This paper improves a previous result on the multi-agent assignment problem, in which a group of agents has to reach a consensus on an optimal distribution of tasks, under communication and assignment constraints. Indeed, the drawbacks of the previously proposed algorithm are that it must start from an initial feasible assignment state and that an analysis on the goodness of the reached consensus is not given. In this paper, starting from any unfeasible solution, we develop an extended version of the previous gossip algorithm able to iteratively find an initial feasible assignment state in order to reach the consensus state among the agents. Moreover, a study is performed in order to evaluate how the solution obtained by the discrete consensus algorithm is close to the optimal distribution of tasks. Finally, an example shows the application of the distributed discrete consensus algorithm.
