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Find k disjoint cycles of distinct lengths or delete O(k^6) vertices

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2026-07-09 23:49 UTC pith:E2JWU3DM

load-bearing objection First Erdős-Pósa theorem for cycles of distinct lengths; clean modular proof, honest about the gap between O(k^6 polylog(k)) and the conjectured O(k log k).

arxiv 2607.06869 v1 pith:E2JWU3DM submitted 2026-07-08 math.CO cs.DM

An ErdH{o}s-P\'osa theorem for cycles and faces of distinct lengths

J. Pascal Gollin , Maximilian Gorsky , Meike Hatzel , Kevin Hendrey , Tony Huynh , Caleb McFarland , Marek Soko{\l}owski , Sebastian Wiederrecht
show 1 more author
Paul Wollan
This is my paper
classification math.CO cs.DM
keywords Erdős-Pósa theoremcycle spectrumvertex-disjoint cyclestreewidthprismgraph minorssurface embeddingadditive combinatorics
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 proves an Erdős-Pósa-type theorem for cycles of distinct lengths. The classic Erdős-Pósa theorem says that every graph either contains k vertex-disjoint cycles or has a small set of vertices whose removal kills all cycles. Here the requirement is stronger: the k cycles must all have different lengths. The authors show that every graph either contains k vertex-disjoint cycles of pairwise distinct lengths, or there is a set of O(k^6 polylog(k)) vertices whose removal leaves a graph with at most k-1 different cycle lengths. The proof splits into two regimes. For graphs of low treewidth, a combinatorial lemma on tree-decompositions provides the packing or the hitting set. For graphs of high treewidth, the authors use the fact that high treewidth forces a subdivision of a large prism (a cylinder-like graph), and they prove that every subdivision of a 6k-prism contains at least k distinct cycle lengths via a weight argument on symmetric differences of cycles. A high-treewidth partition theorem then yields k disjoint subgraphs each contributing distinct lengths. The paper also proves an analogous result for coloured faces of graphs embedded on surfaces, and constructs lower bounds using additive combinatorics showing that subdivided ladders can have surprisingly few cycle lengths.

Core claim

The central discovery is that the Erdős-Pósa duality between packing and covering extends to the setting where cycles must have distinct lengths, with the cycle spectrum Λ(G) (the set of all cycle lengths appearing in G) serving as the covering certificate. The key mechanism is Lemma 2.4: every subdivision of the 6k-prism contains at least k distinct cycle lengths. The proof partitions the 3k square cycles of the prism into three classes based on whether their symmetric difference with one of the two base cycles has positive, negative, or zero weight differential. A pigeonhole argument guarantees one class has size at least k, and within that class, nested symmetric differences produce a mon

What carries the argument

The proof combines three ingredients: (1) a tree-decomposition lemma (Lemma 2.1) showing that coloured subtrees of a tree either contain k disjoint members of distinct colours or admit a hitting set of size k-1; (2) the prism subdivision lemma (Lemma 2.4), which uses a weight argument on symmetric differences of cycles in the 6k-prism to guarantee k distinct cycle lengths; and (3) the Chekuri-Chuzhoy high-treewidth partition theorem, which decomposes a high-treewidth graph into k disjoint subgraphs each of treewidth Ω(k^2), each containing a subdivided 6k-prism. The lower bound construction uses a result of Hennecart-Robert-Yudin from additive combinatorics about sets whose sumset is much sm

Load-bearing premise

The load-bearing lemma is that every subdivision of the 6k-prism contains at least k distinct cycle lengths. The proof is short and uses a pigeonhole argument on the signs of weight differentials of symmetric differences between square cycles and a base cycle. If this lemma failed, the high-treewidth case of the main theorem would collapse.

What would settle it

Construct a subdivision of the 6k-prism with fewer than k distinct cycle lengths, which would refute Lemma 2.4 and break the proof of Corollary 2.6 and hence the main theorem. The authors' lower bound construction (Theorem 1.5) shows that ladders can have few cycle lengths, but prisms have more structure (two base cycles connected by rungs), and the lemma exploits this.

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

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If this is right

  • The theorem provides a structural certificate for graphs lacking many vertex-disjoint cycles of distinct lengths: the cycle spectrum of the remaining graph is small, which is a different kind of dual certificate than the usual forest condition in the classic Erdős-Pósa theorem.
  • The lower bound construction using additive combinatorics (Theorem 1.5) shows that subdivided ladders can have as few as k^α cycle lengths for α ≈ 0.79, which means the prism-based approach cannot be directly improved to give better bounds without fundamentally different subgraph certificates.
  • The surface embedding result (Theorem 1.4) extends the packing-covering duality to coloured faces at prescribed distances, with bounds depending polynomially on the genus, which could find applications in topological graph theory problems involving face packings.
  • The conjectured optimal bound of O(k log k) for the hitting set size, if true, would simultaneously match the lower bound from the classic Erdős-Pósa theorem and the lower bound on the cycle spectrum, making the result tight in both parameters.

Where Pith is reading between the lines

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

  • The fact that the problem does not reduce to a standard hypergraph packing-covering problem (as the authors note) suggests that the cycle spectrum serves as a genuinely new type of dual certificate, and the framework might extend to other packing problems where the objects to be packed are constrained by a global property like distinctness.
  • The gap between the current O(k^6 polylog(k)) bound and the conjectured O(k log k) is substantial. The bottleneck is the high-treewidth regime and the use of the 6k-prism. A subgraph certificate that guarantees k distinct cycle lengths at lower treewidth than Ω(k^2) would directly improve the bound.
  • The additive combinatorics construction showing that sumset-difference set gaps produce ladders with few cycle lengths hints at a deeper connection between additive structure in graphs and cycle length diversity, which could be a productive direction for related extremal questions.

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 / 6 minor

Summary. The paper proves an Erdős-Pósa-type theorem for vertex-disjoint cycles of distinct lengths: for every k, every graph G either contains k vertex-disjoint cycles of pairwise different lengths, or admits a hitting set X of size O(k^6 polylog(k)) such that G−X has at most k−1 cycle lengths (Theorem 1.3). An analogous result for coloured faces of embedded graphs is also proved (Theorem 1.4), and a construction from additive combinatorics shows that subdivided ladders can have surprisingly small cycle spectra (Theorem 1.5), suggesting barriers to improving the main bound. The proof of the main theorem proceeds via a clean modular chain: a coloured-subtree lemma (Lemma 2.1) handles bounded-treewidth graphs; a prism cycle-length lemma (Lemma 2.4) combined with the Chekuri-Chuzhoy high-treewidth partition theorem handles the high-treewidth case. Lower bounds (Theorems 3.1, 3.2) show that both the k−1 bound on remaining cycle lengths and an Ω(k log k) lower bound on the hitting set size are best possible.

Significance. This is a significant contribution to the Erdős-Pósa literature. The problem of packing cycles of distinct lengths does not fit into the standard packing-covering hypergraph framework, making the dual certificate (bounding the cycle spectrum after deletion) a novel conceptual contribution. The proof is modular and relies on well-established external results (Birmelé-Bondy-Reed, Chekuri-Chuzhoy, Eppstein, Hennecart-Robert-Yudin) without circularity. The lower bound construction using additive combinatorics (Theorem 1.5) is a notable feature: it provides a concrete barrier showing that the prism-based approach cannot be straightforwardly improved via ladders, and it identifies a genuine obstruction. The algorithmic versions (Appendices A, B) add practical value. The surface embedding result (Theorem 1.4) extends the framework to a natural topological setting with a clean proof via radial graphs and local treewidth.

minor comments (6)
  1. The abstract states the hitting set bound as O(k^2 d g) while Theorem 1.4 states O(k^2 d(g+1)). These should be made consistent; the (g+1) form is more precise for g=0.
  2. In the proof of Lemma 2.4, the notation w(C1△Ci) is introduced without explicitly defining the symmetric difference of cycles as an edge set. A brief clarifying sentence would help the reader.
  3. Theorem 5.5 introduces the constant D = C/(1+C) < 0.4403, but the value of C is defined earlier in Section 5 as log 2 / log(1+√2) < 0.7865. A forward reference or restatement would improve readability.
  4. The reference [CM26] (Chudnovsky and Maier) is cited as a 2026 arXiv preprint; the authors should verify and update this reference if a published version is available.
  5. In Theorem 1.3, the algorithmic runtime O(18^{k^5} poly(k) |V(G)|^7) is stated but the derivation is deferred to Appendix A. A brief forward pointer in the main text would be helpful.
  6. Conjecture 6.1 proposes an optimal bound of O(k log k), which is a natural and important target. This conjecture is currently somewhat buried at the end of Section 6; the authors might consider highlighting it more prominently.

Circularity Check

0 steps flagged

No circularity found; derivation chain is self-contained against external benchmarks

full rationale

The paper's main theorem (Theorem 1.3) derives from a clean chain: Lemma 2.1 (proved from first principles via induction on trees) → Lemma 2.2 (application to tree-decompositions), combined with Corollary 2.6, which chains Theorem 2.3 (Birmelé-Bondy-Reed 2007, external), Lemma 2.4 (proved from first principles via a weight argument on prism symmetric differences), and Theorem 2.5 (Chekuri-Chuzhoy 2013, external). Theorem 2.7 simply combines Lemma 2.2 and Corollary 2.6. The lower bounds (Theorems 3.1, 3.2) are elementary constructions. Theorem 1.4 derives from Lemma 4.4 (proved in Appendix B) and Theorem 4.5 (proved from first principles), using Theorem 4.3 (Eppstein 2000, external) and Theorem 4.1 (Four Color Theorem + Heawood, external). Theorem 1.5 uses Theorem 5.3 (Hennecart-Robert-Yudin 1999, external). The algorithmic version (Appendix A) uses Theorem A.1 (Visser-Bodlaender 2025, external) and Theorem B.1 (Korhonen-Lokshtanov 2023, external). No load-bearing step reduces to a self-citation: all external results cited in the derivation chain are by non-overlapping author sets. The self-citations that do appear ([AGHK25], [CGH+26], [GHH26], [GHK+24], [GHK+25], [GKKW24]) occur only in the introductory literature survey and Section 6's discussion of extensions, never in the proof of any main result. No parameter is fitted to data and renamed as a prediction. No result is defined in terms of its own conclusion. The derivation is genuinely self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No new entities are introduced. The paper works entirely within standard graph-theoretic objects (cycles, tree-decompositions, prisms, ladders, radial graphs, surfaces).

axioms (5)
  • standard math Chekuri-Chuzhoy high treewidth partition theorem (Theorem 2.5): graphs of treewidth k contain h vertex-disjoint subgraphs of treewidth ≥ r when hr² ≤ k/polylog(k).
    External, independently refereed result from 2013. Used in Corollary 2.6 to partition high-treewidth graphs.
  • standard math Birmelé-Bondy-Reed prism theorem (Theorem 2.3): graphs of treewidth Ω(k²) contain a subdivision of the k-prism.
    External result from 2007. Used in Corollary 2.6 to find prism subdivisions in each high-treewidth piece.
  • standard math Eppstein's local treewidth bound (Theorem 4.3): graphs of genus g with radius r have treewidth O(r(g+1)).
    External result. Used in Theorem 1.4 proof to bound local treewidth of radial graphs.
  • standard math Hennecart-Robert-Yudin theorem (Theorem 5.3): for every n and α > C ≈ 0.7865, there exists A ⊆ ℤ with |A| ≥ n and |A+A| ≤ |A−A|^α.
    External result from additive combinatorics (1999). Used in Theorem 1.5 to construct ladders with few cycle lengths.
  • standard math Four Colour Theorem and Heawood's theorem (Theorem 4.1): graphs on surfaces of Euler genus g are ⌊(7+√(1+24g))/2⌋-colourable.
    Classical results. Used in Theorem 4.2 for the edge-disjoint face packing warm-up.

pith-pipeline@v1.1.0-glm · 25448 in / 4471 out tokens · 186567 ms · 2026-07-09T23:49:34.089791+00:00 · methodology

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Cite this review

Pith. "Pith review of An Erd\H{o}s-P\'osa theorem for cycles and faces of distinct lengths." pith.science (2026). https://pith.science/paper/E2JWU3DM

@misc{pith2026260706869,
  author       = {Pith},
  title        = {Pith review of: An Erd\Hos-P\'osa theorem for cycles and faces of distinct lengths},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E2JWU3DM}},
  note         = {Machine review of arXiv:2607.06869}
}
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read the original abstract

We show that for every $k \in \mathbb{N}$, every graph $G$ contains $k$ vertex-disjoint cycles of different lengths, or there exists a set $X \subseteq V(G)$ with $|X| \in \mathcal{O}(k^6\mathsf{polylog}(k))$ such that $G-X$ has at most $k-1$ cycle lengths. We also prove analogous results for facial lengths of embedded graphs. Let $G$ be a graph with a closed 2-cell embedding $\psi$ on a surface $\Sigma$ of Euler genus $g$, let $c$ be a colouring of the faces $\mathcal{F}(\psi)$ of $\psi$, and let $R(G,\psi)$ be the radial graph of $(G, \psi)$. Then there exist $k$ faces $F_1, \ldots , F_k \in \mathcal{F}(\psi)$ that are given pairwise distinct colours by $c$ and are pairwise at distance at least $d$ in $\psi$, or there exists a set $X \subseteq V(G)$ of order at most $\mathcal{O}(k^2dg)$ such that $|\{ c(F) \mid F \in \mathcal{F}(\psi) \text{ and } V(F) \cap \bigcup_{x \in X} N^d_{R(G,\psi)}(x) = \emptyset \}| \leq k(k+2)$. Finally, using a result from additive combinatorics, we show that there are subdivided ladders with only a small number of cycle lengths. This suggests that it may be difficult to improve our bounds.

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Works this paper leans on

19 extracted references · 19 canonical work pages · 4 internal anchors

  1. [1]

    Kothari, Yang P

    [AGHK25] Jungho Ahn, J. Pascal Gollin, Tony Huynh, and O-joung Kwon. A coarse Erdős-Pósa theorem. InProceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3363–3381. SIAM, Philadelphia, PA, 2025.doi:10.1137/1. 9781611978322.109. 16 [AH76a] K. Appel and W. Haken. The existence of unavoidable sets of geographically good confi...

    work page doi:10.1137/1 2025

  2. [2]

    [BBR07a] E

    doi:10.1215/ijm/1256049012. [BBR07a] E. Birmelé, J. A. Bondy, and B. A. Reed. Brambles, prisms and grids. InGraph theory in Paris, Trends Math., pages 37–44. Birkhäuser, Basel,

    work page doi:10.1215/ijm/1256049012

  3. [3]

    [BBR07b] Etienne Birmelé, J

    URL:https: //doi.org/10.1007/978-3-7643-7400-6_4,doi:10.1007/978-3-7643-7400-6\_4. [BBR07b] Etienne Birmelé, J. Adrian Bondy, and Bruce A. Reed. The Erdős-Pósa property for long circuits.Combinatorica, 27(2):135–145, 2007.doi:10.1007/s00493-007-0047-0. [BGS22] Matija Bucić, Lior Gishboliner, and Benny Sudakov. Cycles of many lengths in Hamiltonian graphs....

    work page doi:10.1007/978-3-7643-7400-6_4 2007

  4. [4]

    [CC13] Chandra Chekuri and Julia Chuzhoy

    doi:10.1016/0095-8956(71)90016-5. [CC13] Chandra Chekuri and Julia Chuzhoy. Large-treewidth graph decompositions and applications,

    work page doi:10.1016/0095-8956(71)90016-5

  5. [5]

    Large-Treewidth Graph Decompositions and Applications

    URL:https://arxiv.org/abs/1304.1577,arXiv:1304.1577. [CGH+26] Rutger Campbell, J. Pascal Gollin, Meike Hatzel, O-joung Kwon, Rose McCarty, Sang-il Oum, and Sebastian Wiederrecht. The Erdős-Pósa property for circle graphs as vertex-minors. InProceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 4930–4952. SIAM, Philadelphia...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1137/1.9781611978971.178 2026

  6. [6]

    [CM26] Maria Chudnovsky and Ilya Maier

    International Workshop on Computing by Graph Transformation (Bordeaux, 1991).doi:10.1016/0304-3975(93) 90064-Z. [CM26] Maria Chudnovsky and Ilya Maier. Induced cycles of many lengths.arXiv preprint arXiv:2602.06874,

    work page doi:10.1016/0304-3975(93 1991

  7. [7]

    [CvBJU20] Wouter Cames van Batenburg, Gwenaël Joret, and Arthur Ulmer

    doi:10.1016/j.jctb.2020.09.010. [CvBJU20] Wouter Cames van Batenburg, Gwenaël Joret, and Arthur Ulmer. Erdős-Pósa from ball packing.SIAM J. Discrete Math., 34(3):1609–1619, 2020.doi:10.1137/19M1309225. [DJMM24] Vida Dujmović, Gwenaël Joret, Piotr Micek, and Pat Morin. Erdős-Pósa property of cycles that are far apart.Preprint,

    work page doi:10.1016/j.jctb.2020.09.010 2020

  8. [8]

    Erd\H{o}s--P\'{o}sa property of cycles that are far apart

    URL:https://arxiv.org/abs/2412.13893. [EP65] Paul Erdős and Lajos Pósa. On Independent Circuits Contained in a Graph.Canadian Journal of Mathematics, 17:347–352, 1965.doi:10.4153/CJM-1965-035-8. [Epp00] David Eppstein. Diameter and Treewidth in Minor-Closed Graph Families.Algorithmica, 27(3):275–291, June 2000.doi:10.1007/s004530010020. [GHH26] Maximilian...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.4153/cjm-1965-035-8 1965

  9. [9]

    [Gor24] Maximilian Gorsky.The Structure of (Even) Directed Cycles

    ACM.doi:10.1145/3618260.3649682. [Gor24] Maximilian Gorsky.The Structure of (Even) Directed Cycles. PhD thesis, Technische Universität Berlin, September 2024.doi:10.14279/depositonce-21276. [GP25] Agelos Georgakopoulos and Panos Papasoglu. Graph minors and metric spaces.Com- binatorica, 45(3):Paper No. 33, 29, 2025.doi:10.1007/s00493-025-00150-6. 18 [Gro0...

    work page doi:10.1145/3618260.3649682 2024

  10. [10]

    A Unified Erdős–Pósa Theorem for Con- strained Cycles.Combinatorica, 39(1):91–133, February 2019.doi:10.1007/s00493- 017-3683-z

    [HJW19] Tony Huynh, Felix Joos, and Paul Wollan. A Unified Erdős–Pósa Theorem for Con- strained Cycles.Combinatorica, 39(1):91–133, February 2019.doi:10.1007/s00493- 017-3683-z. [HK24] Tony Huynh and O-Joung Kwon. On the Erdős-Pósa property for long holes inC4-free graphs.SIAM J. Discrete Math., 38(1):19–42, 2024.doi:10.1137/21M1435239. [HRY99] François H...

    work page doi:10.1007/s00493- 2019

  11. [11]

    Polynomial Bounds in the Apex Minor Theorem

    URL:https://www.numdam.org/item/AST_ 1999__258__173_0/. [HW25] Kevin Hendrey and David R. Wood. Polynomial Bounds in the Apex Minor Theorem, March 2025.arXiv:2503.04228,doi:10.48550/arXiv.2503.04228. [KK18] Naonori Kakimura and Ken-ichi Kawarabayashi. The Erdős-Pósa property for edge- disjoint immersions in 4-edge-connected graphs.J. Combin. Theory Ser. B...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2503.04228 2025

  12. [12]

    [KKKX25] Ken-ichi Kawarabayashi, Stephan Kreutzer, O-joung Kwon, and Qiqin Xie

    doi:10.1016/j.jctb.2020.05.002. [KKKX25] Ken-ichi Kawarabayashi, Stephan Kreutzer, O-joung Kwon, and Qiqin Xie. A half- integral Erdős-Pósa theorem for directed odd cycles.Journal of Combinatorial Theory, Series B, 172:115–145, May 2025.doi:10.1016/j.jctb.2024.12.008. [KKM11] Naonori Kakimura, Ken-ichi Kawarabayashi, and Dániel Marx. Packing cycles throug...

    work page doi:10.1016/j.jctb.2020.05.002 2020

  13. [13]

    [Lic14] Nicolas Lichiardopol

    ACM.doi:10.1145/3564246.3585245. [Lic14] Nicolas Lichiardopol. Proof of a conjecture of Henning and Yeo on vertex-disjoint directed cycles.SIAM J. Discrete Math., 28(3):1618–1627,

    work page doi:10.1145/3564246.3585245

  14. [14]

    Lipiecki, K

    doi:10.1137/ 130922653. [Liu21] Chun-Hung Liu. Packing and covering immersions in 4-edge-connected graphs.Journal of Combinatorial Theory, Series B, 151:148–222, November 2021.doi:10.1016/j. jctb.2021.06.005. [Liu22] Chun-Hung Liu. Packing topological minors half-integrally.Journal of the London Mathematical Society, 106(3):2193–2267, October 2022.doi:10....

    work page doi:10.1016/j 2021

  15. [15]

    Journal of Combinatorial Theory, Series B , volume =

    doi:10.1016/j.jctb.2017.01.004. [MT01] Bojan Mohar and Carsten Thomassen.Graphs on Surfaces. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore,

    work page doi:10.1016/j.jctb.2017.01.004 2017

  16. [16]

    Obstructions to Erd\H{o}s-P\'osa Dualities for Minors

    [PPTW24] Christophe Paul, Evangelos Protopapas, Dimitrios M Thilikos, and Sebastian Wieder- recht. Obstructions to Erdős-Pósa Dualities for Minors, July 2024.arXiv:2407.09671. [Ree99] Bruce A Reed. Mangoes and Blueberries.Combinatorica, 19(2):267–296, February 1999.doi:10.1007/s004930050056. [RRST96] Bruce A Reed, Neil Robertson, Paul D. Seymour, and Robi...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s004930050056 2024

  17. [17]

    [Tho88] Carsten Thomassen

    doi:10.1016/j.dam.2016.12.025. [Tho88] Carsten Thomassen. On the presence of disjoint subgraphs of a specified type.Journal of Graph Theory, 12(1):101–111, March 1988.doi:10.1002/jgt.3190120111. [Tuz90] Zsolt Tuza. A conjecture on triangles of graphs.Graphs Combin., 6(4):373–380,

    work page doi:10.1016/j.dam.2016.12.025 2016

  18. [18]

    [TY23] Robin Thomas and Youngho Yoo

    doi:10.1007/BF01787705. [TY23] Robin Thomas and Youngho Yoo. Packing cycles in undirected group-labelled graphs. Journal of Combinatorial Theory, Series B, 161:228–267, July 2023.doi:10.1016/j. jctb.2023.02.011. [VB25] Jonne Visser and Hans L. Bodlaender. Deterministically countingk-paths and trees parameterized by treewidth in single-exponential time. In...

    work page doi:10.1007/bf01787705 2023

  19. [19]

    [Ver16] Jacques Verstraëte

    doi:10.4230/lipics.ipec.2025.20. [Ver16] Jacques Verstraëte. Extremal problems for cycles in graphs. InRecent trends in combinatorics, volume 159 ofIMA Vol. Math. Appl., pages 83–116. Springer, [Cham], 2016.doi:10.1007/978-3-319-24298-9_4. 20 A. An algorithmic proof of Theorem 1.3 We now present an algorithmic proof of Theorem 1.3. The key ingredient is t...

    work page doi:10.4230/lipics.ipec.2025.20 2025