment benefit. In this paper, we consider the linear assignment problem in the context of networked systems, where the main challenge is dealing with the lack of global information due to the limited communication capabilities of the agents. We address this challenge by means of a distributed auction algorithm, where the agents are able to bid for the task to which they wish to be assigned. The desired assignment relies on an appropriate selection of bids that determine the prices of the tasks and render them more or less attractive for the agents to bid for. Up to date pricing information, necessary for accurate bidding, can be obtained in a multi-hop fashion by means of local communication between adjacent agents. Our algorithm is an extension to the parallel auction algorithm proposed by Bertsekas et al to the case where only local information is available and it is shown to always converge to an assignment that maximizes the total assignment benefit within a linear approximation of the optimal one.