Distributed Consensus with Finite Message Passing

Inspired by distributed resource allocation problems in dynamic topology networks, we initiate the study of distributed consensus with finite messaging passing. We first find a sufficient condition on the network graph for which no distributed protocol can guarantee a conflict-free allocation after $R$ rounds of message passing. Secondly we fully characterize the conflict minimizing zero-round protocol for path graphs, namely random allocation, which partitions the graph into small conflict groups. Thirdly, we enumerate all one-round protocols for path graphs and show that the best one further partitions each of the smaller groups. Finally, we show that the number of conflicts decrease to zero as the number of available resources increase.