EFX Allocations and Orientations on Bipartite Multi-graphs: A Complete Picture
Mahyar Afshinmehr, Alireza Danaei, Mehrafarin Kazemi, Kurt Mehlhorn, Nidhi Rathi
TL;DR
This work studies fair division of indivisible items represented by multi-graphs, where each item is valued by at most its two endpoints, and seeks $ extsf{EFX}$ allocations. It introduces a configuration-based methodology using $T$-cut configurations, defines six structural properties, and constructs a partial $ extsf{EFX}$ orientation that can be extended to a complete allocation, yielding $ extsf{EFX}$ allocations for bipartite multi-graphs with monotone valuations (pseudo-polynomial time) and for cancelable valuations (polynomial time). It also proves existence of $ extsf{EFX}$ allocations on multi-cycles under $ extsf{MMS}$-feasible valuations and characterizes the existence of $ extsf{EFX}$ orientations in bipartite multi-graphs via the parameters $q$ (edge multiplicity) and $d(G)$ (diameter), along with NP-hardness of the orientation decision problem. The results broaden the range of graph-structured fair division problems where $ extsf{EFX}$ allocations are guaranteed and efficiently computable, while highlighting inherent waste when exact orientation is impossible. The paper further delineates limitations and future directions for extending these techniques to general multi-graphs.
Abstract
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 vertices and items as 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. The main contribution of this work is to establish the existence of EFX allocations on bipartite multi-graphs for monotone valuations and on multi-cycles for MMS-feasible valuations. We also present pseudo-polynomial time algorithms to compute EFX allocations for the above settings. Furthermore, we show that for bipartite multi-graphs with cancelable valuations, EFX allocations can be computed in polynomial time. We thus widen the spectrum where EFX allocations are guaranteed to exist. Next, we study EFX orientations (allocations where every item is assigned to one of its two endpoint agents) and provide a complete characterization of their existence on bipartite multi-graphs in terms of two key parameters: (i) the number of edges shared between any two agents and (ii) 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, even with a constant number of agents.