ef2x allocation construction

This work investigates the existence and construction of envy-free up to any good (EFX) allocations among agents with heterogeneous monotone valuations under hypergraph and multi-hypergraph structures. Focusing on settings where only endpoints of hyperedges derive non-zero marginal value from items, the study establishes—for the first time—that EFX allocations always exist for hypergraphs with girth at least four under general monotone valuations, extending beyond the previously studied additive case. The result is further generalized to multi-hypergraphs satisfying specific multiplicity constraints. By integrating tools from combinatorial graph theory and algorithm design, and leveraging girth restrictions together with edge multiplicity analysis, the authors present a polynomial-time algorithm for constructing EFX allocations in hypergraphs with girth ≥ 4, and a quasi-polynomial-time algorithm for the broader class of generalized multi-hypergraphs.

This paper studies fair allocation of indivisible items—goods, chores, or mixed items—under generalized utility functions that may be non-monotonic, non-objective, and even yield negative utilities, while incorporating structural constraints such as path connectivity. Methodologically, it introduces the first multi-color Sperner lemma variant for non-positive utilities, integrated with fixed-point theory, local search, and dynamic programming. Key contributions are: (1) the first existence proofs for EF1* and EQ1* allocations under non-negative, and more generally arbitrary, utility functions; (2) polynomial-time algorithms for computing EQ1 and EQ1* allocations, and a pseudo-polynomial algorithm for EQX*; and (3) a systematic characterization of the existence and computational tractability boundaries for approximately fair (EFx/EQx) and approximately equitable allocations, substantially advancing the theoretical foundations and algorithmic toolkit for fair division under non-standard utility models.

This paper addresses the existence and constructive computation of EFX (envy-free up to any good) allocations for indivisible goods. We propose a simulated annealing–based local search algorithm that employs a single-item transfer neighborhood structure and minimizes the total number of EFX violations as the objective function, under additive utility assumptions. The method enables efficient exploration of the allocation space and yields a novel constructive proof technique for EFX existence—bypassing prior reliance on intricate combinatorial constructions or restricted problem instances. Empirical evaluation demonstrates 100% success rate in finding EFX allocations across thousands of randomly generated instances; the algorithm scales to large settings with hundreds of agents and thousands of goods, substantially outperforming existing heuristics. Our approach thus bridges theoretical insight and practical efficacy, advancing both the understanding and computability of EFX fairness.

This paper studies fair allocation of indivisible goods when a subset of items is pre-allocated, focusing on the computational tractability of envy-free (EF) and envy-free up to any good (EFX) allocations for the remaining items. We introduce a systematic parameterized complexity classification framework for extended fair allocation, designing fixed-parameter tractable (FPT) algorithms parameterized by either the number of agents or the number of distinct item types. We complement these with tight W[1]-hardness lower bounds, demonstrating that the identified tractability boundaries are asymptotically optimal and not generalizable. Notably, we provide the first complete characterization of EFX existence in this extended setting—resolving it definitively—and thereby achieve a theoretical breakthrough in completeness. Our approach integrates parameterized algorithm design, problem reductions, fairness modeling, and lower-bound analysis, significantly advancing the algorithmic feasibility frontier for fair allocation under structural constraints.

This paper studies the EF1 fairness restoration problem for indivisible resources in dynamic environments: given an existing EF1 allocation that becomes violated due to item removal or agent arrival, how to restore EF1 with minimal item reassignments while maintaining near-EF1 throughout the process? We first formalize a variable-input fair allocation model and introduce the EF1-Restoration problem. Methodologically, we develop a combinatorial optimization framework based on EF1-graph orientation: for additive utilities, we design an optimal algorithm achieving $O(m/n)$ transfers; for graphical utility models, we provide an exact algorithm. We further prove that feasibility checking for EF1 restoration under monotone binary utilities is PSPACE-complete. Our core contributions are (i) establishing the first theoretical foundation for dynamic fairness repair, and (ii) providing an efficient algorithmic framework with tight guarantees for diverse utility classes.

This work addresses the fair allocation of indivisible goods under a hypergraph structure, where hyperedges represent items and vertices represent agents, with non-zero marginal value for an item only assigned to its incident agents. For heterogeneous monotone valuation functions, the study combines combinatorial constructions with greedy strategies to achieve various approximate envy-free up to any good (EFX) allocations in polynomial or pseudo-polynomial time. Key contributions include the first existence proof of 2/3-EFX allocations for hypergraphs with edge multiplicity two, along with a simpler construction achieving √2/2-EFX; for hypergraphs of girth at least three, it establishes EF2X under general monotone valuations and √2/2-EFX under subadditive valuations; and for edge multiplicity two, it further obtains EF3X and 2/3-EFX under additive valuations.

This work addresses the problem of envy-free allocation of indivisible items—comprising goods, chores, or a mixture thereof—among multiple agents with additive valuations. It presents the first subexponential-time algorithm for this setting by modeling feasible allocations as a convex polytope in ℝ³ and recursively partitioning the agent set using Miller’s planar separator theorem. The resulting divide-and-conquer framework integrates geometric and graph-theoretic techniques to determine, in $(n \cdot m)^{O(\sqrt{n})}$ time, whether an envy-free allocation exists and to construct such an allocation when it does. The algorithm also accommodates instances where parts of the allocation are pre-fixed, thereby enabling the first efficient treatment of high-multiplicity mixed instances.

This work addresses the challenge of online learning fair allocations of indivisible goods under limited and potentially adversarial fairness feedback. Focusing on achieving EF1 or PROP1 fairness criteria, the paper proposes a general framework that integrates the geometry of valuation polytopes with the ellipsoid method, leveraging separation hyperplanes to handle unknown agent preferences. The authors establish that an EF1 allocation necessarily exists under the “interval plus one” preference structure. Their approach combines polytope maintenance, robust handling of adversarial feedback, and structured search to compute EF1 or PROP1 allocations in polynomial time for additive valuations. Furthermore, for general monotone valuations, the algorithm converges to an EF1 allocation within a polynomial number of rounds.

This study investigates the compatibility between envy-freeness up to one item (EF1) and Pareto optimality (PO) in the allocation of indivisible goods among two or more agents with strictly increasing valuations. For the two-agent case, the authors precisely characterize the threshold: an EF1 and PO allocation always exists when there are at most seven items, yet a counterexample with eight items demonstrates that EF1 and PO may be incompatible. Moreover, even under restrictions on the placement of zero marginal values, deciding the existence of an EF1 and PO allocation remains NP-hard for three agents. The analysis combines combinatorial constructions, complexity reductions, and normalized submodular valuation functions to delineate the structural boundaries governing the trade-off between fairness and efficiency.