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

2024-10-22

Adaptive Agents and Multi-Agent Systems

AI Summary

This paper investigates the existence of EFX (envy-freeness up to any good) allocations for monotone and MMS-feasible valuations on bipartite multigraphs and multicycles, where agents correspond to vertices, indivisible goods to edges, and valuations depend only on incident edges. Using combinatorial game-theoretic and graph-theoretic techniques, we establish, for the first time, that EFX allocations always exist and are constructible in polynomial time on both graph classes. We derive necessary and sufficient conditions for the existence of an EFX orientation—i.e., an edge orientation inducing an EFX allocation—and fully characterize its structural properties. Furthermore, we prove that deciding whether an EFX orientation exists is NP-complete. This work extends the known boundaries of EFX existence, and provides the first exact existence criterion and complexity classification for fair allocations under explicit graph-structural constraints.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Bridging structured and unstructured data

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 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.

Problem

• Extending EFX allocations from simple graphs to bipartite multi-graphs
• Establishing EFX existence for monotone and MMS-feasible valuations
• Characterizing EFX orientations on bipartite multi-graphs with parameters

Innovation

• EFX allocations exist on bipartite multi-graphs for monotone valuations
• Pseudo-polynomial algorithms compute EFX for bipartite multi-graphs
• Characterized EFX orientations using edge count and graph diameter