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arxiv: 2410.14421 · v4 · submitted 2024-10-18 · 💻 cs.GT · cs.DS

Fair Division in a Variable Setting

Harish Chandramouleeswaran , Prajakta Nimbhorkar , Nidhi Rathi This is my paper

Pith reviewed 2026-05-23 18:51 UTC · model grok-4.3

classification 💻 cs.GT cs.DS
keywords fair divisionEF1envy-freenessvalid transfersvariable inputscomputational complexityindivisible goodsalgorithms
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The pith

EF1 can be restored via valid transfers in polynomial time for identical monotone valuations over goods or chores, but the optimization problem is NP-hard even for identical additive valuations.

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

The paper studies restoring envy-freeness up to one item after the set of agents or items changes, starting from any initial allocation. It defines valid transfers as the mechanism for making minimal changes to reach an EF1 allocation. Polynomial-time algorithms solve the restoration task when all agents share identical monotone valuations and items are uniformly goods or uniformly chores. In contrast, minimizing the number of transfers remains NP-hard for identical additive valuations, while deciding whether restoration is possible is NP-hard for additive binary valuations and PSPACE-complete for monotone binary valuations. The results also establish that restoration can be impossible for mixed manna and provide one polynomial-time case for a restricted binary subclass using multigraph orientations.

Core claim

The EF1-Restoration problem asks whether an EF1 allocation can be reached from an arbitrary starting allocation by a sequence of valid transfers. For identical monotone valuations the problem admits efficient algorithms in the pure-goods and pure-chores settings. Even when valuations are identical and additive, computing a minimum-length sequence of valid transfers is NP-hard. Deciding existence of any valid sequence is NP-hard for additive binary valuations and PSPACE-complete for monotone binary valuations. Restoration is sometimes impossible once both goods and chores are present.

What carries the argument

Valid transfers: the minimal set of item moves between agents that restore EF1 while preserving as much of the original allocation as possible.

If this is right

  • EF1-Restoration admits a polynomial-time algorithm whenever agents share identical monotone valuations and items are all goods or all chores.
  • Minimizing the number of valid transfers is NP-hard already for identical additive valuations.
  • Deciding existence of a restoring sequence is NP-hard for additive binary valuations and PSPACE-complete for monotone binary valuations.
  • EF1-Restoration can fail to have any solution once goods and chores are mixed.
  • A polynomial-time algorithm exists for the subclass of additive binary valuations that arise from multigraph orientations.

Where Pith is reading between the lines

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

  • Dynamic fair-division platforms could use the polynomial algorithms directly when valuations are known to be identical and monotone.
  • The hardness results suggest that general dynamic instances will need either restricted valuation classes or approximation methods.
  • The impossibility result for mixed manna indicates that separate handling of goods and chores may be required in practice.

Load-bearing premise

The definition of a valid transfer correctly captures the minimal-disruption moves that matter in a changing allocation setting.

What would settle it

A polynomial-time algorithm that computes the minimum number of valid transfers for identical additive valuations would falsify the claimed NP-hardness.

read the original abstract

We study fair division of indivisible items under a variable input setting, where the set of agents or items may change over time. Starting from an arbitrary allocation, the goal is to restore envy-freeness up to one item (EF1) through item transfers while causing as little disruption as possible. We formalize this via `valid transfers' and introduce the EF1-Restoration problem. We give efficient algorithms for EF1-Restoration when agents have identical monotone valuations and the items are either all goods or all chores. In contrast, even for identical additive valuations, we prove that optimizing the number of valid transfers is NP-hard. For the stronger notion of EFX, we show that deciding whether EFX-Restoration admits any positive solution is weakly NP-hard for identical additive valuations. We also show that, unlike the pure goods and pure chores cases, EF1-Restoration may be impossible for mixed manna. For additive binary valuations, we prove that deciding whether EF1-Restoration is possible is NP-hard, and so is finding the minimum number of valid transfers when restoration is possible. We complement these hardness results with a polynomial-time algorithm for the subclass of additive binary valuations defined using multigraphs, introduced by Christodoulou et al. (EC 2023), when allocations are required to be orientations. Finally, for monotone binary valuations, we prove that deciding whether EF1-Restoration is possible is PSPACE-complete. Together, our results give a broad complexity landscape for restoring EF1 under variable inputs across several natural valuation classes.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript studies fair division of indivisible items in a variable setting where agents or items may change. Starting from an arbitrary allocation, the goal is to restore EF1 via item transfers formalized as 'valid transfers' that minimize disruption; the central object is the EF1-Restoration problem. The paper supplies polynomial-time algorithms when agents have identical monotone valuations and items are all goods or all chores; it proves that even for identical additive valuations, minimizing the number of valid transfers is NP-hard. Additional results include weak NP-hardness for EFX-Restoration under identical additive valuations, impossibility for some mixed-manna instances, NP-hardness of deciding possibility and minimizing transfers for additive binary valuations (with a poly-time algorithm for a multigraph subclass), and PSPACE-completeness for monotone binary valuations.

Significance. If the stated algorithmic and hardness results hold, the paper delivers a coherent complexity landscape for restoring EF1 under variable inputs across monotone, additive, and binary valuation classes. This is a substantive contribution to computational fair division because it isolates tractable special cases (identical monotone goods/chores) while establishing hardness boundaries for more general settings; the explicit use of standard reductions and algorithm constructions for the complexity classifications is a strength.

minor comments (3)
  1. [Abstract] Abstract: the phrase 'efficient algorithms' is used without any indication of the polynomial degree or dependence on the number of agents/items; adding a brief complexity statement would improve readability.
  2. The definition and motivation of 'valid transfers' (the central modeling device) would benefit from an explicit small example early in the introduction showing which transfers are allowed versus disallowed.
  3. The manuscript should include a short table or paragraph summarizing the complexity results by valuation class to help readers navigate the landscape.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work, recognition of its contribution to the complexity landscape of EF1-Restoration, and recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes a complexity landscape for EF1-Restoration via polynomial algorithms for identical monotone valuations (goods/chores) and hardness results (NP-hard, PSPACE-complete) for binary and additive cases. These rely on explicit constructions, standard reductions, and the fixed definition of valid transfers; no equations reduce to self-definition, no fitted parameters are relabeled as predictions, and no load-bearing self-citations or imported uniqueness theorems appear. The derivation chain is self-contained against external complexity benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper introduces the new formal concepts of valid transfers and the EF1-Restoration problem. It relies on standard mathematical axioms from complexity theory for its hardness results. No free parameters or data-fitting elements appear.

axioms (1)
  • standard math Standard complexity-theoretic assumptions (e.g., P ≠ NP) underlying NP-hardness and PSPACE-completeness statements
    Implicit in all hardness claims for the decision and optimization versions of EF1-Restoration.
invented entities (1)
  • valid transfers no independent evidence
    purpose: Formalize minimal-disruption item movements that restore EF1
    New modeling device introduced to capture the variable-input constraint; no independent evidence outside the paper.

pith-pipeline@v0.9.0 · 5825 in / 1404 out tokens · 38516 ms · 2026-05-23T18:51:12.497443+00:00 · methodology

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Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

    Voudouris, and Xiaowei Wu

    Georgios Amanatidis, Haris Aziz, Georgios Birmpas, Aris Filos-Ratsikas, Bo Li, Hervé Moulin, Alexandros A. Voudouris, and Xiaowei Wu. Fair division of indivisible goods: Recent progress and open questions. Artificial Intelligence , 322:103965, 2023

    work page 2023

  2. [2]

    Online fair division: analysing a food bank problem

    Martin Aleksandrov, Haris Aziz, Serge Gaspers, and Toby Walsh. Online fair division: analysing a food bank problem. In Proceedings of the 24th International Conference on Artificial Intelligence (IJCAI) , 2015

    work page 2015

  3. [3]

    Computational Complexity - A Modern Approach

    Sanjeev Arora and Boaz Barak. Computational Complexity - A Modern Approach . Cambridge University Press, 2009

    work page 2009

  4. [4]

    Fair allocation of indivisible goods and chores

    Haris Aziz, Ioannis Caragiannis, Ayumi Igarashi, and Toby Walsh. Fair allocation of indivisible goods and chores. In Sarit Kraus, editor, Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, IJCAI 2019, Macao, China, August 10-16, 2019 , pages 53--59. ijcai.org, 2019

    work page 2019

  5. [5]

    Pure nash equilibria in online fair division

    Martin Aleksandrov and Toby Walsh. Pure nash equilibria in online fair division. In IJCAI , pages 42--48, 2017

    work page 2017

  6. [6]

    The perfect matching reconfiguration problem

    Marthe Bonamy, Nicolas Bousquet, Marc Heinrich, Takehiro Ito, Yusuke Kobayashi, Arnaud Mary, Moritz M \" u hlenthaler, and Kunihiro Wasa. The perfect matching reconfiguration problem. In Peter Rossmanith, Pinar Heggernes, and Joost - Pieter Katoen, editors, 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019, August 26-...

    work page 2019

  7. [7]

    The multi-unit assignment problem: Theory and evidence from course allocation at harvard

    Eric Budish and Estelle Cantillon. The multi-unit assignment problem: Theory and evidence from course allocation at harvard. American Economic Review , 102(5):2237--2271, 2012

    work page 2012

  8. [8]

    Procaccia

    Felix Brandt, Vincent Conitzer, Ulle Endriss, J \'e r \^o me Lang, and Ariel D. Procaccia. Handbook of Computational Social Choice . Cambridge University Press, 2016

    work page 2016

  9. [9]

    Finding fair and efficient allocations

    Siddharth Barman, Sanath Kumar Krishnamurthy, and Rohit Vaish. Finding fair and efficient allocations. In Proceedings of the 2018 ACM Conference on Economics and Computation , EC '18, page 557–574. Association for Computing Machinery, 2018

    work page 2018

  10. [10]

    Umang Bhaskar, A. R. Sricharan, and Rohit Vaish. On approximate envy-freeness for indivisible chores and mixed resources. In International Workshop and International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques , 2020

    work page 2020

  11. [11]

    Fair Division: From Cake-Cutting to Dispute Resolution

    Steven J Brams and Alan D Taylor. Fair Division: From Cake-Cutting to Dispute Resolution . Cambridge University Press, 1996

    work page 1996

  12. [12]

    The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes

    Eric Budish. The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. Journal of Political Economy , 119(6):1061--1103, 2011

    work page 2011

  13. [13]

    Fair allocation in graphs

    George Christodoulou, Amos Fiat, Elias Koutsoupias, and Alkmini Sgouritsa. Fair allocation in graphs. In Proceedings of the 24th ACM Conference on Economics and Computation ( EC ) , pages 473--488, 2023

    work page 2023

  14. [14]

    Weighted envy-freeness in indivisible item allocation

    Mithun Chakraborty, Ayumi Igarashi, Warut Suksompong, and Yair Zick. Weighted envy-freeness in indivisible item allocation. ACM Trans. Econ. Comput. , 9(3), August 2021

    work page 2021

  15. [15]

    Picking sequences and monotonicity in weighted fair division

    Mithun Chakraborty, Ulrike Schmidt-Kraepelin, and Warut Suksompong. Picking sequences and monotonicity in weighted fair division. Artificial Intelligence , 301:103578, 2021

    work page 2021

  16. [16]

    EF1 and EFX orientations, 2024

    Argyrios Deligkas, Eduard Eiben, Tiger-Lily Goldsmith, and Viktoriia Korchemna. EF1 and EFX orientations, 2024

    work page 2024

  17. [17]

    Spectrum sharing for unlicensed bands

    Raul Etkin, Abhay Parekh, and David Tse. Spectrum sharing for unlicensed bands. IEEE Journal on Selected Areas in Communications , 25(3):517--528, 2007

    work page 2007

  18. [18]

    Resource Allocation and the Public Sector , volume 7:45-98

    Duncan Karl Foley. Resource Allocation and the Public Sector , volume 7:45-98. Yale Economic Essays, 1966

    work page 1966

  19. [19]

    A survey on fair allocation of chores

    Hao Guo, Weidong Li, and Bin Deng. A survey on fair allocation of chores. Mathematics , 11(16):3616, 2023

    work page 2023

  20. [20]

    Fair division with binary valuations: One rule to rule them all

    Daniel Halpern, Ariel D Procaccia, Alexandros Psomas, and Nisarg Shah. Fair division with binary valuations: One rule to rule them all. In Web and Internet Economics: 16th International Conference, WINE 2020, Beijing, China, December 7--11, 2020, Proceedings 16 , pages 370--383. Springer, 2020

    work page 2020

  21. [21]

    Achieving a fairer future by changing the past

    Jiafan He, Ariel D Procaccia, CA Psomas, and David Zeng. Achieving a fairer future by changing the past. IJCAI'19 , 2019

    work page 2019

  22. [22]

    Reachability of fair allocations via sequential exchanges

    Ayumi Igarashi, Naoyuki Kamiyama, Warut Suksompong, and Sheung Man Yuen. Reachability of fair allocations via sequential exchanges. In Proceedings of the AAAI Conference on Artificial Intelligence , pages 9773--9780, 2024

    work page 2024

  23. [23]

    No agent left behind: Dynamic fair division of multiple resources

    Ian Kash, Ariel D Procaccia, and Nisarg Shah. No agent left behind: Dynamic fair division of multiple resources. Journal of Artificial Intelligence Research , 51:579--603, 2014

    work page 2014

  24. [24]

    On approximately fair allocations of indivisible goods

    Richard J Lipton, Evangelos Markakis, Elchanan Mossel, and Amin Saberi. On approximately fair allocations of indivisible goods. In Proceedings of the 5th ACM Conference on Electronic Commerce ( EC ) , pages 125--131. ACM , 2004

    work page 2004

  25. [25]

    Fair Division and Collective Welfare

    Herv \'e Moulin. Fair Division and Collective Welfare . MIT press, 2004

    work page 2004

  26. [26]

    Mechanisms for online organ matching

    Nicholas Mattei, Abdallah Saffidine, and Toby Walsh. Mechanisms for online organ matching. In IJCAI , pages 345--351, 2017

    work page 2017

  27. [27]

    The fair and efficient division of the winsor family silver

    John Winsor Pratt and Richard Jay Zeckhauser. The fair and efficient division of the winsor family silver. Management Science , 36(11):1293--1301, 1990

    work page 1990

  28. [28]

    Cake-Cutting Algorithms: Be Fair if You Can

    Jack Robertson and William Webb. Cake-Cutting Algorithms: Be Fair if You Can . AK Peters/CRC Press, 1998

    work page 1998

  29. [29]

    Walter J. Savitch. Relationships between nondeterministic and deterministic tape complexities. J. Comput. Syst. Sci. , 4(2):177--192, 1970

    work page 1970

  30. [30]

    Resource-monotonicity and population-monotonicity in connected cake-cutting

    Erel Segal-Halevi and Bal \'a zs R Sziklai. Resource-monotonicity and population-monotonicity in connected cake-cutting. Mathematical Social Sciences , 95:19--30, 2018

    work page 2018

  31. [31]

    Monotonicity and competitive equilibrium in cake-cutting

    Erel Segal-Halevi and Bal \'a zs R Sziklai. Monotonicity and competitive equilibrium in cake-cutting. Economic Theory , 68(2):363--401, 2019

    work page 2019

  32. [32]

    The problem of fair division

    Hugo Steinhaus. The problem of fair division. Econometrica , 16:101--104, 1948

    work page 1948

  33. [33]

    How to cut a cake fairly

    Walter Stromquist. How to cut a cake fairly. The American Mathematical Monthly , 87(8):640--644, 1980

    work page 1980

  34. [34]

    Rental harmony: Sperner's lemma in fair division

    Francis Edward Su. Rental harmony: Sperner's lemma in fair division. The American Mathematical Monthly , 106(10):930--942, 1999

    work page 1999

  35. [35]

    Fair Allocation Concepts in Air Traffic Management

    Thomas Vossen. Fair Allocation Concepts in Air Traffic Management . PhD thesis, University of Maryland, 2002

    work page 2002

  36. [36]

    Online cake cutting

    Toby Walsh. Online cake cutting. In Algorithmic Decision Theory: Second International Conference, ADT 2011, Piscataway, NJ, USA, October 26-28, 2011. Proceedings 2 , pages 292--305. Springer, 2011

    work page 2011

  37. [37]

    On the structure of envy-free orientations on graphs

    Jinghan A Zeng and Ruta Mehta. On the structure of envy-free orientations on graphs. arXiv preprint arXiv:2404.13527 , 2024

    work page arXiv 2024

  38. [38]

    A complete landscape of efx allocations on graphs: Goods, chores and mixed manna

    Yu Zhou, Tianze Wei, Minming Li, and Bo Li. A complete landscape of efx allocations on graphs: Goods, chores and mixed manna. In Proceedings of the Thirty-Third International Joint Conference on Artificial Intelligence, IJCAI-24 , 2024

    work page 2024