arXiv subjects

Mahyar Afshinmehr

Publications and source records attributed to Mahyar Afshinmehr.

5 recordsLinked to original sources

arXiv2026

EFX Allocations Exist on Multi-Graphs

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.

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arXiv2025

EFX Allocations Exist on Triangle-Free Multi-Graphs

We study the fair allocation of indivisible goods among agents, with a focus on limiting envy. A central open question in this area is the existence of EFX allocations-allocations in which any envy of any agent i towards any agent j vanishes upon the removal of any single good from j's bundle. Establishing the existence of such allocations has proven notoriously difficult in general, but progress has been made for restricted valuation classes. Christodoulou et al. [2023] proved existence for graphical valuations, where goods correspond to edges in a graph, agents to nodes, and each agent values only incident edges. The graph was required to be simple, i.e., for any pair of agents, there could be at most one good that both agents value. The problem remained open, however, for multi-graph valuations, where for a pair of agents several goods may have value to both. In this setting, Sgouritsa and Sotiriou [2025] established existence whenever the shortest cycle with non-parallel edges has length at least six, while Afshinmehr et al. [2025] proved existence when the graph contains no odd cycles. In this paper, we strengthen these results by proving that EFX allocations always exist in multi-graphs that contain no cycle of length three. Assuming monotone valuations, we further provide a pseudo-polynomial time algorithm for computing such an allocation, which runs in polynomial time when agents have cancelable valuations, a strict superclass of additive valuation functions. Accordingly, our results stand as one of the only cases where EFX allocations exist for an arbitrary number of agents.

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arXiv2025

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 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.

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arXiv2024

Approximate EFX and Exact tEFX Allocations for Indivisible Chores: Improved Algorithms

We explore the fair distribution of a set of $m$ indivisible chores among $n$ agents, where each agent's costs are evaluated using a monotone cost function. Our focus lies on two fairness criteria: envy-freeness up to any item (EFX) and a relaxed notion, namely envy-freeness up to the transfer of any item (tEFX). We demonstrate that a 2-approximate EFX allocation exists and is computable in polynomial time for three agents with subadditive cost functions, improving upon the previous $(2 + \sqrt{6})$ approximation for additive cost functions. This result requires extensive case analysis. Christoforidis et al. (IJCAI'24) independently claim the same approximation for additive cost functions; however, we provide a counter-example to their algorithm. We expand the number of agents to any number to get the same approximation guarantee with the assumption of partially identical ordering (IDO) for the cost functions. Additionally, we establish that a tEFX allocation is achievable for three agents if one has an additive 2-ratio bounded cost function, while the others may have general monotone cost functions. This is an improvement from the prior requirement of two agents with additive 2-ratio bounded cost functions. This allocation can also be extended to agent groups with identical valuations. Further, we show various analyses of EFX allocations for chores, such as the relaxations for additive $α$-ratio-bounded cost functions.

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arXiv2024

MMS Approximations Under Additive Leveled Valuations

We study the problem of fairly allocating indivisible goods to a set of agents with additive leveled valuations. A valuation function is called leveled if and only if bundles of larger size have larger value than bundles of smaller size. The economics literature has well studied such valuations. We use the maximin-share (MMS) and EFX as standard notions of fairness. We show that an algorithm introduced by Christodoulou et al. ([11]) constructs an allocation that is EFX and $\frac{\lfloor \frac{m}{n} \rfloor}{\lfloor \frac{m}{n} \rfloor + 1}\text{-MMS}$. In the paper, it was claimed that the allocation is EFX and $\frac{2}{3}\text{-MMS}$. However, the proof of the MMS-bound is incorrect. We give a counter-example to their proof and then prove a stronger approximation of MMS.

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