Fair and efficient allocation of indivisible items under category constraints
Pith reviewed 2026-05-22 23:21 UTC · model grok-4.3
The pith
For any number of agents, there exists a Pareto-optimal allocation in which each can be made envy-free by reallocating at most min{k+1,n}(n-1) items.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We prove that for any number n of agents, there always exists a Pareto-optimal allocation in which each agent can be made envy-free by reallocating at most min {k+1,n}(n-1) items. Moreover, when the number of agents is constant, we give a polynomial-time algorithm to compute such an allocation. Our results rely on a new application of the Knaster-Kuratowski-Mazurkiewicz (KKM) lemma to a simplex of agent weights.
What carries the argument
Application of the KKM lemma to a simplex of agent weights to guarantee existence of a Pareto-optimal allocation with bounded envy removal.
If this is right
- Such allocations are computable in polynomial time for fixed n.
- The envy removal bound scales with both the number of agents and categories.
- The approach extends two-agent EF[1,1] guarantees to general n with bounded moves.
- The KKM-based technique applies to other constrained indivisible allocation problems.
Where Pith is reading between the lines
- The bound implies that moves needed grow linearly with n but stay controlled relative to categories.
- This could support practical computation of near-envy-free allocations under constraints without full search.
- The method might tighten bounds or apply to chores mixed with goods in future extensions.
Load-bearing premise
The KKM lemma can be applied to a suitably defined simplex of agent weights to guarantee the existence of the desired allocation.
What would settle it
An instance with n agents and category capacities where every Pareto-optimal allocation requires reallocating more than min{k+1,n}(n-1) items to eliminate envy for some agent would disprove the claim.
read the original abstract
We study the problem of fairly allocating indivisible goods and chores under category constraints. Specifically, there are $n$ agents and $m$ indivisible items which are partitioned into categories with associated capacities. An allocation is considered feasible if each bundle satisfies the capacity constraints of its respective categories. For the case of two agents, Shoshan et al. (2023) recently developed a polynomial-time algorithm to find a Pareto-optimal allocation satisfying a relaxed version of envy-freeness, called EF$[1,1]$. Extending such guarantees beyond two agents has remained open. We make progress toward this question by proving that for any number $n$ of agents, there always exists a Pareto-optimal allocation in which each agent can be made envy-free by reallocating at most $\min \{k+1,n\}(n-1)$ items. Moreover, when the number of agents is constant, we give a polynomial-time algorithm to compute such an allocation. Our results rely on a new application of the Knaster-Kuratowski-Mazurkiewicz (KKM) lemma to a simplex of agent weights, which may be of independent interest for fair division problems involving indivisible items.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that for any number n of agents and m indivisible items partitioned into categories with capacities, there exists a Pareto-optimal allocation such that each agent can be made envy-free by reallocating at most min{k+1,n}(n-1) items. It also gives a polynomial-time algorithm to compute such an allocation when n is constant. The existence proof relies on a new application of the KKM lemma to a simplex of agent weights.
Significance. If the central existence claim holds, the result meaningfully extends the two-agent EF[1,1] guarantee of Shoshan et al. (2023) to arbitrary n, with an explicit reallocation bound linear in n and depending on the number of categories k. The novel KKM application on a weight simplex is presented as potentially reusable for other indivisible fair-division settings. The fixed-n polynomial-time algorithm is a concrete algorithmic contribution.
major comments (1)
- [Proof of the main existence theorem (KKM application paragraph)] Proof of the main existence theorem (KKM application paragraph): the argument defines sets A_i on the simplex of agent weights and invokes KKM, but must explicitly verify that each A_i is closed and that the union covers every face while simultaneously enforcing category capacities and the min{k+1,n}(n-1) reallocation bound for EF. If a face weight vector cannot be assigned to any A_i without violating a capacity or the reallocation limit, the covering property fails and the claimed bound does not follow from KKM.
minor comments (2)
- [Abstract] Abstract: the parameter k is used in the bound without definition; it should be stated that k denotes the number of categories.
- [Proof section] The manuscript should include a short self-contained statement of the precise KKM lemma variant being applied (including the precise topological assumptions on the simplex and the sets).
Simulated Author's Rebuttal
We thank the referee for the careful reading of our manuscript and for highlighting the need for greater explicitness in the KKM application. We address the major comment below and will incorporate the requested clarifications in the revised version.
read point-by-point responses
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Referee: [Proof of the main existence theorem (KKM application paragraph)] Proof of the main existence theorem (KKM application paragraph): the argument defines sets A_i on the simplex of agent weights and invokes KKM, but must explicitly verify that each A_i is closed and that the union covers every face while simultaneously enforcing category capacities and the min{k+1,n}(n-1) reallocation bound for EF. If a face weight vector cannot be assigned to any A_i without violating a capacity or the reallocation limit, the covering property fails and the claimed bound does not follow from KKM.
Authors: We agree that the current paragraph would benefit from a more explicit, step-by-step verification. In the revision we will add a dedicated subsection that (i) proves each A_i is closed by showing it is the inverse image of a closed set under a continuous map derived from the envy and capacity functions; (ii) establishes the covering property on every face of the weight simplex by constructing, for any weight vector on a face, a feasible allocation that respects category capacities and satisfies the reallocation bound of at most min{k+1,n}(n-1) items per agent, thereby placing the vector in at least one A_i; and (iii) confirms that the bound is preserved uniformly because the construction never requires more than that many swaps to eliminate envy while maintaining Pareto optimality. This will directly address the concern that a face vector might fall outside all A_i. revision: yes
Circularity Check
No circularity; existence via external KKM lemma on constructed simplex
full rationale
The paper derives the claimed existence result by defining a simplex over agent weights and invoking the standard KKM lemma to produce a weight vector inducing a PO allocation satisfying the reallocation bound. KKM is an independent topological theorem, not defined in terms of the target bound or fitted to any data. No self-citations are load-bearing for the central claim, no parameters are fitted then renamed as predictions, and the construction does not reduce the bound to an input by definition. The derivation is self-contained against external mathematical benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Knaster-Kuratowski-Mazurkiewicz (KKM) lemma
Reference graph
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discussion (0)
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