ef3x allocation construction

This paper addresses the long-standing open problem of existence of envy-free up to any good (EFX) allocations for indivisible goods. While prior work established EFX existence only for at most three agents or for settings with at most two distinct additive valuation types, the general case remained unresolved. We prove, for the first time, that an EFX allocation always exists when agents’ additive valuations belong to at most three distinct types—regardless of the number of agents. This result unifies and strictly generalizes previous results for three agents and for two valuation types, thereby resolving the existence question in this broad setting. Technically, our proof integrates tools from combinatorial game theory, inductive construction, valuation-type classification, and pairwise reallocation strategies, yielding the first existence framework for EFX that accommodates arbitrarily many agents under a bounded number (≤3) of valuation types—filling a fundamental gap in fair division theory.

This paper investigates EFX (envy-freeness up to any good) allocations of indivisible goods among heterogeneous agents. For $n$ agents with valuations drawn from $k$ distinct types, we establish—constructively—that an EFX allocation always exists with at most $k-2$ unallocated goods; in particular, when all but two agents share identical valuations, a fully allocated EFX solution is guaranteed. Our approach integrates combinatorial game-theoretic reasoning, valuation-type clustering, and pairwise analysis to yield a constructive existence proof. The primary contribution is breaking the long-standing restriction on the number of agent types in EFX existence results: whereas prior work was limited to at most two valuation types, our framework extends feasibility to arbitrary $k geq 2$, providing the first general sufficient condition for EFX existence in settings with three or more heterogeneous agent types. This significantly advances the theoretical foundations and practical applicability of discrete fair division.

This paper investigates fair allocation of indivisible goods among four agents under cancellation-free utility functions, focusing on the existence and construction of EF2X allocations—allocations that are envy-free up to the removal of any two goods. Employing combinatorial construction, envy-graph analysis, local reallocation, and pseudopolynomial algorithm design, the authors establish, for the first time, that an EF2X allocation always exists for any number of goods and any four agents with cancellation-free valuations—resolving a long-standing open problem concerning EF2X existence where EFX remained elusive for four agents. They further present the first polynomial-time algorithm for EF2X in the three-agent setting and a pseudopolynomial-time constructive algorithm for four agents. These results significantly advance the theoretical frontier of fairness in indivisible resource allocation.

This study addresses a central open problem in fair division: the simultaneous achievement of envy-freeness up to any good (EFX) and Pareto optimality (PO) in the allocation of indivisible goods. Focusing on agents with additive positive valuations and instances restricted to only two distinct types of items, the paper establishes—for the first time—the existence of allocations that satisfy both EFX and PO. Moreover, it presents a quasi-linear time algorithm that efficiently constructs such allocations. This work not only strengthens prior theoretical results that guaranteed EFX existence without ensuring PO but also demonstrates, through combinatorial optimization techniques, the computational tractability and constructive feasibility of finding allocations meeting both fairness and efficiency criteria.

This paper studies the existence of α-envy-free up to any good (α-EFX) allocations for indivisible goods under additive valuations. Addressing a long-standing approximation-ratio bottleneck—where the best prior guarantee was ≈0.618—we establish, for the first time, that a 2/3-EFX allocation always exists under three broad generalizations: (1) at most seven agents; (2) trivalued valuation functions (i.e., each agent’s values for goods take only three distinct values); and (3) valuations representable via multigraph structures. Methodologically, we integrate combinatorial game theory, discrete optimization, and constructive existence proofs to develop a novel analytical framework grounded in valuation-structure modeling. Our results not only break the previous approximation barrier but also systematically extend exact EFX existence theory—from restricted settings (e.g., two agents or binary valuations)—to more realistic multi-agent, multivalued, and structured valuation domains, thereby significantly broadening the applicability frontier of EFX fairness.

This study addresses the problem of allocating divisible goods among multiple agents under additive valuations, with the goal of achieving approximate envy-freeness up to k items (α-EFkX): for any agent, after removing at most k items from another agent’s bundle, the remaining envy is bounded by a factor α. By generalizing the 3PA algorithm and integrating greedy strategies with matching techniques, the work establishes—for the first time—that a (k+1)/(k+2)-EFkX allocation always exists and can be computed in polynomial time for any number of agents and any k ≥ 1, yielding in particular a 3/4-EF2X guarantee. It also extends the known 2/3-EFX result from seven to eight agents. Furthermore, the paper shows that an EFkX graph orientation does not always exist and proves that deciding its existence is NP-complete.

This work addresses the central challenge in fair allocation of indivisible goods: simultaneously achieving EF1 (envy-freeness up to one good) fairness and maximizing social welfare. The authors propose a heuristic strategy that jointly optimizes both item assignment and recipient selection, integrated within the envy-cycle elimination framework, thereby overcoming the limitations of conventional approaches that optimize only a single dimension. Theoretical analysis demonstrates that this method substantially improves the lower bound on utilitarian welfare. Empirical evaluations further confirm its effectiveness in reducing welfare loss on average, successfully balancing fairness and efficiency in practical settings.

This work investigates the existence of envy-freeness up to any good (EFX) allocations for indivisible goods under multi-graph valuation models, where multiple items may be jointly valued by the same pair of agents. Focusing on cancelable valuation functions, it establishes—for the first time—the guaranteed existence of EFX allocations for any number of agents in multi-graph instances, thereby overcoming prior limitations restricted to simple graphs or approximate solutions. Building upon this theoretical guarantee, the authors devise a polynomial-time algorithm that efficiently constructs such EFX allocations. This result not only affirms the existence of EFX allocations in multi-graph settings but also provides a practical computational method, significantly broadening the applicability of EFX fairness theory.

This study investigates the existence of complete EFX (envy-free up to any good) allocations for four agents with additive valuations over at most nine indivisible goods. By integrating hand-crafted reduction lemmas with a machine-verified certificate library, the valuation space is partitioned into polyhedral subregions, each associated with a family of candidate EFX allocations whose validity is confirmed by an independent verifier. The work advances the known boundary for guaranteed complete EFX existence from \(m \leq n+3\) to \(m = n+5\) (i.e., nine goods), revealing that the most challenging instances arise when agents have nearly identical valuations. It further establishes that all such instances admit a complete EFX allocation, with the case of eight goods doubly verified. Empirical analysis indicates that only approximately 0.14% of all allocations satisfy the EFX criterion.

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.