EFX Allocation In (Multi)Hypergraphs

We study whether strictly positive marginal values restore the compatibility of envy-freeness up to one good (EF1) and Pareto optimality (PO) for indivisible goods. For two agents, we identify the exact threshold in the number of goods. Every instance with at most seven goods and strictly increasing valuations admits an allocation that is both EF1 and PO, without any submodularity assumption. In contrast, we construct an eight-good instance with normalized, integer-valued, strictly increasing, submodular valuations in which every EF1 allocation is strictly Pareto dominated. Thus, eight goods are necessary and sufficient for a two-agent counterexample. Finally, we strengthen the three-agent NP-hardness result of Chandramouleeswaran and Nimbhorkar (2026): deciding whether an EF1 and PO allocation exists remains NP-hard for normalized, integer-valued, monotone submodular valuations even when zero marginals are confined to eight fixed agent-good pairs, all involving a single agent.

Recent work in fair division has focused on either simultaneously satisfying closely related fairness notions or achieving a single notion across the ex-ante and ex-post worlds. We study the compatibility of two fundamentally different fairness notions: envy-freeness and equitability. For indivisible goods-only and chores-only settings, we study the existence and complexity of simultaneously satisfying their relaxations, revealing sharp contrasts between the two settings. We show that EF1+EQ1 may fail to exist even for normalized binary goods: we construct an instance with 113 agents and 341 goods in which every agent approves exactly 165 goods, but no complete allocation satisfies both notions. Our main algorithmic result computes an EF1+EQ1 allocation for every normalized binary goods instance with at most seven agents. Thus, the smallest number of agents admitting a counterexample lies between 8 and 113, leaving the cases from 8 through 112 unresolved. In sharp contrast, binary chores admit the stronger EFX+EQX guarantee for any number of agents, even without normalization. We further initiate the study of cross-notion ex-ante and ex-post guarantees, asking whether randomized allocations can provide ex-ante guarantees for one notion while preserving ex-post guarantees for another.

Many allocation problems are intrinsically multidimensional, since an item may contribute differently to several criteria, and optimizing a single aggregate objective can hide severe losses in other dimensions. We study how much efficiency can be guaranteed simultaneously when indivisible items have multiple attributes. To this end, we introduce the \emph{multidimensional efficient allocation} (MDEA) model, where each agent has an additive valuation in each dimension, and investigate simultaneous efficiency under utilitarian social welfare (USW) and egalitarian social welfare (ESW). Our results reveal a sharp worst-case frontier. For exact efficiency, maximizing the number of dimensions attaining the USW optimum admits a $c/\ell$-approximation for every fixed constant $c$, and this dependence on the number $\ell$ of dimensions is essentially unavoidable; for ESW, even deciding whether two dimensions can be optimized simultaneously is NP-hard with binary valuations. For approximate simultaneous efficiency in every dimension, we identify a tight threshold of order $1/\ell$, showing that such guarantees always exist for both USW and ESW, while any asymptotically better dependence on $\ell$ is impossible, even for binary valuations. Finally, we introduce three natural multidimensional Pareto notions and characterize both their relationships and their computational complexity.