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REVIEW 4 major objections 4 minor 32 references

Decremental Greedy Polygons and Polyhedra Without Sharp Angles

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Every monotone bottleneck subset problem has a unique maximal optimal subset, and decremental greedy always finds it.

desk verdict The 2D max-min-angle algorithm and monotone bottleneck framework are solid and new, but the 3D solid-angle theorem rests on an unproved monotonicity claim and the appendix contains a false intermediate equality, so it deserves serious review but needs revision. read the letter →

arxiv 2507.04538 v1 pith:R6YADY3W submitted 2025-07-06 cs.CG cs.DS

David Eppstein This is my paper · ORCID
classification cs.CGcs.DS MSC 68U0552B5505C85
keywords max-minanglepolygondecrementalgreedyalgorithmmonotonebottlenecksubsetconvexpolyhedronsolidcyclegraphdegeneracyNP-hardnesscomputationalgeometry
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

This paper identifies a wide class of optimization problems—monotone bottleneck subset problems—in which a single greedy rule is provably optimal: repeatedly delete the current worst-quality element and return the best intermediate set. The central theorem states that every such problem has a unique maximal optimal subset and that, no matter how ties are broken, both the decremental greedy algorithm and the known-threshold version return it. In geometry this gives an $O(n\log n)$ algorithm for the polygon through given planar points whose sharpest interior angle is as large as possible, and an $O(n^2)$ algorithm for the convex polyhedron through given 3D points whose minimum solid angle is maximized. The same framework subsumes the classical linear-time algorithm for graph degeneracy and yields new bottleneck-cycle algorithms that find max-min-angle closed curves in 3D. A constrained version—a simple, knot-free polygon in 3D with max-min angle—is shown NP-complete, so the greedy tractability has a sharp boundary.

What carries the argument

The load-bearing object is the monotone bottleneck subset problem: given a set $U$, a quality $q(x,S)$ for each $x \in S \subseteq U$ that is monotone non-increasing as $S$ shrinks, and a goal of maximizing $Q(S)=\min_{x\in S} q(x,S)$ over nonempty subsets. The proof mechanism is the observation that the union $M$ of all optimal subsets is itself optimal: for any $x$ in that union, $q(x,M) \ge q(x,X) \ge \beta$ because monotonicity says an element's quality can only be better in a larger set. Hence every element outside $M$ has quality below the optimum in any superset of $M$, so deleting elements outside $M$ is always safe. The decremental greedy algorithm—delete any element with quality at most the current best and record improvements—and the known-$\beta$ variant both terminate at $M$. In the geometric applications this is instantiated by dynamic convex hull data structures that report the sharpest-angle hull vertex; in the graph applications, by bottleneck cycles in directed, mixed, and polar graphs.

What would settle it

Take a random set of 6 to 8 points in $\mathbb{R}^3$, compute the convex hull solid angle at every hull vertex before and after deleting each other hull vertex, and compare the greedy output with a brute-force search over all subsets for the maximum minimum solid angle; any deletion that increases a survivor's solid angle, or any greedy-versus-brute-force mismatch, would falsify the monotonicity behind the $O(n^2)$ polyhedron algorithm.

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Extended reading notes

Core claim

The paper's central claim is that when the quality of each element is monotone non-increasing under deletion of other elements, the bottleneck subset problem—find a nonempty set maximizing the minimum quality of its members—can be solved by deleting low-quality elements. Theorem 1 proves that the union of all optimal subsets is itself optimal, so optimal subsets have a unique maximal member, and that both the decremental greedy algorithm and the known-$\beta$ algorithm reach this maximal optimal subset regardless of tie-breaking. Applied to geometry, the quality of a point is its interior angle when it is a convex hull vertex and $2\pi$ otherwise, or in 3D its solid angle, and monotonicity makes it safe to delete the sharpest remaining hull vertex; the best hull seen during deletion is a max-min-angle polygon or a max-min-solid-angle polyhedron. The paper also formalizes bottleneck cycles in directed, mixed, and polar graphs, giving deterministic algorithms for max-min-angle closed polygonal curves in 3D and proving hardness for the simple-curve restriction.

Load-bearing premise

The entire framework rests on the monotonicity axiom: deleting points can only keep a remaining hull vertex's angle, or solid angle, the same or smaller, so a sharp corner never appears at a survivor; this is obvious for 2D interior angles, but for 3D solid angles it is asserted rather than proved.

Editorial extensions

If this is right

  • A max-min-angle closed polygonal curve through $n$ planar points can be found in $O(n\log n)$, and an optimal curve can be assumed convex.
  • A max-min-solid-angle non-self-intersecting polyhedral surface through $n$ 3D points can be found in $O(n^2)$, and an optimal surface can be assumed a convex polyhedron.
  • Max-min-angle closed 3D curves allowing repeated vertices can be found in near-cubic time: $O(n^3\log^* n)$ with repeated segments, and $O(n^3(\log n\log\log n)^2)$ without repeated segments.
  • Requiring the 3D curve to be simple, or unknotted, makes the max-min-angle problem NP-complete.
  • Graph degeneracy, computed by repeatedly deleting a minimum-degree vertex, is a special case of the same framework.

Reading between the lines

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

  • The same greedy template could be applied to other monotone geometric quality measures, such as minimum edge length or minimum circumradius, wherever deleting points cannot improve the quality of a survivor.
  • The 3D solid-angle monotonicity is the most exposed assumption: a proof of it would close the only gap between the abstract theorem and the polyhedron algorithm, while a small counterexample would invalidate the $O(n^2)$ result.
  • The antimatroid characterization suggests that the correctness of the algorithm depends only on the monotone availability of elements, not on the geometric origin of the quality measure, so the framework may extend to other bottleneck objectives beyond angles and degrees.
  • The NP-hardness boundary for 3D curves indicates that similar greedy methods are unlikely to exist for curve classes carrying global non-self-intersection constraints.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper introduces a formal framework of monotone bottleneck subset problems, in which a quality measure q(x,S) is monotone nondecreasing as the ambient set S grows. It proves that every such problem has a unique maximal bottleneck subset, and that a decremental greedy algorithm that repeatedly removes an element whose quality is no better than the best quality seen so far always finds this subset. The framework is applied to three geometric settings: (1) computing a max-min-angle polygon in a planar point set in O(n log n) time, (2) computing a max-min-solid-angle convex polyhedron in a 3D point set in O(n^2) time, and (3) computing max-min-angle closed polygonal curves in 3D via reductions to bottleneck cycles in directed, mixed, and polar graphs. The paper also shows NP-hardness for the non-self-intersecting variant of the 3D curve problem, and connects the removal sequences to antimatroids. Appendix A is devoted to the antimatroid property of the decremental greedy algorithm, Appendix B to the NP-hardness proof, and Appendix C to the graph bottleneck-cycle algorithms.

Significance. If the claims are fully substantiated, the paper provides an elegant and useful unification of several known algorithms, including graph degeneracy, with new applications. The 2D max-min-angle polygon result is a clean improvement over dynamic programming approaches, and the bottleneck-cycle algorithms for directed, mixed, and polar graphs are likely to be of independent interest. The paper is generally clearly written, and Theorem 1 is a genuinely useful structural insight. However, the 3D polyhedron result rests on an unproved monotonicity assertion, the dynamic hull data structure is presented only as a sketch, and Appendix A contains a false intermediate claim. These issues prevent the manuscript from being acceptable in its present form, though they appear to be repairable within the scope of the paper.

major comments (4)
  1. [Section 3.2, Lemma 4] The monotonicity of the solid-angle quality function is asserted but not proved. The correctness of the decremental greedy algorithm in Theorem 5 relies on q(x,S) ≤ q(x,T) for x∈S⊂T; please provide a proof, e.g., via the fact that conv(S)⊆conv(T) implies the tangent cone at a surviving hull vertex of the smaller hull is contained in the corresponding tangent cone of the larger hull, and handle explicitly the cases where x is interior to one of the two hulls. The current one-sentence justification in Lemma 4 ('This can only remove vertices and improve the solid angle at the remaining vertices') is insufficient for the application of Theorem 1.
  2. [Appendix A, Lemma 8] The statement 'Note that Q(X+) = better(Q)' is false in general. For monotone quality functions one can construct instances in which the union of all subsets with quality at least better(Q) has a strictly higher quality than better(Q); only the inequality Q(X+) ≥ better(Q) is guaranteed. Since the subsequent argument only needs Q(S+) > Q(T), the proof can be repaired by replacing the equality with this inequality, but the false claim must be corrected.
  3. [Section 3.2, Theorem 5] The O(n^2) time bound relies on a dynamic hull data structure that is only sketched. Please specify how two child convex hulls are merged in linear time in 3D (or cite a reference for linear-time merging of convex polytopes), and explain how the global hull is queried to find the vertex with minimum solid angle in O(n) time after each deletion. The present description leaves the claimed per-deletion O(n) update time insufficiently supported.
  4. [Appendix B] The NP-hardness reduction for maxmin-angle simple polygons in 3D is presented as a high-level sketch. Several quantitative assertions (the choice of ε, the Ω(1/n^2) spacing, the perturbation argument using O(1/n^3) movements, and the claimed angle gap) need precise statements and proofs to make the reduction rigorous. As it stands, the proof cannot be checked in detail from the text; please expand it or clearly mark the result as conditional on the described construction.
minor comments (4)
  1. [Appendix C.4, Theorem 16] The phrase 'in time O(m(log m log log m)^2 per edge removal' appears to be a typo; if it were per removal the total would be quadratic. Please state the intended total time or per-removal bound.
  2. [Appendix B] The threshold θ is inconsistently given as '2π − Θ(ε)' and later as 'π − Θ(1/n)'; the latter is the correct form since polygonal angles are at most π.
  3. [Conclusions] In the final paragraph, 'given pints' should be 'given points'.
  4. [Appendix A, Lemma 8] The monotonicity statement of t contains a missing parenthesis: it should read Q(t(S)) ≥ Q(t(S')).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation chain is self-contained, with the unproved 3D solid-angle monotonicity a correctness gap rather than a circular step.

full rationale

The paper's central framework is developed from first principles: Theorem 1 is proved directly from the definition of monotone quality, and the geometric algorithms apply that theorem to independently defined angle and solid-angle measures. The 2D angle monotonicity is argued geometrically, and the 3D solid-angle monotonicity is asserted in Lemma 4 without a proof; this is an unsupported correctness-critical claim, but it is not circular, because it is not derived from the target theorem or from any fitted parameter. The polygonal-curve results reduce to independently defined bottleneck-cycle problems in directed and polar graphs, with algorithms proved in Appendix C. References to the author's own prior work are contextual (e.g., curve reconstruction and minimum-area k-gons) and are not load-bearing for the new results. No prediction is equivalent to its input by construction, and no self-citation chain forces the conclusions. The absence of a proof for a monotonicity assertion is a correctness risk, not a circularity, so the appropriate circularity score is 0.

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

No free parameters are fitted to data; the paper is purely theoretical. The load-bearing assumptions are the monotonicity conditions on quality (angles and solid angles) and the availability of dynamic hull data structures. The NP-hardness construction introduces asymptotic spacing parameters but no data-fitted constants.

assumptions (5)
  • domain assumption Monotonicity of quality: for x ∈ S ⊂ T, q(x,S) ≤ q(x,T)
    Defined in Section 2 and required for Theorem 1. Without it the decremental greedy algorithm can remove the optimal element.
  • domain assumption Interior angle at a convex hull vertex is non-increasing when other vertices are deleted
    Stated in Section 3.1 as 'quality is monotonic'; used to apply Theorem 1 to the max-min-angle polygon.
  • domain assumption Solid angle at a vertex of a convex polyhedron is non-increasing when other vertices are deleted
    Stated implicitly in Section 3.2 ('Analogously'); no proof is given, and this is less evident than the 2D case.
  • standard math Dynamic convex hull data structures with O(log n) update (2D) and O(n) per deletion (3D) exist
    Cited to Brodal-Jacob, Hershberger-Suri, and Overmars; the 3D structure is only sketched in the paper.
  • ad hoc to paper In the NP-hardness reduction, points on a unit sphere can be sampled with spacing ε to achieve angle thresholds and gaps
    Appendix B states 'we will choose ε sufficiently smaller' and similar; the proof is an outline without explicit inequalities.

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Pith. "Pith review of Decremental Greedy Polygons and Polyhedra Without Sharp Angles." pith.science (2026). https://pith.science/paper/R6YADY3W

@misc{pith2026250704538,
  author       = {Pith},
  title        = {Pith review of: Decremental Greedy Polygons and Polyhedra Without Sharp Angles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R6YADY3W}},
  note         = {Machine review of arXiv:2507.04538}
}
abstract

We show that the max-min-angle polygon in a planar point set can be found in time $O(n\log n)$ and a max-min-solid-angle convex polyhedron in a three-dimensional point set can be found in time $O(n^2)$. We also study the maxmin-angle polygonal curve in 3d, which we show to be $\mathsf{NP}$-hard to find if repetitions are forbidden but can be found in near-cubic time if repeated vertices or line segments are allowed, by reducing the problem to finding a bottleneck cycle in a graph. We formalize a class of problems on which a decremental greedy algorithm can be guaranteed to find an optimal solution, generalizing our max-min-angle and bottleneck cycle algorithms, together with a known algorithm for graph degeneracy.

Figures

Figures reproduced from arXiv: 2507.04538 by the authors.

Figure 1
Figure 1. The maxmin-angle simple polygon in a given [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. A polar graph with six vertices representing [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. Blocking gadget (left) and crossing gadget [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figures from the paper (3 more)
Figure 3
Figure 3. Figure 3: Overall form of reduction: two horizontal [PITH_FULL_IMAGE:figures/full_fig_p009_3.png]
Figure 5
Figure 5. Figure 5: Expansion of a mixed graph vertex v, with incident undirected edges uv, wv, xv, and yv, into a subgraph of a directed graph (Lemma 13) the increasing path from 0 to d, we number the vertices of the decreasing path in descending order from d−1 to 1, and we assign distin…
Figure 6
Figure 6. Figure 6: Expanding a polar graph into a degree-three [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

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