Balanced and Fair Partitioning of Friends

Abstract: In the recently introduced model of fair partitioning of friends, there is a set of agents located on the vertices of an underlying graph that indicates the friendships between the agents. The task is to partition the graph into $k$ balanced-sized groups, keeping in mind that the value of an agent for a group equals the number of edges they have in that group. The goal is to construct partitions that are "fair", i.e., no agent would like to replace an agent in a different group. We generalize the standard model by considering utilities for the agents that are beyond binary and additive. Having this as our foundation, our contribution is threefold (a) we adapt several fairness notions that have been developed in the fair division literature to our setting; (b) we give several existence guarantees supported by polynomial-time algorithms; (c) we initiate the study of the computational (and parameterized) complexity of the model and provide an almost complete landscape of the (in)tractability frontier for our fairness concepts.