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Matchings in the conflict graph yield EF1 allocations for ordered and tiered valuations when the item count stays below a bound set by the number of agents and the graph degree.

2026-06-27 05:20 UTC pith:6K3IBW4Q

load-bearing objection The paper gives tight EF1 existence results for ordered and tiered valuations under conflict graphs by parametrizing with max degree Δ and reducing to matchings, plus a round-robin plus matching algorithm for additive valuations when m ≤ 2n.

arxiv 2606.13083 v1 pith:6K3IBW4Q submitted 2026-06-11 cs.GT

Leveraging Matchings in Constrained Fair Division with a Conflict Graph

Evangelos Markakis , Michalis Samaris This is my paper
classification cs.GT
keywords fair divisionEF1 allocationsconflict graphindivisible goodsmatching theoryconstrained allocationenvy-free up to one item
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper studies allocation of indivisible goods where a conflict graph forbids certain pairs of items from being given to the same agent. It shows that for ordered valuations and the wider tiered class, envy-free allocations up to one good exist and can be computed by reducing the problem to matchings, provided the number of items does not exceed a threshold depending on the number of agents and the maximum degree of the graph. The threshold is tight once the degree exceeds two-thirds the number of agents. For general additive valuations the same matching technique, paired with round-robin, gives an almost complete characterization of when EF1 allocations exist provided there are at most twice as many items as agents.

Core claim

By reducing constrained allocation to matchings in the conflict graph, complete EF1 allocations exist for agents with ordered or tiered valuations whenever the number of items respects the bound determined by n and Δ; the bound is tight when Δ exceeds 2n/3. For general additive valuations with m at most 2n, combining round-robin with matchings produces an almost complete picture of the instances that admit EF1 allocations.

What carries the argument

Reductions to matchings in the conflict graph, used to select non-adjacent bundles while preserving the EF1 property under ordered or tiered valuations.

Load-bearing premise

The agents' valuations must belong to the ordered or tiered classes, or the instance must have at most twice as many items as agents for general additive valuations.

What would settle it

An explicit instance with ordered valuations, maximum degree greater than 2n/3, and more items than the stated bound that admits no EF1 allocation.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • EF1 allocations exist and are computable for all ordered and tiered instances whose item count respects the n-and-Δ bound.
  • The same bound is tight, so existence can fail once the degree is large enough and the item count exceeds it.
  • An approximation algorithm is available once the item count exceeds the bound.
  • For general additive valuations with m at most 2n, the round-robin plus matching method covers nearly all cases that admit EF1 allocations.

Where Pith is reading between the lines

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

  • The matching reduction may extend to other fairness criteria such as maximin share if the valuation restrictions can be relaxed.
  • When the conflict graph is a collection of small components, the bound on items may become looser in practice.
  • The approach suggests that polynomial-time algorithms for maximum matching directly yield polynomial-time EF1 algorithms under the stated conditions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The manuscript studies fair allocation of indivisible goods subject to conflict-graph constraints (items are vertices; edges forbid co-allocation in one bundle). Parametrized by the maximum degree Δ, it gives tight existence results for EF1 allocations under ordered and tiered valuations, an algorithm returning EF1 when m is at most a bound depending on n and Δ (tight for Δ > 2n/3), an approximation algorithm beyond that bound, and, for general additive valuations restricted to m ≤ 2n, a nearly complete characterization of EF1-admitting instances obtained by combining round-robin with matchings.

Significance. If the stated existence and algorithmic claims hold, the work is a solid contribution to constrained fair division. It supplies parameterized, matching-theoretic guarantees for structured valuations and a constructive approach for the m ≤ 2n additive case, extending prior impossibility results with explicit, tight bounds rather than purely existential statements.

minor comments (2)
  1. [Abstract] Abstract: the sentence beginning 'We present an algorithm...' contains the typographical error 'when then number of items'; correct to 'when the number of items'.
  2. [Abstract / Introduction] The abstract states that the bound on m is 'determined by n and Δ' and is tight for Δ > 2n/3, but does not display the explicit functional form; the main text should state the bound (e.g., in the theorem statement) so that readers can immediately verify the tightness claim without searching the proofs.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on constrained fair division and for recommending minor revision. We appreciate the recognition of the parameterized existence results, matching-based algorithms, and the near-complete characterization for the m ≤ 2n case.

Circularity Check

0 steps flagged

No significant circularity; results rest on external matching theory

full rationale

The derivation relies on reductions to matching theory (external, non-self-cited in a load-bearing way) for ordered/tiered valuations, combined with round-robin for additive cases when m ≤ 2n. No self-definitional steps, no fitted parameters renamed as predictions, and no uniqueness theorems imported from the authors' own prior work. The m-bound in terms of n and Δ is presented as derived from matching facts rather than by construction from the EF1 target. The paper is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard definitions and background results from fair division and graph theory rather than new free parameters or invented entities.

axioms (2)
  • domain assumption EF1 is defined via additive valuations and the standard envy comparison up to one good.
    The paper invokes the established EF1 notion without re-deriving it.
  • standard math Matching theory supplies polynomial-time algorithms and existence guarantees for the relevant subgraphs induced by the conflict graph.
    The algorithms explicitly leverage known matching results.

pith-pipeline@v0.9.1-grok · 5756 in / 1373 out tokens · 32499 ms · 2026-06-27T05:20:43.383308+00:00 · methodology

0 comments
read the original abstract

We study the problem of allocating indivisible goods under constraints, expressed via a conflict graph $G$. In such an instance, the $m$ items are the vertices of $G$ and connected items cannot be allocated in the same bundle. Under this model, it is already known that EF1 allocations may not exist. Our main contribution is an analysis parametrized by the maximum degree $\Delta(G)=\Delta$ on the existence and computation of complete EF1 allocations. We address this question in various cases by leveraging results from matching theory. First, we provide a tight existence result for agents with ordered valuations and for the broader class of tiered valuations. We present an algorithm that returns an EF1 allocation when then number of items does not exceed a specific bound. This bound is determined by $n$ and $\Delta$, and it is tight when $\Delta$ is greater than $2n/3$. We also construct an approximation algorithm when $m$ exceeds this bound. For general additive valuations the problem becomes more challenging. Given the current impossibility results, we focus on the case where the number of items is at most $2n$. For this case, we provide an almost complete picture for the instances that admit EF1 allocations, by combining Round Robin with matchings.

discussion (0)

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Fair Allocation under Conflict Constraints via Strong Colorability

    cs.GT 2026-07 unverdicted novelty 7.0

    The paper introduces a hierarchy of strong chromatic number variants to characterize and algorithmically guarantee SD-EF1, EF1, and EF[1,1] allocations under graph conflict constraints, showing 3Δ-1 agents suffice for...

Reference graph

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