arxiv: 2605.10371 · v1 · submitted 2026-05-11 · 💻 cs.GT

Recognition: 2 theorem links

· Lean Theorem

Approximate Envy-Free Allocations up to any k Goods

Aris Filos-Ratsikas , Georgios Kalantzis , Fangxiao Wang

Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:18 UTC · model grok-4.3

classification 💻 cs.GT

keywords approximate envy-freenessEFkXfair divisionadditive valuationspolynomial-time algorithmsNP-completenessgraph orientationsgame theory

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The pith

For any k>2, (k+1)/(k+2)-EFkX allocations exist for any number of agents and can be computed in polynomial time.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that approximate envy-free allocations up to any k goods achieve a ratio of (k+1)/(k+2) for additive valuations when k exceeds 2. This guarantee holds no matter how many agents are present. The allocations are found efficiently by extending an existing iterative assignment procedure. One immediate payoff is the existence of 3/4-EF2X allocations for arbitrarily many agents, which strengthens what was known for the two-good case. The work further improves the bound for exact EFX to eight agents and shows that certain structural variants of the problem are intractable to decide.

Core claim

For any k>2, (k+1)/(k+2)-EFkX allocations exist for any number of agents with additive valuations over goods, and such allocations can be computed in polynomial time via a generalization of the 3PA algorithm. This yields 3/4-EF2X allocations for arbitrary numbers of agents. Additionally, 2/3-EFX allocations exist for eight agents, and EFkX graph orientations do not always exist with their existence decision being NP-complete.

What carries the argument

Generalization of the 3PA algorithm that iteratively assigns goods while ensuring envy can be eliminated by removing at most k goods from any bundle.

If this is right

3/4-EF2X allocations exist for any number of agents.
2/3-EFX allocations exist for eight agents.
EFkX graph orientations do not always exist.
Deciding existence of EFkX graph orientations is NP-complete.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The polynomial-time result opens the door to implementing these fairness guarantees in automated resource allocation systems.
The same algorithmic template may extend to related fairness notions such as proportionality or maximin share.
The NP-completeness for orientations suggests that structural variants of approximate fairness remain hard even when the basic existence question is settled.

Load-bearing premise

The generalized 3PA procedure produces allocations meeting the (k+1)/(k+2) ratio for EFkX under only additive valuations and no further hidden constraints.

What would settle it

A concrete instance of additive valuations for which the generalized algorithm outputs an allocation violating (k+1)/(k+2)-EFkX, or for which no such allocation exists at all.

Figures

Figures reproduced from arXiv: 2605.10371 by Aris Filos-Ratsikas, Fangxiao Wang, Georgios Kalantzis.

**Figure 1.** Figure 1: The graph 𝐻 = (𝑁 , 𝑀) used in the proof of Theorem 4.2, when 𝑛 = 10. The black edges denote the heavy edges and the green edges denote the light edges. 𝑁𝐻 is the set of nodes incident to the inner 𝐾5 graph of light edges. We can easily verify that for any orientation of the highlighted inner 𝐾5 graph, there exists a node in 𝑁ℎ with at least two incoming edges. Next, we move on to our NP-completeness result… view at source ↗

**Figure 2.** Figure 2: The funnel gadget 𝐹 used in proof of Theorem 4.4. The 𝐻𝑒 box represents the envy enhancer gadget. The black edges are the solid edges and the red are the transit edges that connect to 2𝑘 vertices 𝑁𝐾 of the envy enhancer gadget. In an EFkX orientation, the black (solid) edges would be oriented towards𝑠 and 𝑘 of the transit edges would be oriented towards𝑣𝑖 for𝑖 ∈ [𝑘] from nodes 𝑁 ℎ 𝐾 ⊂ 𝑁𝐾, and the other 𝑘 t… view at source ↗

read the original abstract

We study the problem of finding approximate envy-free allocations up to any $k$ goods ($\alpha$-EFkX), when agents have additive values over goods in a bundle. As our main result, we show that for any $k>2$, $\frac{k+1}{k+2}$-EFkX allocations exist for any number of agents, and can be computed in polynomial time, via an appropriate generalization of the 3PA algorithm of [Amanatidis et al., 2024]. An immediate corollary of this result is that $3/4$-EF2X allocations exist for any number of agents; in contrast, $2/3$-EFX allocations are only known to exist for up to 7 agents. We improve this latter result by devising an algorithm that achieves $2/3$-EFX for 8 agents. We also consider EFkX graph orientations; we prove that such orientations do not always exist, and that deciding their existence is NP-complete, thereby generalizing the corresponding result of [Christodoulou et., 2023] for $k=1$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a poly-time algorithm for (k+1)/(k+2)-EFkX with any number of agents by generalizing 3PA, plus a 3/4-EF2X corollary, an 8-agent 2/3-EFX improvement, and NP-completeness for EFkX orientations.

read the letter

The key takeaway is that for any k>2 the authors produce (k+1)/(k+2)-EFkX allocations in polynomial time for arbitrary numbers of agents by generalizing the 3PA algorithm from Amanatidis et al. 2024. As a direct corollary they get 3/4-EF2X for every number of agents, and they push the known 2/3-EFX existence result from 7 to 8 agents. They also show that EFkX orientations do not always exist and that deciding their existence is NP-complete, extending the k=1 hardness from Christodoulou et al. 2023. These are the concrete new pieces. The algorithmic result is the main one; the hardness is a clean but unsurprising generalization. The paper does well by stating explicit approximation ratios that scale with k and by keeping the computational claim polynomial. It builds directly on the cited prior work without obvious circularity. The soft spot is the generalization step itself. The (k+1)/(k+2) guarantee rests on the claim that the 3PA construction extends inductively or recursively while preserving the exact ratio for every k and every number of agents. If the analysis introduces any hidden restriction on bundle sizes or valuation structure that only holds for small k, the ratio could slip for larger k. A referee would need to see the full inductive argument to confirm it carries through without degradation. The NP-completeness reduction looks routine and does not raise separate issues. This work is for researchers in fair division and algorithmic game theory who track envy-free approximations up to k goods. Anyone already following the EFkX or EFX literature will get immediate use from the new ratios and the general-k algorithm. It deserves a serious referee because the claims are precise, the algorithmic contribution is stated clearly, and the hardness result is a straightforward extension that still needs verification. Send it to review.

Referee Report

1 major / 3 minor

Summary. The manuscript studies approximate envy-free allocations up to any k goods (α-EFkX) under additive valuations. Its central claim is that for any k>2, (k+1)/(k+2)-EFkX allocations exist for arbitrary numbers of agents and can be computed in polynomial time via a generalization of the 3PA algorithm from Amanatidis et al. (2024). It also derives the corollary that 3/4-EF2X allocations exist for any number of agents, presents an algorithm achieving 2/3-EFX for 8 agents, and proves that EFkX graph orientations do not always exist with NP-completeness for deciding their existence (generalizing Christodoulou et al. for k=1).

Significance. If the generalization of 3PA preserves the stated ratio without degradation, the result would be significant for providing the first poly-time constant-factor approximation to EFkX that improves toward 1 as k increases and holds for any number of agents, in contrast to the more limited existence results known for exact EFX. The explicit improvement to 8 agents for 2/3-EFX and the structural negative result on orientations would further strengthen the contribution to fair division.

major comments (1)

[Generalization of 3PA algorithm] The section describing the generalization of the 3PA algorithm: the claim that the generalized procedure achieves exactly the (k+1)/(k+2) ratio for EFkX for arbitrary k>2 rests on an inductive or recursive extension of the k=3 base case. The analysis must explicitly confirm that the envy bound (when each agent compares her bundle to any other after removing at most k goods) is preserved without introducing hidden restrictions on the number of goods, bundle sizes, or valuation additivity assumptions, as any weakening of the ratio for larger k would invalidate the main existence and algorithmic result.

minor comments (3)

[Abstract] Abstract: the statement of the main result could briefly indicate the key structural property of the 3PA generalization that enables the ratio to hold for all k>2.
[EFX for 8 agents] The 8-agent 2/3-EFX algorithm: the description should clarify whether the construction is a direct specialization of the generalized 3PA or requires additional case analysis.
[EFkX graph orientations] The NP-completeness reduction for EFkX orientations: the proof should explicitly note how it extends the k=1 case from Christodoulou et al. (2023) to highlight the generalization.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need for greater explicitness in the analysis of the generalized 3PA algorithm. We address the major comment below.

read point-by-point responses

Referee: The section describing the generalization of the 3PA algorithm: the claim that the generalized procedure achieves exactly the (k+1)/(k+2) ratio for EFkX for arbitrary k>2 rests on an inductive or recursive extension of the k=3 base case. The analysis must explicitly confirm that the envy bound (when each agent compares her bundle to any other after removing at most k goods) is preserved without introducing hidden restrictions on the number of goods, bundle sizes, or valuation additivity assumptions, as any weakening of the ratio for larger k would invalidate the main existence and algorithmic result.

Authors: We appreciate the referee highlighting this point. In Section 3, the manuscript defines the generalized 3PA algorithm recursively, extending the k=3 procedure of Amanatidis et al. (2024) to arbitrary k>2. The approximation guarantee is established by induction on k: the base case reuses the known 3/4-EF3X bound, and the inductive step assumes a suitable (k/(k+1))-EF(k-1)X allocation and shows that the next iteration produces an allocation satisfying the (k+1)/(k+2)-EFkX bound. Because valuations are additive, the value of any bundle after removal of at most k goods is controlled by the sum of the k highest-valued goods in the compared bundle; the algorithm's selection rule (which prioritizes bundles that minimize maximum envy after such removals) ensures the factor is preserved. The argument places no restrictions on the total number of goods, the sizes of individual bundles, or the number of agents, and the polynomial-time bound holds uniformly. We agree, however, that the current exposition of the inductive step could be expanded with an explicit lemma verifying that the envy ratio does not degrade. We will therefore revise the proof to include these additional calculations and a self-contained verification that the bound holds under the stated assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central result generalizes the 3PA algorithm from the independent prior work of Amanatidis et al. (2024) to establish the existence of (k+1)/(k+2)-EFkX allocations for any k>2 and any number of agents, along with a polynomial-time algorithm. This generalization is presented as a new constructive contribution rather than reducing by definition or construction to the paper's own inputs. The corollary for 3/4-EF2X, the improvement to 2/3-EFX for 8 agents, and the NP-completeness result for EFkX orientations (generalizing Christodoulou et al. 2023) are likewise independent extensions that do not rely on self-definitional equivalences, fitted parameters renamed as predictions, or load-bearing self-citation chains. The derivation chain remains self-contained against the external benchmarks of the cited prior algorithms and standard additive valuation assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard domain assumptions in fair division plus the claim that a prior algorithm generalizes appropriately; no free parameters or invented entities are introduced.

axioms (2)

domain assumption Agents have additive valuations over bundles of goods.
Explicitly stated in the abstract as the problem setting.
ad hoc to paper The 3PA algorithm of Amanatidis et al. 2024 can be generalized to achieve the claimed approximation for EFkX.
This is the key step invoked for the main existence and polynomial-time result.

pith-pipeline@v0.9.0 · 5501 in / 1500 out tokens · 65907 ms · 2026-05-12T04:18:06.648576+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We show that for any k>2, (k+1)/(k+2)-EFkX allocations exist ... via an appropriate generalization of the 3PA algorithm
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

the modified envy graph ... proxy valuation function

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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