EFX Allocations and Orientations on Bipartite Multi-graphs: A Complete Picture

We consider the fundamental problem of fairly allocating a set of indivisible items among agents having valuations that are represented by a multi-graph -- here, agents appear as the vertices and items as the edges between them and each vertex (agent) only values the set of its incident edges (items). The goal is to find a fair, i.e., envy-free up to any item (EFX) allocation. This model has recently been introduced by Christodoulou et al. (EC'23) where they show that EFX allocations always exist on simple graphs for monotone valuations, i.e., where any two agents can share at most one edge (item). A natural question arises as to what happens when we go beyond simple graphs and study various classes of multi-graphs? We answer the above question affirmatively for the valuation class of bipartite multi-graphs and multi-cycles. Our main positive result is that EFX allocations on bipartite multi-graphs (and multi-cycles) always exist and can be computed in polynomial time for additive valuations. We, therefore, push the frontiers of our understanding of EFX allocations and expand the scenarios where they are known to exist for an arbitrary number of agents. Next, we study EFX orientations (i.e., allocations where every item is allocated to one of its two endpoint agents) and give a complete picture of when they exist for bipartite multi-graphs dependent on two parameters -- the number of edges shared between any two agents and the diameter of the graph. Finally, we prove that it is NP-complete to determine whether a given fair division instance on a bipartite multi-graph admits an EFX orientation.