Pith. sign in

REVIEW 1 major objections 3 minor 19 references

Where Treewidth and Pathwidth Diverge: Towards a Uniform Kernel for Pathwidth-$\eta$ Deletion

T0 review · 1 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Pathwidth-η Deletion admits uniform polynomial kernels on modulator parameterizations where Treewidth-η Deletion provably cannot.

desk verdict Strong uniform kernelization result for pathwidth deletion, but the β=0 case has a patchable gap that must be fixed. read the letter →

arxiv 2608.09800 v1 pith:UIQL35Q3 submitted 2026-08-10 cs.DS cs.CC

Ahmed Ghazy , Jakob Greilhuber , Tim A. Hartmann , Roohani Sharma This is my paper · ORCID
classification cs.DScs.CC MSC 05C8568Q2568R10
keywords kernelizationpathwidthuniformpolynomialkerneleliminationdistancevertexdeletionparameterizedcomplexitycoverparameterizationminor-minimalobstructions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Pathwidth-η Deletion asks whether deleting at most k vertices leaves a graph that can be laid out in a sequence of overlapping bags of size at most η+1. This paper establishes that, in kernelization, this problem behaves like Treedepth-η Deletion rather than like Treewidth-η Deletion: for every fixed η and β, the version parameterized by a modulator M whose removal leaves a graph of pathwidth η and elimination distance β to pathwidth-1 graphs admits a uniform polynomial kernel of size $g(\eta,\beta)\cdot|M|^{60}$, meaning the exponent 60 does not depend on η or β. The same machinery yields a cubic kernel in the vertex cover number, a parameterization for which the treewidth analogue is known to have no uniform polynomial kernel. The paper reads this as evidence that Pathwidth-η Deletion may admit a uniform polynomial kernel in the solution size k, placing it on the treedepth side of the treewidth-versus-treedepth divide.

What carries the argument

The argument is carried by interacting reduction engines. The first is stability in path decompositions: a connected subgraph is stable when every bag that meets it contains exactly the same boundary vertices, and a marking scheme over pairs of modulator neighbors guarantees that one of many similar components of $G-M$ can be deleted safely. The second is polishing sets and near-protrusions, which ensure that any optimal solution touches at most $3(\eta+1)$ vertices of each remaining low-width piece, together with virtual cliques that make modulator neighbors of a piece behave like a clique without adding edges. The third is a nice $G_{\mathrm{pw}\le 1}$-elimination tree of depth at most $\beta$, whose leaves are caterpillars and whose sibling and ancestor-type structure lets the algorithm identify an irrelevant edge incident to the modulator and delete it. A final protrusion-shrinking step lifts small bags into the modulator and bounds each remaining component by a constant depending only on η and β, yielding the uniform $|M|^{60}$ bound.

What would settle it

Fix β=1 and η=1 and apply the kernel exhaustively to the infinite family of irreducible inputs in $G^1_{\mathrm{pw}\le 1}$; if any instance has more than $g(1,1)\cdot|M|^{60}$ vertices, Main Theorem 1 is false. A more direct test of the load-bearing premise is to search for graphs of elimination distance 1 whose only polynomial-time-computable nice elimination trees have depth growing with the number of vertices, which would destroy the depth-sensitive component bounds.

Watch

Extended reading notes

Core claim

On its own terms, the paper proves that uniform kernelization for Pathwidth-η Deletion is governed by elimination distance to pathwidth-1 graphs rather than by pathwidth alone. Main Theorem 1 gives a kernel on $g(\eta,\beta)\cdot|M|^{60}$ vertices for Pathwidth-η Deletion on the class of graphs of pathwidth at most η with elimination distance at most β to pathwidth-1 graphs, and the kernel outputs an equivalent instance that is a minor of the input, with $k' = k$ whenever the input is a no-instance. The proof fully resolves one of the two bottlenecks, uniformly bounding the number of connected components of $G-M$; it then makes substantial progress on the other by bounding the size of a single component under the bounded-elimination-distance assumption, through degree reductions on modulator vertices and a caterpillar reduction. From this, Corollary 1.5 extracts a cubic kernel in the vertex cover number, and the minor-output property yields polynomial bounds on minor-minimal obstructions to k-apices of pathwidth-η graphs.

Load-bearing premise

Every depth-sensitive bound in the proof assumes that, for constant β, a nice $G_{\mathrm{pw}\le 1}$-elimination tree of depth at most β can be computed in polynomial time; if the cited algorithm returns only approximate elimination distance or a forest whose node subtrees are not connected, the sibling-and-stability arguments collapse.

Editorial extensions

If this is right

  • For each fixed η, Pathwidth-η Deletion parameterized by the size of a given vertex cover M has a kernel with $O(vc^3)$ vertices, so the lower bounds that rule out uniform treewidth kernels cannot apply to it.
  • Uniform polynomial kernels follow for the parameterizations listed in Corollary 1.4, including modulators to bounded-treedepth graphs, distance to pathwidth-1 graphs, distance to graphs of treedepth at most $\eta+1$, and pathwidth-η graphs without long paths.
  • Minor-minimal obstructions to the class of k-apices of pathwidth-≤η graphs have size polynomial in their vertex cover number, while the analogous treewidth classes have obstructions whose size depends on η in the exponent.
  • Because every reduction outputs a minor and keeps k unchanged on no-instances, the kernelization can be applied exhaustively and also serves as a combinatorial size bound for irreducible instances.
  • If the remaining component-size obstacles are overcome, the same strategy would yield a uniform polynomial kernel parameterized by the solution size k, as Conjecture 1.2 proposes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The cubic vertex-cover kernel suggests the $|M|^{60}$ exponent is far from tight; a refined version of the same marking scheme might yield a uniform kernel with a much smaller fixed exponent.
  • The route from minor-output kernels to obstruction-size bounds may be a general principle: for any minor-closed class, a uniform kernel parameterized by vertex cover should imply that minor-minimal obstructions to the k-apex class have polynomial size in the vertex cover number.
  • If Conjecture 1.2 is true but hard, the obstacle is isolated: the components that resist reduction are exactly those with unbounded treedepth and no long paths of degree-2 vertices, so future algorithms or counterexamples can focus on that family.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 3 minor

Summary. The paper studies Pathwidth-η Deletion parameterized by the size of a modulator M such that G−M has pathwidth at most η and elimination distance at most β to pathwidth-1 graphs. The main result (Main Theorem 1) is a kernel on g(η,β)·|M|^60 vertices with the exponent independent of η and β, and the kernel outputs a minor of the input graph. As corollaries, the paper obtains uniform polynomial kernels for parameterization by vertex cover number (Corollary 1.5), by distance to pathwidth-1 graphs, by distance to treedepth η+1, and others. The paper also derives bounds on the size of minor-minimal obstructions to k-apices of pathwidth-η graphs, contrasting with a new lower-bound construction for treewidth (Theorem 1.9). The proof is built from a reduction rule for the number of connected components (Theorem 2.1), a degree-reduction rule for modulator vertices using elimination forests (Lemma 2.4), a caterpillar reduction rule (Lemma 2.5), and a protrusion-reduction step (Section 6).

Significance. If the proof is correct, this is a substantial contribution: it provides the first uniform polynomial kernel for Pathwidth-η Deletion under several structural parameterizations, resolving the question in the affirmative for these parameterizations and showing a rare algorithmic divergence between pathwidth and treewidth. The paper is technically detailed, states exact functions and bounds for each reduction rule, and the kernel outputs a minor of the input graph, which yields elegant obstruction-size consequences. However, as discussed in the major comments, the proof currently has a load-bearing gap for the case β=0, which is precisely the parameterization advertised in Corollary 1.4(2). The gap appears repairable by adjusting the constants in Definition 5.12, but the current manuscript does not prove Main Theorem 1 for all stated values of β.

major comments (1)
  1. [Definition 5.12, Lemma 5.16, Algorithm 3, Claim 5.20, Algorithm 2, Corollary 1.4(2)] The proof of Main Theorem 1 is incomplete for β=0, which is included in the theorem statement and is exactly the case used by Corollary 1.4(2) (Pathwidth-η Deletion/dist-G_{pw≤1}). In Definition 5.12, B2 is defined as 2β·B1, so for β=0 we have B2=0 and consequently B#marked=f5.11(B2,1)=0. Lemma 5.16 then justifies Line 16 of Algorithm 3 by applying Lemma 5.11 with c=B#marked=0, but Lemma 5.11 is stated only for integers c≥2 and its proof divides by c. Furthermore, in the soundness proof of Algorithm 3, Claim 5.20 requires the inequality B2≥f4.3(B0+β)+B0+1, which is false for β=0 because the right-hand side is positive while B2=0. At the level of Algorithm 2, the branch condition in Line 12 tests whether m has more than BF(|NG(C)|) F-neighbors; since BF(x)=(β+1)·B3(x)=0 for β=0, any modulator vertex with at least one neighbor in C enters the invalid ReduceDegreeTree branch, making the caterpillar-reduction branch (Lines 15–17) unreachable even though for β=0 the elimination tree F consists of a single leaf and the intended reduction would be the caterpillar rule. This gap is load-bearing and should be repaired, for instance by redefining B2 to be at least max(2, 2βB1, f4.3(B0+β)+B0+1) and rechecking the downstream inequalities, or by handling β=0 as a separate case.
minor comments (3)
  1. [Section 3.5, Theorem 3.15] The statement that a nice G_{pw≤1}-elimination tree of depth at most β can be computed in polynomial time is justified only by a brief reference to [14, Corollary 1] and a short guessing argument; since all depth-sensitive bounds in Sections 5.3 and 6.2 depend on the depth bound and niceness, please expand this into a self-contained proof or a precise extraction from the cited work.
  2. [Throughout (e.g., Definition 5.12)] Many mathematical expressions have lost their superscripts in typesetting, e.g., f5.11(c,x)=(β+1)c^{β+1}x and the definitions of B#marked and B3 in Definition 5.12 appear without the exponent on B2; please ensure the camera-ready version uses correct math typesetting.
  3. [Section 4] The sentence 'Note that we have fixed η≥1 earlier at the start of this section as this global assumption is useful later in the paper' is confusing because Theorem 2.1 is then stated and proved for η=0 as well; consider clarifying the scoping of the fixed constants.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the kernel bounds are proved from structural reduction lemmas, and the only self-citation [48] is background, not load-bearing.

full rationale

The paper's central claim is derived, not assumed. Main Theorem 1 is assembled from independently proved reduction blocks: ReduceComponents (Theorem 2.1), ReduceModulatorDegree (Lemma 5.10), and CreateProtrusions plus ReduceProtrusions (Theorem 6.1 and Lemma 6.3). Each rule is proved safe by explicit path-decomposition surgery (Theorem 4.5, Corollary 5.15, Claims 5.18–5.22 and 5.31–5.40, and Lemma 5.41); no rule is justified solely by citing an earlier paper. The constants Bpolish, BF, Bdeg, and B6.3 are thresholds chosen to trigger these rules, not fitted parameters, and they are not renamed as outputs. The only self-citation is [48] (Lochet–Sharma) in the related-work paragraph; it is background and is never used in the proof of Main Theorem 1, so it is not load-bearing. The external result [14, Corollary 1] (Bulian–Dawar) is used to obtain nice Gpw≤1-elimination trees of depth at most beta; this is independent support, not circularity. One non-circular fragility should be recorded: in the beta=0 case, Definition 5.12 sets B2 = 2*beta*B1 = 0 and hence B#marked = B3 = BF = 0, while Lemma 5.11 is stated only for c >= 2 and is invoked with c = B#marked in Lemma 5.16; this is a correctness gap for the beta=0 corollary, not a circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No empirical free parameters appear: the many bounds (f2.1, f5.11, BF, Bcat, Bdeg, BPM) are explicit functions of eta, beta, k, and |M| derived in Appendix A, not fitted to data. The central claim rests on standard results in graph structure theory and on several black-box algorithms from prior work. No new physical or structural entities are postulated; concepts such as virtual cliques and polishing sets are proof devices.

assumptions (5)
  • standard math Minor-closed graph classes are well-quasi-ordered, so each minor-closed class has a finite obstruction set (Robertson-Seymour Graph Minor Theorem).
    Used in Section 1 to frame F-Minor-free Deletion and in Section 7 to define minor-minimal obstructions and apply kernel-based bounds.
  • domain assumption For every constant eta, a minimum-width path decomposition of a graph of pathwidth at most eta can be computed in linear time.
    Theorem 3.8 is used throughout to compute pathwidths of components and to implement marking schemes and the lifting step in Theorem 6.1.
  • domain assumption For every constant beta, a nice G_{pw<=1}-elimination tree of depth at most beta can be computed in polynomial time for graphs with elimination distance at most beta to pathwidth-1 graphs.
    Theorem 3.15, cited to Bulian-Dawar [14, Corollary 1], is the load-bearing black box behind Lemmas 2.4 and 6.3.
  • domain assumption Polynomial-time minor testing and constant-factor approximation algorithms for Pathwidth-eta Deletion exist.
    Minor testing [44,54] is used for membership in minor-closed classes; the approximation [31,34] is used in Corollary 1.4(1) to compute a modulator of size O(k).
  • standard math Helly's theorem for paths and Menger's theorem hold.
    Used in Observation 3.7, Claim 4.7, and Lemma 5.4 to force vertex sets into a bag via many internally disjoint paths.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Where Treewidth and Pathwidth Diverge: Towards a Uniform Kernel for Pathwidth-$\eta$ Deletion." pith.science (2026). https://pith.science/paper/UIQL35Q3

@misc{pith2026260809800,
  author       = {Pith},
  title        = {Pith review of: Where Treewidth and Pathwidth Diverge: Towards a Uniform Kernel for Pathwidth-$\eta$ Deletion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UIQL35Q3}},
  note         = {Machine review of arXiv:2608.09800}
}
abstract

For a constant $\eta \geq 0$, Pathwidth-$\eta$ Deletion is the problem of deciding whether, for a given graph $G$ and integer $k$, there is a set $S \subseteq V(G)$ of size at most $k$ such that the pathwidth of $G - S$ is at most $\eta$. The problems Treewidth-$\eta$ Deletion and Treedepth-$\eta$ Deletion are defined similarly for the parameters treewidth and treedepth, respectively. A landmark result of Fomin et al. [FOCS, 2012] shows that, for any constant $\eta$, all three problems admit a kernel on $O(k^{c(\eta)})$ vertices, where $c(\eta)$ is a constant depending on $\eta$. Giannopoulou et al. [ACM TALG, 2017] show that, in some sense, this result is optimal for Treewidth-$\eta$ Deletion: for $\eta \geq 2$ and even when parameterizing by the size of a vertex cover $M$ of the input graph, there is no kernel of size $O(|M|^{\frac{\eta+1}{2}-\varepsilon})$, for any $\varepsilon > 0$. Contrasting this result, they prove that Treedepth-$\eta$ Deletion admits a uniform polynomial kernel, that is, a kernel of size $O(k^c)$ for a constant $c$ that is independent of $\eta$. In comparison, the question whether Pathwidth-$\eta$ Deletion admits a uniform polynomial kernel has been neglected in the literature. As treewidth and pathwidth tend to behave similarly, it is natural to expect that no uniform kernel exists when parameterizing by the size of a vertex cover. Surprisingly, we show this not to be the case. More concretely, we prove the existence of a uniform polynomial kernel for Pathwidth-$\eta$ Deletion when parameterizing by (1) the solution size $k$ plus the size of a set $M$ such that $G - M$ has bounded treedepth; (2) the (vertex-deletion) distance to pathwidth-$1$ graphs; (3) the distance to the class of graphs with treedepth at most $\eta + 1$. This leads us to conjecture that Pathwidth-$\eta$ Deletion admits a uniform kernel when parameterizing by the solution size $k$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 11 canonical work pages

  1. [9]

    27 Michael R

    doi:10.1007/S00453-022-00984-2. 27 Michael R. Fellows, Lars Jaffke, Aliz Izabella Király, Frances A. Rosamond, and Mathias Weller. What is known about vertex cover kernelization? In Hans-Joachim Böckenhauer, Dennis Komm, and Walter Unger, editors,Adventures Between Lower Bounds and Higher Altitudes - Essays Dedicated to Juraj Hromkovič on the Occasion of ...

    doi:10.1007/s00453-022-00984-2

  2. [12]

    37 Eva-Maria C

    doi:10.1137/130907392. 37 Eva-Maria C. Hols, Stefan Kratsch, and Astrid Pieterse. Elimination distances, blocking sets, and kernels for vertex cover.SIAM J. Discret. Math., 36(3):1955–1990,

    work page doi:10.1137/130907392 1955

  3. [13]

    doi: 10.1137/20M1335285. 38 W. A. Horn. Three results for trees, using mathematical induction.Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences, 76B(1-2):39–43,

    doi:10.1137/20m1335285

  4. [15]

    40 Bart M

    doi:10.1007/S00224-012-9393-4. 40 Bart M. P. Jansen and Astrid Pieterse. Polynomial kernels for hitting forbidden minors under structural parameterizations.Theor. Comput. Sci., 841:124–166, 2020.doi:10.1016/J.TCS. 2020.07.009. 41 Bart M. P. Jansen and Michal Wlodarczyk. Lossy planarization: A constant-factor approximate kernelization for planar vertex del...

    doi:10.1007/s00224-012-9393-4 2020

  5. [17]

    doi:10.4230/LIPICS.ISAAC.2024.46. 58 Towards a Uniform Kernel for Pathwidth-ηDeletion 49 Jaroslav Nesetril and Patrice Ossona de Mendez.Sparsity - Graphs, Structures, and Al- gorithms, volume 28 ofAlgorithms and Combinatorics. Springer,

    work page doi:10.4230/lipics.isaac.2024.46 2024

  6. [18]

    50 Geevarghese Philip, Venkatesh Raman, and Yngve Villanger

    doi:10.1007/ 978-3-642-27875-4. 50 Geevarghese Philip, Venkatesh Raman, and Yngve Villanger. A quartic kernel for pathwidth- one vertex deletion. In Dimitrios M. Thilikos, editor,Graph Theoretic Concepts in Computer Science - 36th International Workshop, WG 2010, Zarós, Crete, Greece, June 28-30, 2010, Revised Papers, volume 6410 ofLecture Notes in Comput...

    work page doi:10.1007/978-3-642-16926-7_19

  7. [19]

    52 Neil Robertson and Paul D

    doi: 10.1007/978-3-662-43948-7_77. 52 Neil Robertson and Paul D. Seymour. Graph minors. I. Excluding a forest.J. Comb. Theory B, 35(1):39–61, 1983.doi:10.1016/0095-8956(83)90079-5. 53 Neil Robertson and Paul D. Seymour. Graph minors. V. Excluding a planar graph.J. Comb. Theory B, 41(1):92–114, 1986.doi:10.1016/0095-8956(86)90030-4. 54 Neil Robertson and P...

    arXiv 1983

  8. [1972]

    39 Bart M

    doi:10.6028/JRES.076B.002. 39 Bart M. P. Jansen and Hans L. Bodlaender. Vertex cover kernelization revisited - upper and lower bounds for a refined parameter.Theory Comput. Syst., 53(2):263–299,

    work page doi:10.6028/jres.076b.002

Show all 19 references
  1. [1987]

    Graph isomorphism parameterized by elimination distance to bounded degree.Algorithmica, 75(2):363–382, 2016.doi:10.1007/S00453-015-0045-3

    13 Jannis Bulian and Anuj Dawar. Graph isomorphism parameterized by elimination distance to bounded degree.Algorithmica, 75(2):363–382, 2016.doi:10.1007/S00453-015-0045-3. 14 Jannis Bulian and Anuj Dawar. Fixed-parameter tractable distances to sparse graph classes. Algorithmic...

    2016

  2. [2007]

    2 Stefan Arnborg, Andrzej Proskurowski, and Detlef Seese

    doi:10.1007/S00224-007-1328-0. 2 Stefan Arnborg, Andrzej Proskurowski, and Detlef Seese. Monadic second order logic, tree automata and forbidden minors. In Egon Börger, Hans Kleine Büning, Michael M. Richter, and Wolfgang Schönfeld, editors,Computer Science Logic, 4th Workshop...

    doi

  3. [2008]

    29 Fedor V

    doi:10.1007/S00453-007-9146-Y. 29 Fedor V. Fomin, Bart M. P. Jansen, and Michal Pilipczuk. Preprocessing subgraph and minor problems: When does a small vertex cover help?J. Comput. Syst. Sci., 80(2):468–495,

    doi

  4. [2012]

    Springer, 2012.doi:10.1007/978-3-642-31155-0_31

    Proceedings, volume 7357 ofLecture Notes in Computer Science, pages 352–363. Springer, 2012.doi:10.1007/978-3-642-31155-0_31. 8 Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Preprocessing for treewidth: A combinatorial analysis through kernelization.SIAM J. Discre...

    2012 doi

  5. [2013]

    9 Hans L

    doi:10.1137/120903518. 9 Hans L. Bodlaender, Arie M. C. A. Koster, and Frank van den Eijkhof. Preprocessing rules for triangulation of probabilistic networks.Comput. Intell., 21(3):286–305, 2005.doi: 10.1111/J.1467-8640.2005.00274.X. 10 Hans L. Bodlaender and Rolf H. Möhring. ...

    2005

  6. [2014]

    30 Fedor V

    doi:10.1016/J.JCSS.2013.09.004. 30 Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, Geevarghese Philip, and Saket Saurabh. Hittingforbiddenminors: Approximationandkernelization.SIAM J. Discret. Math., 30(1):383– 410, 2016.doi:10.1137/140997889. 31 Fedor V. Fomin, Daniel Lok...

    2013 doi

  7. [2015]

    21 Marek Cygan, Daniel Lokshtanov, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh

    doi:10.1007/978-3-319-21275-3. 21 Marek Cygan, Daniel Lokshtanov, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. On the hardness of losing width.Theory Comput. Syst., 54(1):73–82, 2014.doi:10.1007/ S00224-013-9480-1. 22 Marek Cygan, Marcin Pilipczuk, Michal Pilipczuk, ...

    2014 arXiv

  8. [2016]

    7 Hans L

    doi: 10.1145/2973749. 7 Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernel bounds for structural parameterizations of pathwidth. In Fedor V. Fomin and Petteri Kaski, editors,Algorithm Theory - SWAT 2012 - 13th Scandinavian Symposium and Workshops, Helsinki, Finl...

    2012 doi

  9. [2022]

    4 Daniel Bienstock and Michael A

    doi:10.1137/20M1385779. 4 Daniel Bienstock and Michael A. Langston. Algorithmic implications of the graph minor theorem. InNetwork Models, volume 7 ofHandbooks in Operations Research and Management Science, pages 481–502. Elsevier, 1995.doi:10.1016/S0927-0507(05)80125-2. 5 Han...

    doi

  10. [2024]

    45 Stefan Kratsch and Magnus Wahlström

    doi:10.1109/FOCS61266.2024.00014. 45 Stefan Kratsch and Magnus Wahlström. Representative sets and irrelevant vertices: New tools for kernelization.J. ACM, 67(3):16:1–16:50, 2020.doi:10.1145/3390887. 46 Mithilesh Kumar and Daniel Lokshtanov. A2ℓk kernel forℓ-component order con...

    2024

  11. [2026]

    doi:10.4230/LIPICS.STACS.2026.17. 12 R. L. Bryant, Michael R. Fellows, N. G. Kinnersley, and Michael A. Langston. On finding obstruction sets and polynomial-time algorithms for gate matrix layout. InProc. 25th Allerton Conf. on Communication, Control and Computing, pages 397–398,

    2026 doi

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.