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Every abelian Cayley graph has an optimal spectral sparsifier with only O(log |G|) generators.

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2026-07-10 10:26 UTC pith:ZZBBSD6F

load-bearing objection Clean optimal abelian Cayley sparsifiers via a short character-symmetry volume argument; the result is real and the proof holds up.

arxiv 2607.08261 v1 pith:ZZBBSD6F submitted 2026-07-09 cs.DS math.CO

Optimal Sparsifiers for Abelian Cayley Graphs

Arpon Basu , Pravesh K. Kothari , Raghu Meka , Stefan Tudose This is my paper
classification cs.DS math.CO MSC 05C5068R1068W20
keywords Cayley graphsspectral sparsificationabelian groupscode sparsifiersvolume estimatescharacter symmetrypartial coloring
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 shows that any weighted Cayley graph on a finite abelian group G can be replaced by another weighted Cayley graph on the same group that uses only O(ε⁻² log |G|) generators and still approximates every Laplacian eigenvalue of the original graph within a (1 ± ε) factor. The size bound is optimal: some abelian Cayley graphs simply cannot be sparsified further while remaining Cayley. When the group is the vector space F_{2}^{n} the same statement yields optimally sized sparsifiers for linear codes, removing the extra polylog factors that earlier algorithms left behind. The argument works by iteratively shrinking the generating set; each shrinkage step is possible once a certain convex body (the sparsification polytope) is shown to have large volume, and that volume lower bound follows from a short symmetry argument with group characters.

Core claim

For every finite abelian group G, every weighted Cayley graph Cay(G, S, w) admits an ε-spectral sparsifier that is itself a weighted Cayley graph and uses only O(ε⁻² log |G|) generators; the sparsifier can be found in randomized polynomial time in |G| and 1/ε. The bound matches known lower bounds and is therefore optimal for the abelian setting.

What carries the argument

The sparsification polytope Q_ε of admissible re-weightings, together with the elementary volume lower bound Vol(Q^{+}_{0}) ≥ (1/|G|) · 2^{|S|} obtained from character symmetry and a rotationally invariant measure on the plane. Large volume plus Minkowski-type partial-coloring theorems guarantee a constant-factor support reduction at each iteration.

Load-bearing premise

The volume of the one-sided sparsification polytope is at least a constant fraction of the volume of the unit cube, which the paper derives from character symmetry; if that volume estimate fails for some groups or weights, the iterative reduction no longer works.

What would settle it

Exhibit a concrete abelian group G and a weight function w on a generating set S for which the measured volume of the corresponding polytope Q^{+}_{0} is o(2^{|S|}/|G|), or produce an abelian Cayley graph that provably requires ω(log |G|) generators in every ε-sparsifier.

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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 / 5 minor

Summary. The paper proves that every weighted Cayley graph over a finite abelian group G admits an ε-spectral sparsifier that is itself a weighted Cayley graph on O(ε^{-2} log |G|) generators (Theorem 1.5), which is optimal by a cited lower bound. The argument reduces sparsification to a volume lower bound on a natural sparsification polytope Q_ε via the partial-coloring theorems of Reis–Rothvoss; the novel step is an elementary character-symmetry argument (Theorem 3.3 / Lemma 4.2) showing Vol(Q^{+}_0) ≥ (1/N)·2^{|S|}. An iterative pruning schedule with geometrically decreasing ε_t then yields the claimed size while controlling multiplicative error. As a corollary one obtains optimal-size code sparsifiers for F_2-linear codes, removing the polylog(n) factor of prior work.

Significance. The result settles the optimal size of abelian Cayley sparsifiers and immediately improves the best known code sparsifiers for F_2-linear codes to the information-theoretic bound O(n/ε^{2}). The proof is short, elementary once the Reis–Rothvoss black boxes are granted, and cleanly isolates the new volume estimate that exploits character symmetry. The same bound recovers the Alon–Roichman theorem for abelian groups as a special case and clarifies the remaining barrier (deterministic poly(n)-time construction) for optimal ε-biased codes. These contributions are of clear interest to spectral graph theory, coding theory and discrepancy.

minor comments (5)
  1. The manuscript repeatedly cites Theorems 3.1 and 3.2 of Reis–Rothvoss [RR26] as black boxes. A one-sentence restatement of the precise hypotheses (central symmetry, volume lower bound on all sections, etc.) would make the reduction self-contained for readers who have not yet read that preprint.
  2. In the error-accumulation argument of Section 4 the constant C_0 is required only to satisfy C_0 ≫ α^{2}; an explicit numerical choice (or a short calculation showing that any C_0 > 4α^{2} works) would remove the last “large enough” quantifier.
  3. Fact 2.3 supplies a concrete density for the rotationally invariant measure; a parenthetical remark that any rotationally invariant measure with uniform marginals would suffice would clarify that the particular formula is only for concreteness.
  4. The final symmetrization step (averaging w(g) and w(-g)) is correct but appears only at the very end; a sentence earlier in the iterative construction noting that one may always restore symmetry at a factor-2 cost would improve readability.
  5. A few typographical inconsistencies remain (e.g., “polylog(n)loss” missing a space, occasional missing spaces around “ε_t”). A light copy-edit pass would polish the presentation.

Circularity Check

0 steps flagged

No significant circularity: volume lower bound is derived from character symmetry and an explicit rotationally-invariant measure, then fed into independent Reis-Rothvoss theorems.

full rationale

The central claim (Theorem 1.5) is obtained by iteratively applying the Reis-Rothvoss partial-coloring theorems (Theorems 3.1 and 3.2, cited from [RR26]) to the sparsification polytope Q_ε. Those theorems are external black boxes whose hypotheses are verified by an independent volume estimate: Theorem 3.3 asserts Vol_S(Q^{+}_{0}) ≥ (1/N)·2^{|S|}. The estimate is proved from first principles in Lemma 4.2 and Proposition 4.1 by constructing a bG-invariant process via the rotationally-invariant measure of Fact 2.3, applying elementary averaging, and transferring the resulting probability bound to the real marginals. No parameter is fitted to data; no uniqueness theorem is imported from the authors’ prior work; the only self-citations ([BKLM26] for optimality, [KPS24] for the code-sparsifier corollary) are non-load-bearing. The derivation is therefore self-contained against its external benchmarks and exhibits no circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 1 invented entities

The argument rests on standard abelian Fourier analysis, two black-box theorems from Reis-Rothvoss, and one elementary probabilistic fact about rotationally invariant measures. No free parameters are fitted; the only invented objects are the sparsification polytopes Q_ε that are defined directly from the Laplacian eigenvalues.

axioms (4)
  • standard math Characters of a finite abelian group form an eigenbasis of any Cayley Laplacian with eigenvalues ∑_s w(s)(1-Re χ(s)) (Fact 2.1).
    Classical representation theory of abelian groups; used throughout to reduce spectral approximation to scalar inequalities.
  • domain assumption Reis-Rothvoss partial-coloring theorem: any centrally symmetric convex body K \subseteq [-1,1]^m of volume ≥ c^m contains a point in αK with at least m/4 coordinates equal to -1 (Theorem 3.1).
    Black-box citation of [RR26, Thm 7]; supplies the iterative support reduction.
  • domain assumption Reis-Rothvoss section-volume theorem: sufficiently large volumes of all coordinate sections of K imply Vol(K igcap -K) ≥ 2^{-5m} (Theorem 3.2).
    Black-box citation of [RR26, Thm 16]; converts the asymmetric volume bound into a symmetric one.
  • standard math Existence of a rotationally invariant probability measure on R^{2} whose marginal is uniform on [-1,1] (Fact 2.3).
    Elementary measure-theory fact used to construct the group-invariant process in Lemma 4.2.
invented entities (1)
  • sparsification polytope Q_ε (and its one-sided versions Q^{+}_ε, Q^{-}_ε) no independent evidence
    purpose: Encodes all weight adjustments that keep the Cayley Laplacian within (1±ε) of the original; volume lower bounds on these bodies drive the partial-coloring argument.
    Defined directly from the eigenvalue inequalities; no independent physical or combinatorial existence claim beyond the volume calculation.

pith-pipeline@v1.1.0-grok45 · 16844 in / 2777 out tokens · 22582 ms · 2026-07-10T10:26:20.009071+00:00 · methodology

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Cite this review

Pith. "Pith review of Optimal Sparsifiers for Abelian Cayley Graphs." pith.science (2026). https://pith.science/paper/ZZBBSD6F

@misc{pith2026260708261,
  author       = {Pith},
  title        = {Pith review of: Optimal Sparsifiers for Abelian Cayley Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZZBBSD6F}},
  note         = {Machine review of arXiv:2607.08261}
}
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read the original abstract

We prove that for every Cayley graph $\mathcal{G}$ over any finite abelian group $G$, there is a weighted Cayley graph with $O(\log |G|)$ generators that is a spectral sparsifier for $\mathcal{G}$. This bound is optimal. Applying our bound to the group $G = \mathbb{F}_2^n$, yields, as a corollary, $O(n/\varepsilon^2)$-sized code sparsifiers for $\mathbb{F}_2$-linear codes, improving on the work of Khanna, Putterman and Sudan (SODA'24) who obtained a similar result with an additional $\mathrm{polylog}(n)$ loss. Our proof is strongly inspired by a recent work of Reis and Rothvoss for the construction of $\ell_1$-sparsifiers. Following their work, the abelian Cayley sparsification problem can be reduced to establishing a lower bound for the volume of a certain natural convex body. This volume bound follows from a short, elementary argument that relies on character symmetry.

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Works this paper leans on

23 extracted references · 23 canonical work pages · 1 internal anchor

  1. [1]

    On fully dynamic graph sparsifiers

    [ADK+16] Ittai Abraham, David Durfee, Ioannis Koutis, Sebastian Krinninger, and Richard Peng. On fully dynamic graph sparsifiers. In Irit Dinur, editor,IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, Hyatt Regency, New Brunswick, New Jersey, USA, October 9-11, 2016, pages 335–344. IEEE Computer Society, 2016.doi:10.1109/FOCS.2016.44. (pg

    work page doi:10.1109/focs.2016.44 2016

  2. [2]

    [AGM12] KookJinAhn,SudiptoGuha,andAndrewMcGregor.Graphsketches: sparsification, spanners, and subgraphs. In Michael Benedikt, Markus Krötzsch, and Maurizio Lenzerini,editors,Proceedingsofthe31stACMSIGMOD-SIGACT-SIGARTSymposium on Principles of Database Systems, PODS 2012, Scottsdale, AZ, USA, May 20-24, 2012, pages 5–14. ACM, 2012.doi:10.1145/2213556.2213560. (pg

    work page doi:10.1145/2213556.2213560 2012

  3. [3]

    doi:10.1145/3717823.3718212

    Association for Computing Machinery. doi:10.1145/3717823.3718212. (pg

    work page doi:10.1145/3717823.3718212

  4. [4]

    Association for Computing Machinery.doi:10.1145/237814.237827. (pg

    work page doi:10.1145/237814.237827

  5. [5]

    Kothari, Yang P

    [BKLM26] Arpon Basu, Pravesh K. Kothari, Yang P. Liu, and Raghu Meka.Sparsifying Sums of Positive Semidefinite Matrices, pages 6042–6064. 2026.doi:10.1137/1. 9781611978971.216. (pg. 2,

    work page doi:10.1137/1 2026

  6. [6]

    Spielman, and Nikhil Srivastava

    [BSS14] Joshua Batson, Daniel A. Spielman, and Nikhil Srivastava. Twice-ramanujan sparsifiers.SIAM Review, 56(2):315–334, 2014.doi:10.1137/130949117. (pg. 1,

    work page doi:10.1137/130949117 2014

  7. [7]

    Sparsification of Binary CSPs.SIAM Journal on Discrete Mathematics, 34(1):825–842, January 2020.doi:10.1137/19M1242446

    [BŽ20] Silvia Butti and Stanislav Živný. Sparsification of Binary CSPs.SIAM Journal on Discrete Mathematics, 34(1):825–842, January 2020.doi:10.1137/19M1242446. (pg

    work page doi:10.1137/19m1242446 2020

  8. [8]

    Roberson , editor =

    [CKN20] Yu Chen, Sanjeev Khanna, and Ansh Nagda. Near-linear size hypergraph cut sparsifiers. In Sandy Irani, editor,61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 61–72. IEEE, 2020.doi:10.1109/FOCS46700.2020.00015. (pg

    work page doi:10.1109/focs46700.2020.00015 2020

  9. [9]

    doi:10.1137/15M1046186. (pg

    work page doi:10.1137/15m1046186

  10. [10]

    Sparsifying cayley graphs on every group

    [HLM+26] Jun-TingHsieh,DanielZ.Lee,SidhanthMohanty,AaronPutterman,andRachelYun Zhang. Sparsifying cayley graphs on every group. In Kasper Green Larsen and BarnaSaha,editors,Proceedingsofthe2026AnnualACM-SIAMSymposiumonDiscrete Algorithms, SODA 2026, Vancouver, BC, Canada, January 11-14, 2026, pages 6029–6041. SIAM, 2026.doi:10.1137/1.9781611978971.215. (pg

    work page doi:10.1137/1.9781611978971.215 2026

  11. [11]

    Liu, and Aaron Sidford

    [JLS23] Arun Jambulapati, Yang P. Liu, and Aaron Sidford. Chaining, group leverage score overestimates, and fast spectral hypergraph sparsification. In Barna Saha and 10 Rocco A. Servedio, editors,Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 196–206. ACM, 2023.doi:10.1145/3564246...

    work page doi:10.1145/3564246.3585136 2023

  12. [12]

    Near-Optimal Cayley Expanders for Abelian Groups

    [JM21] Akhil Jalan and Dana Moshkovitz. Near-Optimal Cayley Expanders for Abelian Groups. InMikołajBojańczykandChandraChekuri,editors,41stIARCSAnnualCon- ference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021), volume 213 ofLeibniz International Proceedings in Informatics (LIPIcs), pages 24:1–24:23, Dagstuhl, Germany,

    work page 2021

  13. [13]

    FSTTCS.2021.24,doi:10.4230/LIPIcs.FSTTCS.2021.24

    Schloss Dagstuhl – Leibniz-Zentrum für Infor- matik.URL: https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs. FSTTCS.2021.24,doi:10.4230/LIPIcs.FSTTCS.2021.24. (pg

    work page doi:10.4230/lipics 2021

  14. [14]

    Sketchingcutsingraphsandhypergraphs

    [KK15] DmitryKoganandRobertKrauthgamer. Sketchingcutsingraphsandhypergraphs. In Tim Roughgarden, editor,Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, ITCS 2015, Rehovot, Israel, January 11-13, 2015, pages 367–376. ACM, 2015.doi:10.1145/2688073.2688093. (pg

    work page doi:10.1145/2688073.2688093 2015

  15. [15]

    Spectral hypergraph sparsifiers of nearly linear size

    [KKTY21a] Michael Kapralov, Robert Krauthgamer, Jakab Tardos, and Yuichi Yoshida. Spectral hypergraph sparsifiers of nearly linear size. In62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 1159–1170. IEEE, 2021.doi:10.1109/FOCS52979.2021.00114. (pg

    work page doi:10.1109/focs52979.2021.00114 2021

  16. [16]

    Towards tight bounds for spectral sparsification of hypergraphs

    [KKTY21b] MichaelKapralov, RobertKrauthgamer, JakabTardos, andYuichiYoshida. Towards tight bounds for spectral sparsification of hypergraphs. In Samir Khuller and Virginia Vassilevska Williams, editors,STOC ’21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 598–611. ACM, 2021.doi:10.1145/3406325.345...

    work page doi:10.1145/3406325.3451061 2021

  17. [17]

    Code sparsification and its applications

    [KPS24] Sanjeev Khanna, Aaron Putterman, and Madhu Sudan. Code sparsification and its applications. In David P. Woodruff, editor,Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, Alexandria, VA, USA, January 7-10, 2024, pages 5145–5168. SIAM, 2024.doi:10.1137/1.9781611977912.185. (pg. 1, 2,

    work page doi:10.1137/1.9781611977912.185 2024

  18. [18]

    Association for Computing Machinery.doi:10.1145/3717823.3718205. (pg. 2,

    work page doi:10.1145/3717823.3718205

  19. [19]

    [Lee23] James R. Lee. Spectral hypergraph sparsification via chaining. In Barna Saha and Rocco A. Servedio, editors,Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 207–218. ACM, 2023.doi:10.1145/3564246.3585165. (pg

    work page doi:10.1145/3564246.3585165 2023

  20. [20]

    URL:https: //arxiv.org/abs/2606.28147,arXiv:2606.28147. (pg. 1, 5, 6,

    work page internal anchor Pith review Pith/arXiv arXiv

  21. [21]

    Spielman and Nikhil Srivastava

    [SS11] Daniel A. Spielman and Nikhil Srivastava. Graph sparsification by effective resis- tances.SIAMJournalonComputing,40(6):1913–1926,2011. doi:10.1137/080734029. (pg. 1,

    work page doi:10.1137/080734029 1913

  22. [22]

    doi:10.1145/1007352.1007372

    Association for Computing Machinery. doi:10.1145/1007352.1007372. (pg

    work page doi:10.1145/1007352.1007372

  23. [23]

    Spielman and Shang-Hua Teng

    [ST11] Daniel A. Spielman and Shang-Hua Teng. Spectral sparsification of graphs.SIAM Journal on Computing, 40(4):981–1025, 2011.doi:10.1137/08074489X. (pg

    work page doi:10.1137/08074489x 2011