Connectedness in weighted consensus division of graphical cakes between two agents (2312.12260v2)

Abstract: Austin's moving knife procedure was originally introduced to find a consensus division of a cake between two agents: each agent believes that they receive exactly half of the cake. Here, we adapt it to the case when the two agents have arbitrary entitlements of the cake and seek a weighted consensus division -- one where each agent believes that they received exactly the share that they are entitled to -- which also minimizes the number of connected components that each agent receives. First, we find a weighted consensus division of a circular cake, which gives exactly one connected piece to each agent: this gives an explicit protocol for the Stromquist-Woodall theorem for two agents. Next, by judiciously mapping a circle to a graph, we produce a weighted consensus division of a star graph cake that gives at most two connected pieces to each agent -- and show that this bound on the number of connected pieces is tight. For trees, each agent receives at most h+1 connected pieces, where h is the height of the tree, and for general (connected) graphical cakes each agent receives r+2 connected pieces, where r is the radius of the graph.