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REVIEW 4 major objections 4 minor 3 cited by

A General Solution to Bellman's Lost-in-a-forest Problem

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that Bellman's lost-in-a-forest problem, for any known forest boundary, can be solved by discretizing starting points and orientations and solving a traveling-salesman-type optimization.

desk verdict A genuinely interesting TSP reformulation of Bellman's problem, but the 'general solution' theorem is a sketch and the paper's own numerics show the discretization can undercut the true optimum. read the letter →

arxiv 2412.10686 v3 pith:F64QVYNT submitted 2024-12-14 math.OC

Zhipeng Deng This is my paper · ORCID
classification math.OC MSC 49K3049Q1052A4090C27
keywords Bellman'slost-in-a-forestproblemminimaxescapepathtravelingsalesmanHamiltonianGamma-convergenceMoser'swormshortestopaquesetdiscretegeometry
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

Bellman's lost-in-a-forest problem asks for the shortest path that guarantees escape from a forest of known shape when the hiker's starting point and facing direction are unknown. This paper aims to turn that continuous minimax problem into a finite optimization problem by fixing a fine grid of possible starting points and orientations and requiring the escape path to hit one point on each rotated and translated copy of the forest boundary. The discrete version becomes a variant of the traveling salesman problem, and the paper argues that as the grid and orientation steps shrink to zero, the discrete optima converge to the true optimal escape path. If the argument is right, any forest shape given by a boundary equation can in principle be solved by refining a grid and running discrete optimization, and the same machinery transfers to Moser's worm problem and shortest opaque set problems. The paper also reproduces known optimal paths for lines, circles, strips, and triangles, which supports the method.

What carries the argument

The central device is to freeze the escape path and rotate and translate the forest boundary instead: for each sampled starting point and initial heading, the boundary is moved so that the hiker's coordinate frame stays fixed, and the point where the path first hits the boundary becomes a constrained escape point lying on the transformed boundary Fki(x,y)=0. Collecting all such escape points turns the problem into finding the shortest polygonal path, starting from the origin, that visits one escape point per transformed boundary; this is an open traveling salesman or Hamiltonian path problem, which the paper writes with binary order variables and Miller–Tucker–Zemlin subtour-elimination constraints. The limiting step is the assertion that these discrete optima Γ-converge to the continuous minimax solution.

What would settle it

Take the unit circle, whose escape optimum is known to be the diameter of length 2. Solve the Weak Form II mixed-integer program to certifiable global optimality for growing M and N; if the optimal values do not converge to 2, or if the only optima require an escape path that meets a rotated boundary more than once, the general-solution claim collapses.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3: when M starting points are evenly distributed in the region like grid points and N orientations are evenly distributed over [0, 2π], the solution to Weak Form II—the shortest path through the MN escape points on the corresponding rotated and translated boundary copies—yields the solution to the original Bellman problem as M and N tend to infinity. Weak Form I treats a known starting point with unknown orientation, while Weak Form II treats finitely many possible starting points. The escape path is sought as the shortest Hamiltonian path through escape points constrained to lie on the transformed boundaries, formulated with Miller–Tucker–Zemlin subtour elimination. The proof of convergence invokes Weierstrass existence, uniform continuity, Riemann-sum convergence, and Γ-convergence.

Load-bearing premise

The whole argument rests on the assumption that the shortest escape path is an open, simple, piecewise-smooth curve that crosses each rotated or translated forest boundary exactly once, and that the discrete optima genuinely converge to that continuous optimum; the convergence proof states the two Gamma-convergence inequalities but does not construct the recovery sequence.

Editorial extensions

If this is right

  • For any boundary expressible as F(x,y)=0, an escape path can be approximated by solving a mixed-integer program whose size grows with the number of sampled starting points and orientations.
  • Known optimal results—Isbell's half-plane search, Zalgaller's strip, and the circle and point-search cases—are recovered by the same formulation rather than by shape-specific geometry.
  • The framework gives a route to upper bounds for Moser's worm problem: any solved escape path of length L for a region of area A yields A/L^2 as an upper bound on the minimal covering area.
  • The same discrete formulation can be applied to opaque sets, replacing escape points with intersection points on all sampled lines, and to three-dimensional analogues.
  • Closed escape paths that return to the starting point fit the same framework by adding a return segment to the objective, connecting the method to closed-worm variants.

Reading between the lines

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

  • The paper's examples fix the visiting order of escape points, for instance assuming points are visited in increasing index; the full generality claim therefore depends on solving the complete permutation search, not the order-restricted runs shown, so global optima of the full Miller–Tucker–Zemlin problem are needed as the grid refines.
  • If the convergence theorem survives scrutiny, a natural next step is to quantify the discretization error—how large M and N must be to certify an epsilon-optimal escape path—which the paper leaves open.
  • The opaque-set connection suggests that the beam-detection constant for the unit circle could be attacked as a sequence of multiple-path mixed-integer programs, but only if the unconnected multi-curve variant is solved globally rather than by an imposed order.
  • The proposed extension to non-Euclidean geometry is speculative because the rotation and translation constraints rely on Euclidean distance; a testable extension would replace planar rigid motions with spherical or hyperbolic ones.
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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 proposes a discretization framework for Bellman's lost-in-a-forest problem. The original minimax problem over starting points and orientations is converted into a constrained shortest-path problem through 'escape points' on rotated and translated copies of the forest boundary. Upon discretization this becomes a Hamiltonian-path/TSP-type optimization over the locations and visiting order of the escape points. The paper claims in Theorem 3 that as the numbers of orientation samples N and starting-point samples M tend to infinity, the discrete optima converge to the solution of the original continuous problem. The bulk of the paper consists of numerical examples for lines, circles, strips, triangles, sectors, opaque sets, and related problems, with Mathematica notebooks in the appendices. The examples reproduce several known optimal or near-optimal shapes and values.

Significance. If the claimed general solution were established, this would be a substantial contribution: it would reduce a long-standing minimax problem over paths with continuous uncertainty to a sequence of discrete TSP-type optimizations and would provide a computational route to Moser's worm problem and opaque-set variants. The paper also has a genuine methodological idea—the translation/rotation reduction to escape points—and the supplied notebooks make the discrete computations reproducible. However, the central convergence claim is not proved, and the paper's own numerics indicate that the discrete problem can produce paths shorter than the known continuum optimum, so the paper does not currently deliver a verified general solution. Its value at this stage is that of a heuristic framework with several consistency checks against known results.

major comments (4)
  1. [Section 6.1, Theorem 3, Eqs. (60)-(61)] The proof of Theorem 3 is only a sketch and is load-bearing for the paper's main claim. The liminf inequality (60) and the recovery sequence (61) are asserted from 'lower semicontinuity' and 'continuity' without being constructed, and the discrete functionals live on spaces of dimension 3MN plus a permutation variable, so standard Gamma-convergence requires a common topological space, equi-coercivity, and control of the order variable; none of these is established. As stated, the theorem does not follow from the cited Weierstrass and uniform-continuity arguments.
  2. [Appendix III and Section 6.2.1] The reported numerical result for the unit strip, length 2.24853 for N=12, M=26, is smaller than Zalgaller's known continuum optimum 2.278292 quoted in Section 4.2.10. Since the continuum problem is a minimax problem over all starting points and orientations, a value below the known optimum cannot correspond to a feasible continuous escape strategy; it is an artifact of the finite discretization. This directly contradicts the convergence asserted in Theorem 3 and shows that the discrete minimizer need not even be a valid approximate escape path for the original problem.
  3. [Sections 4.2.1, 4.2.10, and 6.2.1] Nearly all examples fix the visiting order of the escape points rather than optimizing it. For example, Section 4.2.1 states 'If assuming the order of points ranges from 0 to N-1', and Appendix III hard-codes a permutation BB in the unit-strip computation. The formulations in Definitions 3.15 and 3.22 explicitly include the permutation as an optimization variable, so these examples solve a restricted subproblem and cannot be used as evidence for the claimed global solution of Weak Form I or II.
  4. [Appendix I] The continuous calculus-of-variations derivation for the half-plane example is not a derivation of the optimal path: the Euler-Lagrange equation (29) is left unsolved, and the boundary conditions are imposed by hand ('Approximate boundary value' y[1]=0.57735 and the condition y[nL]+1=0) to match the known line-search solution. The resulting length 6.39724 reproduces the known value only because those boundary values are fitted, not because the variational problem is solved from first principles.
minor comments (4)
  1. [Definition 3.21] The heading reads 'Length of escape path for Weak Form I' but the definition is for Weak Form II; this is confusing and should be corrected.
  2. [Section 4.2, introductory paragraph] The sentence 'Them are nontrivial and consistent with previous papers' contains a grammatical error and should be rewritten.
  3. [References] Several references are Wikipedia pages ([2], [31], [46]) rather than archival sources; for a mathematical paper these should be replaced by standard bibliographic entries.
  4. [Appendix III] The constraint in the Mathematica code uses angle step 2π/(nn-1) while the main text Definition 3.10 uses 2π/N; the discrepancy between N and N-1 should be explained or reconciled.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the discrete reformulation is a genuine, though incompletely proved, approximation, and the examples rely on explicit assumptions rather than on outputs that are definitionally equal to their inputs.

full rationale

The paper's central move (Def. 3.7 to Defs. 3.15/3.22 to Theorem 3) is a reformulation, not a renaming: the original minimax is over continuous escape paths hitting every rotated/translated boundary, while Weak Form II minimizes a polygonal length through escape points on finitely many sampled boundaries. The claimed equality in the limit is nontrivial and is asserted via Gamma-convergence in Sec. 6.1 (Eqs. (60)-(61)) without constructing the liminf inequality or recovery sequence; that is an unproved step and a serious correctness risk, but it is not circular because the conclusion is not already contained in the definitions. The examples do impose external structure: Section 4.2.1 and elsewhere say 'If assuming the order of points ranges from 0 to N-1', Section 4.2.7 fixes a specific visiting order, Section 6.2.1 uses a hand-specified order list, and Appendix I imposes y(0)=0.57735, y(2pi)=-1 as 'Approximate boundary value[s]'. These choices mean the experiments are restricted validations rather than demonstrations of the full unconstrained method, and Appendix III's N=12, M=26 length 2.24853 falling below Zalgaller's 2.278292 confirms that finite runs are not feasible continuous escape strategies. However, the paper labels these as assumptions, and the same line-search value (6.39724 = 7pi/6 + 1 + sqrt(3)) is also obtained in Appendix II by direct NMinimize with no fitted boundary values, so the boundary constants are not load-bearing. Section 10's admission that large instances are NP-hard and that NMinimize may not return global optima is a limitation, not a circular dependency. There is no self-citation chain and no imported uniqueness theorem; the only methodological citation (Braides, Gamma-convergence) is external. On the strict definition of circularity used here, a result equivalent by construction to its inputs, no step in the paper exhibits that reduction.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim depends on discretization parameters N and M, on the assumed visiting order of escape points, and on an unproven Gamma-convergence statement. The numerical examples further rely on boundary values chosen to reproduce known answers, so the framework's independence from the target results is limited.

free parameters (4)
  • N, number of orientation samples = e.g., N=12 in the unit strip example
    The discretization level is chosen by the user; no convergence rate or adaptive criterion is given.
  • M, number of starting point samples = e.g., M=26 in the unit strip example
    The grid resolution is chosen by the user; Theorem 3 only states the limit as M,N go to infinity.
  • Order of escape points a (permutation) = assumed as identity order in most examples; a 312-element list in the unit strip case
    The visiting order is a decision variable in the TSP formulation, but all examples fix it by assumption rather than solving for it.
  • Boundary values in Appendix I = y[1]=0.57735, y[nL]+1=0
    These boundary conditions are chosen to match the known line-search solution and are not derived from the problem.
assumptions (5)
  • standard math Euclidean metric invariance under SE(2) rigid motions
    Used in Remark after Definition 2.1 to justify that rotated and translated boundaries preserve escape-point existence.
  • domain assumption The escape path is an open simple piecewise-C1 curve
    Stated in Definition 3.2; this excludes self-intersecting or fractal curves that might be optimal in degenerate cases.
  • ad hoc to paper The optimal path can be represented as a Hamiltonian path through the discrete escape points
    The TSP formulation in Eqs. (19) and (59) assumes each escape point is visited exactly once in a fixed order; this is not proven for the original continuous problem.
  • ad hoc to paper Gamma-convergence of the discretized minima to the continuous minimum
    Invoked in Theorem 3 but only sketched with a generic liminf/recovery statement; no rigorous construction is provided.
  • standard math NP-hardness of the TSP and related mixed-integer programs
    Cited in Section 10 and used to justify that the formulation is hard to solve exactly.

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Cite this review

Pith. "Pith review of A General Solution to Bellman's Lost-in-a-forest Problem." pith.science (2026). https://pith.science/paper/F64QVYNT

@misc{pith2026241210686,
  author       = {Pith},
  title        = {Pith review of: A General Solution to Bellman's Lost-in-a-forest Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F64QVYNT}},
  note         = {Machine review of arXiv:2412.10686}
}
read the original abstract

We present a general solution and formulation framework to Bellman's lost-in-a-forest problem. The forest boundary is known and may take any shape. The starting point and the orientation are unspecified. We convert the problem into translation and rotation of the forest boundary. This transformation allows us to formulate this problem as a constrained minimization problem. Upon discretization, the problem becomes a variation of the traveling salesman problem or the Hamiltonian path problem. We leverage discrete optimization and derive several nontrivial results consistent with those from previous papers. This method is general, and we also extend the approach to related problems, including Moser's worm problem and the shortest opaque set problem.

Figures

Figures reproduced from arXiv: 2412.10686 by the authors.

Figure 1
Figure 1. Starting from a known point with unknown orientation. We convert the escape path out [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Starting from two possible known points with unknown orientation. We convert the path [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Results of search for one line/half plane with unit distance. (Black curve is escape path, [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: Results of search for a circle from exterior. (Black curve is escape path, and red curve is [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: Results of search for a circle from interior. (Black curve is escape path, and red curve is [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: Results of search for a circle from interior when shortest path is not unique. (Black curve [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Results of search for one point with given distance 1. (Black curve is escape path, and [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: Search for a shape from interior with distance significantly short at certain angle, when [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]
Figure 9
Figure 9. Figure 9: Results of search for two perpendicular lines with [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]
Figure 10
Figure 10. Figure 10: Results of search for two parallel lines with unit distance (unit strip) from the middle. [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]
Figure 11
Figure 11. Figure 11: Search for two lines from the angle bisector in the middle at certain angle [PITH_FULL_IMAGE:figures/full_fig_p023_11.png]
Figure 12
Figure 12. Figure 12: Results of search for two parallel lines with unit distance (unit strip) from edge (Zalgaller [PITH_FULL_IMAGE:figures/full_fig_p024_12.png]
Figure 13
Figure 13. Figure 13: Results of search for two parallel lines with unit distance (unit strip) when [PITH_FULL_IMAGE:figures/full_fig_p028_13.png]
Figure 14
Figure 14. Figure 14: Starting points evenly distributed with the region to escape: (a)unit stripe, (b)unit circle, [PITH_FULL_IMAGE:figures/full_fig_p031_14.png]
Figure 15
Figure 15. Figure 15: Opaque set with one curve for unit circle (Black curve is opaque set, and red curve is [PITH_FULL_IMAGE:figures/full_fig_p036_15.png]
Figure 16
Figure 16. Figure 16: Opaque set with one curve for ellipse with different opening directions (a) [PITH_FULL_IMAGE:figures/full_fig_p036_16.png]
Figure 17
Figure 17. Figure 17: Results of search for (a) one line with unit distance; (b) a circle from exterior; (c) a [PITH_FULL_IMAGE:figures/full_fig_p038_17.png]
Figure 18
Figure 18. Figure 18: Results of search for two lines from the angle bisector in the middle at certain angle [PITH_FULL_IMAGE:figures/full_fig_p039_18.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

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  1. Universal Triangle Covering Curve and Polygonal Chain: Escaping Forest and Fitting Worm

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    An exact support-function inequality is derived that characterizes shortest escape paths from arbitrary triangular forests and dual triangle covers for Moser's worm problem.

  2. Proof and More Variations of Bellman's Lost-in-a-forest Problem

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    The paper asserts a certified computational scheme for Bellman's lost-in-a-forest problem via TSPN discretization and convergence, plus new numerical tables for two-line and closed-path variants, but the proof of the ...

  3. Revisit escape path for infinite unit strip forest and unit broadworm

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    A convex-optimization reformulation reproduces Zalgaller's unit-strip escape path and argues the closed variant is a length-π constant-width curve, but the key equivalence is not rigorously established.

Reference graph

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Pith tools

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