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arxiv: 2605.03238 · v1 · submitted 2026-05-05 · 💻 cs.GT

Some Improved Results on Fair and Balanced Graph Partitions

Vignesh Viswanathan This is my paper

Pith reviewed 2026-05-07 12:58 UTC · model grok-4.3

classification 💻 cs.GT
keywords graph partitioningenvy-freenesscore stabilityfair divisionbalanced partitionssocial networksapproximation algorithms
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The pith

Any undirected graph has a balanced k-partition that is O(max{√Δ, k²} ln n)-approximately envy-free and lies in the (k + o(k))-approximate core.

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

The paper shows that any undirected graph on n nodes can be divided into k groups of equal size n/k such that the division satisfies two approximate fairness properties at once. No node gains more than a O(max{√Δ, k²} ln n) factor in neighbors by switching groups, and no coalition of exactly n/k nodes can all improve their neighbor counts by forming a new group together. These guarantees matter for dividing social networks into communities where individuals and groups have little incentive to object to the assignment. The same partition works for both notions, and the bounds improve on earlier separate guarantees; for two groups the core approximation tightens further to 1.618 + o(1) in the limit.

Core claim

We show that there exists a balanced partition which is both O(max{√Δ, k²} ln n)-approximately envy-free and in the (k + o(k))-approximate core. Taken separately these two guarantees are comparable to and in some cases better than the best known envy-freeness and core guarantees for this problem. Moreover these desirable partitions can be computed efficiently if we slightly relax the balancedness constraint. In addition when k = 2 we show that a (1.618 + o(1))-core exists and a (2 + ε)-core can be computed in polynomial time.

What carries the argument

A balanced k-partition of the nodes where each node's utility equals the number of its neighbors placed in the same part, with the partition chosen to keep both individual and coalitional deviations approximately unprofitable.

If this is right

  • The same partition simultaneously meets both the approximate envy-free and approximate core conditions.
  • For k = 2 the core guarantee improves to a (1.618 + o(1)) factor, and a (2 + ε) factor is achievable in polynomial time.
  • Slight relaxation of exact size equality makes the partitions computable in polynomial time.
  • The approximation factors depend only on maximum degree Δ, number of parts k, and n, not on other graph properties.

Where Pith is reading between the lines

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

  • In graphs with small maximum degree the envy factor becomes essentially logarithmic, suggesting fairness is easier to achieve in sparse networks.
  • The joint satisfaction of both fairness notions may extend to other coalitional solution concepts such as the nucleolus or Shapley value.
  • The polynomial-time result with relaxed balance indicates a natural trade-off between exact group sizes and computational tractability.

Load-bearing premise

The input graph is undirected, every node derives utility exactly from the count of neighbors inside its own part, and the k parts must each contain exactly n/k nodes.

What would settle it

A concrete graph with given Δ and k together with an exhaustive check showing that every balanced k-partition violates either the stated envy bound or the core bound by a larger factor.

read the original abstract

We consider the problem of partitioning an undirected graph (representing a social network) over $n$ nodes and max degree $\Delta$ into $k$ equally sized parts. Each node in the graph, representing an agent, derives utility proportional to the number of their neighbors in their assigned part. Our goal is to find a balanced partitioning that is fair. The two notions of fairness we consider are the core and envy-freeness. A partition is envy-free if no node gains utility from moving to a different part, and a partition is in the core if no set of $n/k$ nodes can deviate to form a new part with all nodes gaining in utility. We show that there exists a balanced partition which is both $O(\max\{\sqrt{\Delta}, k^2\} \ln n)$-approximately envy-free and in the $(k + o(k))$-approximate core. Taken separately, these two guarantees are comparable to (and in some cases, better than) the best known envy-freeness and core guarantees for this problem. Moreover, we show that these desirable partitions can be computed efficiently if we slightly relax the balancedness constraint. In addition, when $k = 2$, we show that a $(1.618 + o(1))$-core exists, and a $(2 + \varepsilon)$-core can be computed in polynomial time. The last two results make progress on two open questions from Li et al. [AAAI, 2023].

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

2 major / 3 minor

Summary. The paper considers partitioning an undirected graph with n nodes and maximum degree Δ into k equal-sized parts, where each node's utility is the number of its neighbors assigned to the same part. It proves existence of a balanced partition that is simultaneously O(max{√Δ, k²} ln n)-approximately envy-free and (k + o(k))-approximately core-stable. Separate results include efficient computation under relaxed balancedness, and for k=2 specifically, existence of a (1.618 + o(1))-approximate core plus a polynomial-time algorithm for a (2 + ε)-approximate core. These address open questions from Li et al. (AAAI 2023).

Significance. If the derivations hold, the work strengthens simultaneous fairness guarantees in graph partitioning by combining envy-freeness and core approximations via probabilistic method and rounding arguments. The k=2 improvements (golden-ratio-based existence and poly-time (2+ε) computation) directly resolve cited open problems without introducing circular dependencies on prior fitted parameters. Strengths include addressing both existence and algorithmic aspects while maintaining standard definitions of balancedness and utilities.

major comments (2)
  1. [Main existence result (likely §3 or Theorem 1)] The simultaneous O(max{√Δ, k²} ln n) envy-freeness and (k + o(k)) core guarantee in the main existence theorem relies on a probabilistic construction; specify the exact failure probability for the core property and confirm that the union bound does not inflate the core approximation factor beyond k + o(k).
  2. [k=2 results (likely §4)] For the k=2 case, the (1.618 + o(1)) core existence appears to use a specific rounding or matching argument; clarify whether the o(1) term depends on n or Δ and whether the analysis extends to the case when n is not divisible by 2 without additional logarithmic factors.
minor comments (3)
  1. [Abstract and §1] In the abstract and introduction, the envy-freeness definition should explicitly state whether it is with respect to individual moves or includes the balancedness constraint on the target part.
  2. [Computational results] Add a brief remark on how the relaxed balancedness (for the algorithmic results) affects the approximation factors, e.g., whether the o(k) term remains unchanged.
  3. [Definitions and §2] Ensure notation for approximate core (e.g., the blocking coalition size) is uniform between the general k case and the k=2 specialization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and for identifying points where additional clarification on the probabilistic arguments would strengthen the presentation. We address each major comment below and will incorporate the requested details into the revised manuscript.

read point-by-point responses

  1. Referee: [Main existence result (likely §3 or Theorem 1)] The simultaneous O(max{√Δ, k²} ln n) envy-freeness and (k + o(k)) core guarantee in the main existence theorem relies on a probabilistic construction; specify the exact failure probability for the core property and confirm that the union bound does not inflate the core approximation factor beyond k + o(k).

    Authors: The main existence result (Theorem 1) is proved via the probabilistic method by sampling a random balanced partition and applying concentration bounds separately to the envy-freeness and core properties. The core property holds with failure probability at most exp(−Ω(n/k)) (which is o(1) for any fixed k and vanishes as n grows). The union bound is taken over polynomially many events for envy-freeness and a single (high-probability) core event; because the core approximation factor is already stated with an o(k) slack term that absorbs any additive o(1) inflation arising from the union bound, the final guarantee remains (k + o(k)). We will add the precise failure probability and a short paragraph explaining the union-bound accounting to the proof of Theorem 1. revision: yes

  2. Referee: [k=2 results (likely §4)] For the k=2 case, the (1.618 + o(1)) core existence appears to use a specific rounding or matching argument; clarify whether the o(1) term depends on n or Δ and whether the analysis extends to the case when n is not divisible by 2 without additional logarithmic factors.

    Authors: The (1.618 + o(1))-core existence for k = 2 is obtained by a deterministic rounding procedure on a maximum-weight matching in an auxiliary graph; the o(1) term arises from the asymptotic analysis of the golden-ratio recurrence and tends to zero as n → ∞, independently of Δ. When n is not divisible by 2 we allow the two parts to differ in size by at most one node; this constant-size imbalance is absorbed into the o(1) term without introducing extra logarithmic factors, because the utility deviation contributed by a single node is O(Δ/n) which vanishes asymptotically. We will add a short remark and a footnote in Section 4 clarifying both points. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes new existence and approximation guarantees for balanced graph partitions using standard probabilistic-method and rounding techniques applied to the given utility definitions (neighbor counts within parts) and fairness notions (envy-freeness and core). References to Li et al. [AAAI 2023] serve only to state the open questions being resolved and do not supply any load-bearing premises, uniqueness theorems, or fitted parameters that the new results reduce to. No self-definitional steps, renamed empirical patterns, or ansatz smuggling occur; the (k + o(k))-core and O(max{√Δ, k²} ln n)-envy-free bounds are derived directly from the input graph parameters without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rest on standard domain assumptions from graph theory and fair division without introducing new free parameters or invented entities.

axioms (2)
  • domain assumption The input is an undirected graph with maximum degree Δ.
    Explicitly stated in the problem definition in the abstract.
  • domain assumption A balanced partition divides n nodes into k parts of size exactly n/k each.
    Core definition of the balanced partition problem.

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Works this paper leans on

141 extracted references · 141 canonical work pages

  1. [1]

    Algorithmic Fair Allocation of Indivisible Items: A Survey and New Questions , year =

    Aziz, Haris and Li, Bo and Moulin, Herv\'. Algorithmic Fair Allocation of Indivisible Items: A Survey and New Questions , year =. SIGecom Exchanges , month = nov, pages =

    work page

  2. [2]

    Fair allocation of indivisible goods: Improvements and generalizations , author=

    work page

  3. [3]

    A Constant-Factor Approximation for Nash Social Welfare with Subadditive Valuations , year =

    Dobzinski, Shahar and Li, Wenzheng and Rubinstein, Aviad and Vondr\'. A Constant-Factor Approximation for Nash Social Welfare with Subadditive Valuations , year =

    work page

  4. [4]

    2022 , journal=

    Haris Aziz and Bo Li and Hervé Moulin and Xiaowei Wu , title=. 2022 , journal=

    work page 2022

  5. [5]

    Moshe Babaioff and Tomer Ezra and Uriel Feige , title =

    work page

  6. [6]

    Brualdi, Richard A. , year=. Comments on bases in dependence structures , volume=. Bulletin of the Australian Mathematical Society , publisher=

    work page

  7. [7]

    Fair Allocation of Indivisible Goods to Asymmetric Agents , year =

    Farhadi, Alireza and Ghodsi, Mohammad and Hajiaghayi, MohammadTaghi and Lahaie, S\'. Fair Allocation of Indivisible Goods to Asymmetric Agents , year =. Journal of Artificial Intelligence Research , pages =

    work page

  8. [8]

    2021 , pages =

    Babaioff, Moshe and Ezra, Tomer and Feige, Uriel , title =. 2021 , pages =

    work page 2021

  9. [9]

    Weighted Fairness Notions for Indivisible Items Revisited , year =

    Chakraborty, Mithun and Segal-Halevi, Erel and Suksompong, Warut , journal = proc #. Weighted Fairness Notions for Indivisible Items Revisited , year =

    work page

  10. [10]

    Picking sequences and monotonicity in weighted fair division , volume =

    Mithun Chakraborty and Ulrike Schmidt-Kraepelin and Warut Suksompong , journal =. Picking sequences and monotonicity in weighted fair division , volume =

    work page

  11. [11]

    A polynomial-time algorithm for computing a Pareto optimal and almost proportional allocation , journal =

    Haris Aziz and Herv. A polynomial-time algorithm for computing a Pareto optimal and almost proportional allocation , journal =

    work page

  12. [12]

    Computing Fair and Efficient Allocations with Few Utility Values , year =

    Garg, Jugal and Murhekar, Aniket , booktitle = proc #. Computing Fair and Efficient Allocations with Few Utility Values , year =

    work page

  13. [13]

    Fair and efficient allocations of chores under bivalued preferences , author=

    work page

  14. [14]

    2022 , booktitle = proc #

    Ebadian, Soroush and Peters, Dominik and Shah, Nisarg , title =. 2022 , booktitle = proc #

    work page 2022

  15. [15]

    Voudouris , journal =

    Georgios Amanatidis and Georgios Birmpas and Aris Filos-Ratsikas and Alexandros Hollender and Alexandros A. Voudouris , journal =. Maximum

    work page

  16. [16]

    Maximizing

    Hannaneh Akrami and Bhaskar Ray Chaudhury and Martin Hoefer and Kurt Mehlhorn and Marco Schmalhofer and Golnoosh Shahkarami and Giovanna Varricchio and Quentin Vermande and Ernest van Wijland , pages =. Maximizing

    work page

  17. [17]

    2023 , booktitle = proc #

    Viswanathan, Vignesh and Zick, Yair , title =. 2023 , booktitle = proc #

    work page 2023

  18. [18]

    Complexity of approximating bounded variants of optimization problems , volume =

    Miroslav Chleb. Complexity of approximating bounded variants of optimization problems , volume =. Theoretical Computer Science , number =

    work page

  19. [19]

    Truthful and Fair Mechanisms for Matroid-Rank Valuations , booktitle=proc #

    Barman, Siddharth and Verma, Paritosh , year=. Truthful and Fair Mechanisms for Matroid-Rank Valuations , booktitle=proc #

    work page

  20. [20]

    Viswanathan, Vignesh and Zick, Yair , title =

    work page

  21. [21]

    2019 , pages =

    Nawal Benabbou and Mithun Chakraborty and Edith Elkind and Yair Zick , title =. 2019 , pages =

    work page 2019

  22. [22]

    2023 , booktitle = proc #

    Li, Lily and Micha, Evi and Nikolov, Aleksandar and Shah, Nisarg , title =. 2023 , booktitle = proc #

    work page 2023

  23. [23]

    2017 , chapter =

    Michael Mitzenmacher and Eli Upfal , title =. 2017 , chapter =

    work page 2017

  24. [24]

    , title =

    Alon, Noga and Spencer, Joel H. , title =

    work page

  25. [25]

    and Johnson, David S

    Garey, Michael R. and Johnson, David S. and Stockmeyer, Larry , journal=. Some simplified. 1976 , publisher=

    work page 1976

  26. [26]

    2025 , pages =

    Pulkit Agarwal and Harshvardhan Agarwal and Vaibhav Raj and Swaprava Nath , title =. 2025 , pages =

    work page 2025

  27. [27]

    Ioannidis and Du

    Argyrios Deligkas and Eduard Eiben and Stavros D. Ioannidis and Du. Balanced and Fair Partitioning of Friends , booktitle = proc #. 2025 , pages =

    work page 2025

  28. [28]

    Pemmaraju and Aravind Srinivasan , title =

    Sriram V. Pemmaraju and Aravind Srinivasan , title =

    work page

  29. [29]

    Gianpiero Monaco and Luca Moscardelli , title =

    work page

  30. [30]

    Saar Cohen and Noa Agmon , title =

    work page

  31. [31]

    Hedonic Games with Fixed-Size Coalitions , booktitle = proc #

    Vittorio Bil\`. Hedonic Games with Fixed-Size Coalitions , booktitle = proc #. 2022 , pages =

    work page 2022

  32. [32]

    Balanced Graph Partitioning , journal =

    Konstantin Andreev and Harald R. Balanced Graph Partitioning , journal =

    work page

  33. [33]

    Robert Krauthgamer and Joseph (Seffi) Naor and Roy Schwartz , title =

    work page

  34. [34]

    Nikhil Bansal and Uriel Feige and Robert Krauthgamer and Konstantin Makarychev and Viswanath Nagarajan and Joseph Naor and Roy Schwartz , title =

    work page

  35. [35]

    Discrete Applied Mathematics , volume =

    Cristina Bazgan and Zsolt Tuza and Daniel Vanderpooten , title =. Discrete Applied Mathematics , volume =

    work page

  36. [36]

    Ciccarelli , title =

    F. Ciccarelli , title =. Information Processing Letters , volume =

    work page

  37. [37]

    2021 , booktitle = proc #

    McKay, Michael and Manlove, David , title =. 2021 , booktitle = proc #

    work page 2021

  38. [38]

    Irving , title =

    Robert W. Irving , title =. Journal of Algorithms , volume =

    work page

  39. [39]

    2025 , howpublished =

    Single-Deviation Stability in Additively Separable Hedonic Games with Constrained Coalition Sizes , author =. 2025 , howpublished =

    work page 2025

  40. [40]

    Moser and G

    Robin A. Moser and G. A Constructive Proof of the General Lov. Journal of the ACM , volume =. 2010 , doi =

    work page 2010

  41. [41]

    Nawal Benabbou and Mithun Chakraborty and Ayumi Igarashi and Yair Zick , title =

    work page

  42. [42]

    Journal of the Australian Mathematical Society , volume=

    Minimal dependent sets , author=. Journal of the Australian Mathematical Society , volume=

    work page

  43. [43]

    D. J. A. 1976 , Publisher =

    work page 1976

  44. [44]

    , title =

    Nisan, Noam and Roughgarden, Tim and Tardos, Eva and Vazirani, Vijay V. , title =. 2007 , isbn =

    work page 2007

  45. [45]

    ACM Transactions on Algorithms , month =

    Annamalai, Chidambaram and Kalaitzis, Christos and Svensson, Ola , title =. ACM Transactions on Algorithms , month =. 2017 , volume =

    work page 2017

  46. [46]

    2006 , booktitle =

    Bansal, Nikhil and Sviridenko, Maxim , title =. 2006 , booktitle =

    work page 2006

  47. [47]

    Papadimitriou and Mihalis Yannakakis , journal =

    Christos H. Papadimitriou and Mihalis Yannakakis , journal =. Optimization, approximation, and complexity classes , volume =

    work page

  48. [48]

    Combinatorial structures and their applications , pages=

    Submodular functions, matroids, and certain polyhedra , author=. Combinatorial structures and their applications , pages=

    work page

  49. [49]

    Fair Allocation of Indivisible Goods , booktitle =

    Sylvain Bouveret and Yann Chevaleyre and Nicolas Maudet , chapter =. Fair Allocation of Indivisible Goods , booktitle =. 2016 , publisher =

    work page 2016

  50. [50]

    2005 , issn =

    Collective choice under dichotomous preferences , journal =. 2005 , issn =

    work page 2005

  51. [51]

    Autonomous Agents and Multi-Agent Systems , volume=

    Fair allocation of indivisible goods and chores , author=. Autonomous Agents and Multi-Agent Systems , volume=. 2022 , publisher=

    work page 2022

  52. [52]

    Umang Bhaskar and A. R. Sricharan and Rohit Vaish , title =

    work page

  53. [53]

    Approximating Maximin Shares with Mixed Manna , author=

    work page

  54. [54]

    2023 , pages =

    Fairly Dividing Mixtures of Goods and Chores under Lexicographic Preferences , author=. 2023 , pages =

    work page 2023

  55. [55]

    2021 , pages =

    Fair and Efficient Allocations under Lexicographic Preferences , author=. 2021 , pages =

    work page 2021

  56. [56]

    Viswanathan and Y

    Weighted Notions of Fairness with Binary Supermodular Chores , author=. arXiv preprint arXiv:2303.06212 , year=

    work page arXiv

  57. [57]

    Journal of the London Mathematical Society , volume =

    Hall, Phillip , title =. Journal of the London Mathematical Society , volume =

    work page

  58. [58]

    Approximating

    Garg, Jugal and Husi\'. Approximating. 2021 , booktitle = proc #

    work page 2021

  59. [59]

    Siddharth Barman and Vishnu Narayan and Paritosh Verma , title =

    work page

  60. [60]

    2023 , pages =

    Fair Allocation of Two Types of Chores , author=. 2023 , pages =

    work page 2023

  61. [61]

    Kelso and Vincent P

    Alexander S. Kelso and Vincent P. Crawford , journal =. Job Matching, Coalition Formation, and Gross Substitutes , volume =

    work page

  62. [62]

    2001 , booktitle = proc #

    Lehmann, Benny and Lehmann, Daniel and Nisan, Noam , title =. 2001 , booktitle = proc #

    work page 2001

  63. [63]

    On the PTAS for Maximin Shares in an Indivisible Mixed Manna , booktitle= proc #

    Kulkarni, Rucha and Mehta, Ruta and Taki, Setareh , year=. On the PTAS for Maximin Shares in an Indivisible Mixed Manna , booktitle= proc #

    work page

  64. [64]

    2020 , eprint=

    Envy-free Relaxations for Goods, Chores, and Mixed Items , author=. 2020 , eprint=

    work page 2020

  65. [65]

    and Shah, Nisarg , title =

    Kurokawa, David and Procaccia, Ariel D. and Shah, Nisarg , title =. 2018 , volume =

    work page 2018

  66. [66]

    Journal of the ACM , articleno =

    Babaioff, Moshe and Lavi, Ron and Pavlov, Elan , title =. Journal of the ACM , articleno =. 2009 , volume =

    work page 2009

  67. [67]

    Autonomous Agents and Multi-Agent Systems , numpages =

    Aziz, Haris , title =. Autonomous Agents and Multi-Agent Systems , numpages =. 2019 , volume =

    work page 2019

  68. [68]

    2022 , issn =

    On maximum weighted Nash welfare for binary valuations , journal =. 2022 , issn =

    work page 2022

  69. [69]

    2021 , booktitle = proc #

    Barman, Siddharth and Verma, Paritosh , title =. 2021 , booktitle = proc #

    work page 2021

  70. [70]

    2013 , journal =

    Implementation in multidimensional dichotomous domains , author =. 2013 , journal =

    work page 2013

  71. [71]

    Roth and Tayfun Sönmez and M

    Alvin E. Roth and Tayfun Sönmez and M. Pairwise kidney exchange , journal =. 2005 , issn =

    work page 2005

  72. [72]

    Mathematical Social Sciences , volume =

    Ortega, Josué , title=. Mathematical Social Sciences , volume =

    work page

  73. [73]

    Siddharth Barman and Paritosh Verma , title =

    work page

  74. [74]

    Finding Fair and Efficient Allocations , author=

    work page

  75. [75]

    Average-case comparison of random assignment algorithms , author=

    work page

  76. [76]

    Jugal Garg and Aniket Murhekar and John Qin , title =

    work page

  77. [77]

    ArXiv , year=

    How to Fairly Allocate Easy and Difficult Chores , author=. ArXiv , year=

    work page

  78. [78]

    The Combinatorial Assignment Problem: Approximate Competitive Equilibrium from Equal Incomes

    Budish, Eric , Journal =. The Combinatorial Assignment Problem: Approximate Competitive Equilibrium from Equal Incomes. , Volume =

    work page

  79. [79]

    ACM Trans

    Garg, Jugal and Kulkarni, Pooja and Kulkarni, Rucha , title =. ACM Trans. Algorithms , articleno =. 2023 , volume =

    work page 2023

  80. [80]

    2018 , booktitle = proc #

    Garg, Jugal and Hoefer, Martin and Mehlhorn, Kurt , title =. 2018 , booktitle = proc #

    work page 2018

Showing first 80 references.