EFX Allocations Exist on Multi-Graphs

Abstract: We study the fair allocation of indivisible goods among agents, with a focus on limiting envy. A central fairness notion is envy-freeness up to any good (EFX), which requires that any envy toward another agent vanishes after the removal of any single good from the latter's bundle. The existence of EFX allocations is considered a major open problem in fair division. So far, it has only been established in limited settings. Christodoulou et al. [2023] proved the existence of EFX allocations for graphical valuations. In this setting, the agents correspond to the nodes of an underlying graph, and the goods correspond to the edges, and any good has positive value only for the endpoints of the corresponding edge. Their proof crucially relies on the restriction that the graph is simple, meaning that for any pair of agents, there is at most one good that has value to both. For multigraph valuations, where multiple goods may be valued by the same pair of agents, only partial results are known. Amanatidis et al. [2024] and Kaviani et al. [2025] obtained 2/3 and sqrt(2)/2 approximations of EFX, respectively; Kaviani et al. [2024] established existence under restricted additive valuations; and Afshinmehr et al. [2025a] proved existence under the assumption that the shortest cycle containing non-parallel edges has length at least 4. In this paper, we resolve this open problem by proving the existence of EFX allocations for multigraph instances under cancelable valuations, a strict superclass of additive valuation functions. Our proof is algorithmic and computes such allocations in polynomial time when the valuation functions are cancelable. This work contributes to the small number of EFX existence results that apply to an arbitrary number of agents.

• The paper presents a breakthrough by proving that EFX allocations exist in multi-graph settings for agents with cancelable valuations.
• It introduces a polynomial-time algorithm that constructs EFX allocations through unit bundle partitioning, greedy allocation, and critical-path elimination.
• The work broadens fair division theory and practice by enabling equitable allocations in complex interdependent environments without restrictive graph constraints.

Existence of EFX Allocations on Multi-Graphs

Introduction and Problem Statement

The paper "EFX Allocations Exist on Multi-Graphs" (2606.18665) addresses the central open problem in discrete fair division regarding the existence of \emph{envy-free up to any good} (EFX) allocations of indivisible goods. The study focuses on the generalization from simple graphs (where at most one good is shared between any pair of agents) to multi-graphs, in which multiple goods may be mutually valued by the same pair of agents. Unlike in the classic setting with additive or monotone valuations, this multi-graph structure allows the modeling of much richer local interdependencies among agents.

The EFX property requires that, for any two agents $i$ and $j$, after the removal of any good from $j$'s bundle, $i$ no longer envies $j$. Although EFX allocations are guaranteed under certain restrictive domains (e.g., for identical or binary valuations, or on simple graphical valuation structures [CFKS23]), the general case for multi-graphs remained unresolved except under strong cycle restrictions or for coarse approximation variants.

Contributions and Main Theoretical Results

This work proves, constructively, the existence of EFX allocations in multi-graph instances under the broad class of \emph{cancelable} valuations, which strictly generalize additive but are more tractable than arbitrary monotone functions. Specifically, the results establish:

• Existence: For any multi-graph, an EFX allocation always exists when agents have cancelable valuations.
• Computability: There is a polynomial-time algorithm for finding such an allocation, given access to polynomial-time cancellation-value oracles.

This framework resolves the generalized graphical EFX existence problem without assumptions on the multi-graph's topology (no bounds on cycles, chromatic number, or edge multiplicities) and for general $n$.

Algorithmic Framework and Technical Structure

The algorithm operates in several carefully reasoned phases that maintain increasingly fine-grained invariants about the structure of partial allocations, with special attention to the multi-graph context:

Unit Bundle Construction

For each pair of agents $i$ and $j$, the set of shared goods $E_{i,j}$ is split via a pair of balanced partitions, called \emph{unit bundles}, constructed so that both agents regard both sides as EFX-feasible. This uses a variant of the Plaut-Roughgarden (PR) local search method [PR20], adapted for cancelable valuations and shown to terminate in polynomial time due to the comparability and monotonicity properties enjoyed by cancelability.

Greedy EFX Partial Allocation & Tree-Breaking

• The initial allocation is built greedily, with each agent sequentially choosing their most valued available unit bundle, which ensures no strong envy arises and the resulting \emph{resent graph} (tracking the direction of resenting preferences) is a forest.
• Any resulting resentment trees with large depth are broken along carefully chosen \emph{critical paths} to reduce all trees to height at most one, yielding a "simple height-one" allocation in which the longest resentment chain is of length one, every resented agent is resented by at most one agent, and each agent receives at most one unit bundle.

Completion to Total Allocation (Dumping Phase)

In the final stage, all disconnected or otherwise unassigned unit bundles are allocated via a combination of deterministic local reassignment rules (including "dumping" of low-value leftovers onto carefully selected non-envied agents), often using the concept of \emph{support pairs}—pairs of non-resented agents whose local bundle placements enable further allocation without introducing strong envy.

Multiple special cases are addressed with specialized lemmas to handle intricate structural configurations of the resentment graph and multi-graph, ensuring that in all situations the EFX condition persists and the final allocation is complete.

Formal Claims and Computational Guarantees

• For every multigraph instance with cancelable valuations, there exists an allocation $X=(X_1,\ldots,X_n)$ such that for every pair $j$0, and for every good $j$1, we have $j$2.
• The proposed algorithm computes such allocations in $j$3 time, making use of dynamic construction and balancing of unit bundles, envy-cycle elimination, and critical-path rerouting, all under the closure properties of cancelable valuations.

Relation to Prior Literature

Previously, EFX existence was achieved under additive valuations on simple graphs (or graphical valuations with no parallel edges) [CFKS23] and under strong restrictions (e.g., triangle-freeness [AAMM25], large girth [AAMM25, BP24], or limited cycles [ADKMR25, SS25]). Only approximate EFX allocations (e.g., $j$4- or $j$5-EFX) were known without such topological constraints [ARS24, KKSS25]. Results for cancelable valuations were known for fewer than four agents [BCFF21], or in further special graph classes.

This work removes all such restrictions for cancelable valuations, generalizing and strictly subsuming prior positive results for multigraphs.

Implications and Future Extensions

Theoretical

The result advances the theoretical understanding of the boundary between tractable and intractable fair division. It signals that local structure, when coupled with moderate valuation axioms (cancelability), is sufficient for full EFX guarantees—contrasting sharply with the known nonexistence of EFX allocations for certain monotone valuations [AMMSW26]. Furthermore, the proof techniques underscore the power of localized bundle partitioning and iterative envy truncation in fair division.

Practical

Since cancelable valuations encompass all additive preferences and many realistic agent models with local complementarities (but exclude pathological monotonicity interactions), the approach yields practical, scalable algorithms for fair allocation in a wide range of discrete market, scheduling, or network resource contexts where local overlaps are the norm.

Open Problems

• Extending the result to even broader classes of valuations (arbitrary monotone, subadditive, or XOS) remains open.
• Investigating the interaction between EFX fairness and varying efficiency objectives (e.g., Nash or utilitarian welfare) in multi-graph settings [FMP24] is a natural next step.
• The complexity and approximability of EFX allocations with non-cancelable valuations on multi-graphs, and tight lower bounds for various subclasses, are also compelling avenues.

Conclusion

This work solves a longstanding open problem in the fair division of indivisible goods by establishing that EFX allocations exist and are efficiently computable in all multi-graph instances with cancelable valuations, regardless of graph structure or number of agents. The techniques introduced—combinatorial partitioning into unit bundles, critical-path elimination, and structured dumping—open new directions for both theory and algorithmic design in fair allocation under complex agent interdependencies.

References

• "Fair allocation in graphs" [CFKS23]
• "Almost Envy-Freeness with General Valuations" [PR20]
• "Almost Full EFX Exists for Four Agents" [BCFF21]
• "EFX Allocations Exist on Triangle-Free Multi-Graphs" [AAMM25]
• "Pushing the Frontier on Approximate EFX Allocations" [ARS24]