The Computational Complexity of the Weak Gravity Conjecture

Stefano Lanza

REVIEW 4 major objections 5 minor 1 cited by

The Computational Complexity of the Weak Gravity Conjecture

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper argues that verifying the Convex Hull Weak Gravity Conjecture is exponentially costly in the number of gauge fields, and that finding minimal realizing sets is NP-hard or NP-complete.

desk verdict Useful bridge between computational geometry and the Swampland, but the headline EXP claim is only a Quickhull upper bound and the paper's own polynomial bypass misuses quadratic programming. read the letter →

arxiv 2505.03868 v1 pith:2XTPG42Q submitted 2025-05-06 hep-th

Stefano Lanza This is my paper · ORCID

classification hep-th

keywords WeakGravityConjectureConvexHullcomputationalcomplexitySwamplandprogramalgorithmsNP-hardnesseffectivefieldtheorymultiplegaugefields

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 asks how hard it is, in practice, to check the Convex Hull Weak Gravity Conjecture, the version of the Weak Gravity Conjecture that applies when an effective theory has many U(1) gauge fields. The author presents an explicit algorithm, WGCHull, that builds the convex hull of the charge-to-mass vectors of superextremal particles and then tests whether that hull contains the black-hole extremality sphere. He argues that the worst-case running time grows exponentially with the number of gauge fields, placing the general constructive check in the complexity class EXP, while noting that a shortcut that skips hull construction runs in polynomial time. He further argues that finding minimal sets of particles that realize the conjecture is NP-hard, and NP-complete in some formulations. If correct, this means that explicit bottom-up verification of the conjecture becomes computationally infeasible in theories with many gauge fields, even though the conjecture itself may be true.

What carries the argument

The central object is the convex hull conv(S ∪ {0}) formed by the charge-to-mass ratio vectors z(i) = M_P q(i)/m(i) of superextremal particles, together with the origin, in the N-dimensional z-space of gauge charges. The Convex Hull Weak Gravity Conjecture holds exactly when this hull contains the black-hole extremality sphere ||Z|| = α. The argument runs on the V-representation/H-representation duality of polytopes: WGCHull(I) builds the hull facet by facet via Quickhull, whose worst-case facet count f_n = O($n^{{⌊N/2⌋}}$/(⌊N/2⌋)!) grows exponentially in N; WGCHull(II) turns the containment check into a linearly constrained quadratic program, solvable in time O(N_s $N^{4}$). Minimal realizations are analyzed by reducing the search to largest-N-simplex problems. The hull construction, not the containment check, is what makes the algorithm exponential.

What would settle it

Run the containment check directly as the quadratic program in Step II on raw superextremal vectors, without building the hull, for randomly generated sets at increasing N; if its runtime stays polynomial while the number of hull facets grows exponentially, the claim that the verification problem itself is in EXP would be refuted. More decisively, a polynomial-time separation-oracle algorithm that decides coverage without enumerating facets would settle the question.

Watch

Extended reading notes

Core claim

The central claim is that verifying the Convex Hull Weak Gravity Conjecture is not just a physics question but a computational one with severe complexity: in general, the verification problem lies in EXP because constructing the convex hull of the superextremal states can require a number of facets exponential in the number N of gauge fields, and the Quickhull-based construction inherits that cost. The author presents this as a fundamental obstruction to concrete, algorithmic realizations of the conjecture. At the same time, he shows that if one only asks whether some hull could cover the extremality sphere, the question reduces to a convex quadratic program solvable in polynomial time, although that shortcut does not identify the realizing particles. For minimal realizations, he argues that enumeration is exponential and that finding the largest N-simplex inside a hull is NP-hard in the V-representation and NP-complete in the H-representation.

Load-bearing premise

The exponential-time conclusion rests on the assumption that verifying the conjecture requires constructing the full convex hull; the paper derives its complexity from the Quickhull algorithm's worst case and does not prove a lower bound applying to every possible algorithm.

Editorial extensions

If this is right

For a fixed small number of gauge fields, the WGCHull algorithm runs in polynomial time, so low-dimensional checks of the Convex Hull Weak Gravity Conjecture remain feasible.
For many gauge fields, any explicit construction of the hull is exponentially expensive, so landscape-scale scans cannot rely on building hulls.
The polynomial-time quadratic-programming shortcut can flag theories where no realization is possible, but cannot name the particles that would realize it.
Finding the smallest realizing set of particles requires an exponential enumeration or solving an NP-hard or NP-complete largest-simplex problem, so minimal realizations are not practically computable in general.
Hull-based Swampland constraints beyond the Weak Gravity Conjecture, such as the hull versions of the Distance Conjecture, should face the same computational barriers.

Reading between the lines

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

The EXP classification is an upper bound derived from one algorithm; the paper has not proven a lower bound, so the obstruction could be an artifact of demanding full hull construction rather than an inherent property of the verification problem.
If the quadratic-programming shortcut is as efficient as claimed, the tractability boundary may fall between deciding existence in polynomial time and constructing realizing states in exponential time, which is a sharper statement than saying verification is simply in EXP.
A testable extension would be to train a machine-learning classifier on random superextremal sets to predict hull coverage; high accuracy with polynomial-time inference could bypass the exact bottleneck at the price of statistical error.
One could try to prove a true lower bound for the containment-decision problem by connecting it to known hardness results for polytope containment; if the decision problem were shown NP-hard, the polynomial-time shortcut would be invalid.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First thing to know: this paper correctly identifies a real computational bottleneck for explicit hull-based checks of the Convex Hull WGC, but it overstates that bottleneck as a problem-level lower bound. The stress-test note's SOCP rescue is not valid as written, so the paper's real flaw is the absence of a lower bound, not the existence of a fast algorithm.

What works: the paper casts the Convex Hull WGC as a clean two-step problem—build the hull of superextremal charge vectors, then check whether it contains the extremality ball—and gives a concrete Quickhull-based algorithm for it. The observation that the facet count (and hence Quickhull's runtime) can blow up exponentially with the number of gauge fields N is correct and useful. That translation of known computational-geometry results into the WGC context is the paper's genuine contribution. The Section 4 discussion of minimal realizations, though informal, points at real NP-hardness results for largest simplices.

The central conclusion, however, does not follow. Section 3.4 claims the problem 'resides in EXP' because T_WGCHull is exponential, but T_WGCHull is a bound on one specific algorithm, and the text itself calls it an upper bound a few lines earlier. Without a matching lower bound or a reduction from an EXP-hard problem, the statement that a Turing machine 'requires' exponential time is unsupported. The stress-test note's proposed SOCP fix is also not right: minimizing over the unit ball gives zero identically, and the correct unit-sphere constraint is nonconvex. So I am not saying the decision problem is easy; I am saying the paper does not prove it is hard.

There is a second, independent error in Section 3.5. The author writes the containment check as a linearly constrained convex-quadratic maximization and cites [41] for polynomial time. But maximizing a convex quadratic is NP-hard in general; [41] is about minimizing. This is load-bearing because the paper uses it to claim that skipping hull construction makes checks tractable—a claim that may be true, but not for the reason given.

The NP-hardness discussion in Section 4 is on firmer ground, but it relies on an informal analogy rather than a formal reduction, so those claims should be treated as suggestive.

Who is this for? People working on the algorithmic Swampland program. The paper is readable and the two-step decomposition is genuinely clarifying. But the abstract and conclusions overstate what is proved. I would send it to peer review, with a clear request: either downgrade the EXP claim to an algorithm-specific upper bound or supply a real lower-bound argument, and fix the convex-quadratic error.

Referee Report

4 major / 5 minor

Summary. The paper argues that checking the Convex Hull Weak Gravity Conjecture (CHWGC) in effective field theories with N gauge fields is computationally hard. It introduces the WGCHull algorithm, which first constructs the convex hull of the charge-to-mass vectors of superextremal particles using the Quickhull algorithm and then checks whether the hull contains the extremality sphere. From the Quickhull runtime bound, the paper concludes in §3.4 that the overall problem 'resides in EXP'. Section 4 then considers minimal realizations and claims, via known results on largest simplices in polytopes, that finding minimal realizing subsets is NP-hard or NP-complete. The paper also discusses a shortcut that bypasses hull construction.

Significance. If the EXP claim were correct, the paper would establish a fundamental computational obstruction for a widely used Swampland criterion, which would be a notable cross-disciplinary result. The paper is clearly written, carefully explains the geometry of the CHWGC, and faithfully imports known results on Quickhull complexity and on the complexity of largest-simplex problems. These imports are genuine strengths: the identification of explicit hull construction as a bottleneck and the connection to the polytope literature are useful. However, the central complexity conclusion is not supported: an algorithm-specific upper bound is presented as a problem-level classification, and the proposed polynomial shortcut is mis-stated. The decision problem can in fact be solved by a polynomial-time second-order cone program, so the claimed 'fundamental obstruction' is not established. The paper contains no machine-checked proofs or numerical benchmarks, and the minimal-realization NP-completeness claim lacks a formal reduction within the manuscript.

major comments (4)

[§3.4, Eq. (3.10)] The statement that the problem of 'constructing the convex hull ... and then checking the Convex Hull Weak Gravity Conjecture resides, in general, in the class EXP' is not supported. Equation (3.10) is an upper bound for the particular Quickhull-based algorithm, not a lower bound on all algorithms for the problem. The decision version of the CHWGC check can be solved without constructing the full hull: since C=conv(S∪{0}) contains the ball B(0,α) iff min_{||u||≤1} max_i u·z_i ≥ α, this is a second-order cone program solvable in polynomial time in N, Ns, and the input bit length. An upper bound in EXP does not imply that a Turing machine 'requires' exponential time, so the claimed fundamental obstruction is not established.
[§3.5, WGCHull (II) bypass] The proposed polynomial-time shortcut is invalid as written. The displayed linearly constrained quadratic program asks to maximize Q = 1/2 x^T P x + q^T x with P positive definite; maximizing a convex quadratic is generally NP-hard, and the polynomial-time result [41] applies to minimizing convex quadratic programs, not maximizing them. The correct convex formulation of the bypass is to minimize t subject to t ≥ u·z_i for all i and ||u||^2 ≤ 1, which is an SOCP solvable in polynomial time. Once this correction is made, the bypass becomes valid and it directly undermines the EXP conclusion of §3.4.
[§4.2] The NP-hardness and NP-completeness claims for minimal realizations are not established by the cited results. Reference [23] concerns the problem of finding the largest j-simplex in an N-polytope, but the paper does not provide a reduction from 3SAT (or any NP-complete problem) to the specific problem of finding the smallest subset of superextremal particles whose convex hull covers the extremality ball. Moreover, the reduction to a largest-simplex problem assumes that the minimal realizing set is a simplex whose vertices are vertices of the full hull; already for N≥3 a ball can be covered by polytopes that are not simplices, so this equivalence requires proof.
[§2.2] The definition of EXP as the class of problems for which a Turing machine 'requires an exponential time to solve' is inaccurate. EXP is the class of decision problems solvable in exponential time, and membership in EXP does not imply an exponential lower bound. This definitional slip is load-bearing because §3.4 uses it to infer 'requires exponential time' from a single algorithmic upper bound. The paper should either prove a matching lower bound for the decision problem or explicitly restrict its claims to the worst-case cost of explicit hull construction by a specific algorithm.

minor comments (5)

[§3.2, Eq. (3.3)] The denominator in Eq. (3.3) appears to be a typographical mangling of (⌊N/2⌋)!, and the provenance of the bound should be stated more clearly in the main text rather than only via the citation to [18].
[§3.1, footnote 2] The claim that if the origin is not included in the set then S cannot realize the Weak Gravity Conjecture is too strong: a set of charge-to-mass vectors surrounding the origin can generate a convex hull that contains the origin even when the origin is not one of the generators.
[§3.3, Step 4] The boundary case d_H = α should be discussed explicitly, since the conjecture is stated for the closed subextremal region ||Z||≤α; with the current wording it is ambiguous whether equality is treated as realizing the conjecture.
[§3.4] The text itself concedes that Eq. (3.10) 'ought to be considered as an upper bound'; the conclusions and the abstract should be rephrased accordingly, since the current abstract claims a 'fundamental obstruction' that the body of the paper does not prove.
[§4.1, Eq. (4.1)] The enumeration runtime T(Ns)=O(Ns^c 2^{Ns}) is stated as exponential in Ns, which is true, but the discussion would benefit from distinguishing the dependence on Ns (which drives 2^{Ns}) from the earlier dependence on N (which drives the Quickhull facet count).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the runtime bounds are imported from external computational-geometry results, and the one self-citation is confined to the outlook.

full rationale

The paper's derivation does not reduce to its own target. The alleged exponential hardness is obtained by combining the Quickhull complexity bound from Barber, Dobkin, and Huhdanpaa (ref. [18], Eq. 3.3) with the quadratic-programming bound from Ye and Tse (ref. [41], Eq. 3.7), both external to the paper and not derived from the WGC itself. The WGCHull algorithm is explicitly described, and its runtime is calculated from that algorithm rather than fitted to any prediction. The only self-citation, ref. [53] on neural-network learning, appears in the conclusions/outlook and is not load-bearing for the complexity claims. The paper itself flags the key limitation of its own argument: it states that Eq. (3.10) 'ought to be considered as an upper bound on the effective time complexity of the problem,' and Section 3.5 explicitly gives a polynomial-time bypass by applying WGCHull (II) directly to the particle set. That undermines the strength of the EXP claim as a problem-level statement, but it is a correctness/rigor concern, not circularity. No fitted parameter is renamed as a prediction, no uniqueness theorem from the author's prior work is invoked, and no ansatz is smuggled in via self-citation. The central derivation is self-contained against external computational-complexity benchmarks, so the circularity score is 0.

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

No free parameters or invented physical entities. The analysis operates entirely with known concepts; the main assumptions are geometric genericity, the existence of a supplied set of superextremal particles, and the choice of Quickhull as the reference algorithm.

assumptions (4)

domain assumption The black hole subextremality region is a sphere in charge-to-mass space, or can be made one by rescaling the charge-to-mass vectors.
Section 3.1 imposes this simplification; with light scalars or moduli the region can be an ellipsoid or more complicated. The paper asserts that rescalings suffice without proof.
domain assumption A finite set S of Ns superextremal particles with |z(i)| > alpha and Ns >= N+1 is given as input.
The algorithm requires enough superextremal states to form an N-dimensional hull; a theory may not provide such a set, and finding it is part of the difficulty.
domain assumption The charge vectors are in general position, so the convex hull is a simplicial polytope.
Appendix A.2 assumes this to ensure that any N+1 selected points form a simplex; degeneracies would require perturbation or symbolic methods.
ad hoc to paper The complexity of the problem is measured by the worst-case runtime of the Quickhull algorithm on the given set.
The paper moves from Quickhull's upper bound in equation (3.3) to the conclusion that the problem "resides in EXP" without proving a matching lower bound for all algorithms; Section 3.4 itself calls the result an upper bound.

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Pith. "Pith review of The Computational Complexity of the Weak Gravity Conjecture." pith.science (2026). https://pith.science/paper/2XTPG42Q

@misc{pith2026250503868,
  author       = {Pith},
  title        = {Pith review of: The Computational Complexity of the Weak Gravity Conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2XTPG42Q}},
  note         = {Machine review of arXiv:2505.03868}
}

read the original abstract

The Weak Gravity Conjecture imposes stringent constraints on effective field theories to allow for an ultraviolet completion within quantum gravity. While substantial evidence supports the conjecture across broad classes of string theory-derived effective field theories, constructing low-dimensional models realizing it explicitly remains highly non-trivial. In this work, we illustrate how the presence of multiple gauge fields in an effective field theory significantly complicates the bottom-up implementation of the Weak Gravity Conjecture. To this end, we introduce a general algorithm that constructs the convex hull associated with a given set of superextremal states and verifies whether it satisfies the Convex Hull version of the Weak Gravity Conjecture. We show that the computational time of this construction grows exponentially with the number of gauge fields, thereby revealing a fundamental obstruction to concrete, algorithmic realizations of the conjecture in theories with many gauge fields.

Figures

Figures reproduced from arXiv: 2505.03868 by the authors.

**Figure 1.** On the left, an example of four particles not realizing the Convex Hull Weak Gravity Conjecture: the four particles (blue dots) in the (z1, z2) plane, generate a convex hull (the light blue rectangle), that does not fully cover the subextremality disk, depicted as the blue disk. On the right, an example of four particles that determine a convex hull that fully covers the subextremality disk, thus realizing the Conve… view at source ↗

**Figure 2.** Intersection of basic computational complexity classes. quires an exponential time to solve; namely, the number of steps before halting behaves as T(n) ∼ O(2n c ), for some c > 0. The problems that belong to this class are instead considered to be intractable by a machine, as their runtime grows too quickly to be feasible for even moderately large inputs. NP, or non-deterministic polynomial time: problems belonging … view at source ↗

**Figure 3.** A two-dimensional depiction of the input data. Furthermore, we assume knowledge of a set of Ns species of superextremal species of particles. As in Section 2.1, we label their charge-to-mass ratio vectors z (i) , with i = 1, . . . , Ns, obeying |z (i) | > α, and they are part of the set S = {z (i) : i = 1, . . . , Ns}. 2 To form an N-dimensional convex polytope with some of the vectors z (i) determining its vertice… view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: The idea behind the iterative construction of the convex hull proposed by the Quickhull algorithm. WGCHull (I) Question: What is the convex hull C that contains the set of superextremal particles S? Input: number of gauge fields N; set of superextremal particles S, wit…

**Figure 5.** Figure 5: On the left, a two-dimensional example of convex hull C not covering the black hole extremality disk, with hyperplane H with all the vertices of C above it crossing the extremality disk; conversely, on the right, a convex hull covering the extremality disk, with hyperp…

**Figure 6.** Figure 6: The steps of the two-dimensional version of the Quickhull algorithm. The two-dimensional Quickhull algorithm proceeds according to the following steps (see [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]

**Figure 7.** Figure 7: A three-dimensional convex hull. Before presenting the more general version of the Quickhull algorithm [18], and to keep this section selfcontained, we first recall key definitions and concepts related to convex hulls, following [55]. Consider a set of points S in R N…

**Figure 8.** Figure 8: A visual representation of the two cases described by the simplified beneath-beyond theorem. In [PITH_FULL_IMAGE:figures/full_fig_p026_8.png]

**Figure 9.** Figure 9: The steps of the generic N-dimensional version of the Quickhull algorithm. 1 Randomly select N + 1 charge-to-mass ratio vectors z (i) within the set of superextremal particles S, and construct the initial N-simplex generated by said vectors. For instance, in three dime…

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Tame Complexity of Effective Field Theories in the Quantum Gravity Landscape
hep-th 2026-01 conditional novelty 7.0 of 10

Effective field theories consistent with quantum gravity are conjectured to have uniformly bounded 'tame complexity', a quantitative measure of the information needed to specify them.

Reference graph

Works this paper leans on

55 extracted references · 27 canonical work pages · cited by 1 Pith paper

[41]

Ye and E

Y. Ye and E. Tse, An extension of karmarkar projective algorithm for convex quadratic programming, Math. Program. 44 (1989) 157–179

work page 1989
[23]

Gritzmann, V

P. Gritzmann, V. Klee and D. Larman, Largest j-simplices in n-polytopes, Discrete & Computational Geometry 13 (1995) 477

work page 1995
[1]

M. R. Douglas, The statistics of string/M-theory vacua , in 2nd String Phenomenology 2003 , pp. 102–113, 7, 2003

work page 2003
[2]

Ashok and M

S. Ashok and M. R. Douglas, Counting flux vacua , JHEP 01 (2004) 060 [hep-th/0307049]

arXiv 2004
[3]

M. R. Douglas, Basic results in vacuum statistics , Comptes Rendus Physique 5 (2004) 965 [hep-th/0409207]

arXiv 2004
[4]

Taylor and Y.-N

W. Taylor and Y.-N. Wang, The F-theory geometry with most flux vacua , JHEP 12 (2015) 164 [1511.03209]

arXiv 2015
[5]

Vafa, The String landscape and the swampland , hep-th/0509212

C. Vafa, The String landscape and the swampland , hep-th/0509212

arXiv
[6]

Arkani-Hamed, L

N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, The String landscape, black holes and gravity as the weakest force , JHEP 06 (2007) 060 [ hep-th/0601001]

arXiv 2007

Show all 55 references

[7]

Ooguri and C

H. Ooguri and C. Vafa, On the Geometry of the String Landscape and the Swampland , Nucl. Phys. B766 (2007) 21 [ hep-th/0605264]

2007 arXiv
[8]

Palti, The Swampland: Introduction and Review , Fortsch

E. Palti, The Swampland: Introduction and Review , Fortsch. Phys. 67 (2019) 1900037 [1903.06239]

2019 arXiv
[9]

N. B. Agmon, A. Bedroya, M. J. Kang and C. Vafa, Lectures on the string landscape and the Swampland, 2212.06187. – 27 –

arXiv
[10]

Palti, The Weak Gravity Conjecture and Scalar Fields , JHEP 08 (2017) 034 [1705.04328]

E. Palti, The Weak Gravity Conjecture and Scalar Fields , JHEP 08 (2017) 034 [1705.04328]

2017 arXiv
[11]

Cheung and G

C. Cheung and G. N. Remmen, Naturalness and the Weak Gravity Conjecture , Phys. Rev. Lett. 113 (2014) 051601 [ 1402.2287]

2014 arXiv
[12]

Palti, A Brief Introduction to the Weak Gravity Conjecture , LHEP 2020 (2020) 176

E. Palti, A Brief Introduction to the Weak Gravity Conjecture , LHEP 2020 (2020) 176

2020
[13]

Harlow, B

D. Harlow, B. Heidenreich, M. Reece and T. Rudelius, Weak gravity conjecture, Rev. Mod. Phys. 95 (2023) 035003 [ 2201.08380]

2023 arXiv
[14]

Denef and M

F. Denef and M. R. Douglas, Computational complexity of the landscape. I. , Annals Phys. 322 (2007) 1096 [ hep-th/0602072]

2007 arXiv
[15]

Denef, M

F. Denef, M. R. Douglas, B. Greene and C. Zukowski, Computational complexity of the landscape II—Cosmological considerations, Annals Phys. 392 (2018) 93 [ 1706.06430]

2018 arXiv
[16]

Halverson and F

J. Halverson and F. Ruehle, Computational Complexity of Vacua and Near-Vacua in Field and String Theory , Phys. Rev. D 99 (2019) 046015 [ 1809.08279]

2019 arXiv
[17]

Chazelle, An optimal convex hull algorithm in any fixed dimension , Discrete & Computational Geometry 10 (1993) 377

B. Chazelle, An optimal convex hull algorithm in any fixed dimension , Discrete & Computational Geometry 10 (1993) 377

1993
[18]

C. B. Barber, D. P. Dobkin and H. Huhdanpaa, The quickhull algorithm for convex hulls , ACM Trans. Math. Softw. 22 (1996) 469–483

1996
[19]

D. Avis, D. Bremner and R. Seidel, How good are convex hull algorithms? , Computational Geometry 7 (1997) 265

1997
[20]

Eaves and R

B. Eaves and R. Freund, Optimal scaling of balls and polyhedra , Mathematical Programming 23 (1982) 138

1982
[21]

Freund and J

R. Freund and J. Orlin, On the complexity of four polyhedral set containment problems , Mathematical Programming 33 (1985)

1985
[22]

Gritzmann and V

P. Gritzmann and V. Klee, On the complexity of some basic problems in computational convexity: I. containment problems , Discrete Mathematics 136 (1994) 129

1994
[24]

Arora and B

S. Arora and B. Barak, Computational Complexity: A Modern Approach . Cambridge University Press, USA, 2009

2009
[25]

Goldreich, P, NP, and NP-Completeness: The Basics of Computational Complexity

O. Goldreich, P, NP, and NP-Completeness: The Basics of Computational Complexity . Cambridge University Press, 2010

2010
[26]

van den Dries, Alfred tarski’s elimination theory for real closed fields , The Journal of Symbolic Logic 53 (1988) 7

L. van den Dries, Alfred tarski’s elimination theory for real closed fields , The Journal of Symbolic Logic 53 (1988) 7

1988
[27]

T. W. Grimm, L. Schlechter and M. van Vliet, Complexity in tame quantum theories , JHEP 05 (2024) 001 [ 2310.01484]. – 28 –

2024 arXiv
[28]

T. W. Grimm and M. van Vliet, On the Complexity of Quantum Field Theory , 2410.23338

arXiv
[29]

T. W. Grimm, D. Prieto and M. van Vliet, Tame Embeddings, Volume Growth, and Complexity of Moduli Spaces , 2503.15601

arXiv
[30]

G. W. Gibbons and K.-i. Maeda, Black Holes and Membranes in Higher Dimensional Theories with Dilaton Fields , Nucl. Phys. B 298 (1988) 741

1988
[31]

Garfinkle, G

D. Garfinkle, G. T. Horowitz and A. Strominger, Charged black holes in string theory , Phys. Rev. D 43 (1991) 3140

1991
[32]

G. T. Horowitz and A. Strominger, Black strings and P-branes , Nucl. Phys. B 360 (1991) 197

1991
[33]

Heidenreich, M

B. Heidenreich, M. Reece and T. Rudelius, Sharpening the Weak Gravity Conjecture with Dimensional Reduction, JHEP 02 (2016) 140 [ 1509.06374]

2016 arXiv
[34]

Gendler and I

N. Gendler and I. Valenzuela, Merging the weak gravity and distance conjectures using BPS extremal black holes , JHEP 01 (2021) 176 [ 2004.10768]

2021 arXiv
[35]

Bastian, T

B. Bastian, T. W. Grimm and D. van de Heisteeg, Weak gravity bounds in asymptotic string compactifications, JHEP 06 (2021) 162 [ 2011.08854]

2021 arXiv
[36]

M. Berg, O. Cheong, M. Kreveld and M. Overmars, Computational Geometry: Algorithms and Applications. Springer Berlin, 2008

2008
[37]

J. S. Greenfield, A proof for a quickhull algorithm , 1990, https://api.semanticscholar.org/CorpusID:117799832

1990
[38]

Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics

G. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics. Springer New York, 2012

2012
[39]

Boyd and L

S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004

2004
[40]

C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity. Prentice Hall, 1982

1982
[42]

Welzl, Smallest enclosing disks (balls and ellipsoids) , in New Results and New Trends in Computer Science, H

E. Welzl, Smallest enclosing disks (balls and ellipsoids) , in New Results and New Trends in Computer Science, H. Maurer, ed., (Berlin, Heidelberg), pp. 359–370, Springer Berlin Heidelberg, 1991

1991
[43]

T. M. Chan, Improved deterministic algorithms for linear programming in low dimensions , ACM Transactions on Algorithms (TALG) 14 (2018)

2018
[44]

Heidenreich, M

B. Heidenreich, M. Reece and T. Rudelius, Evidence for a sublattice weak gravity conjecture, JHEP 08 (2017) 025 [ 1606.08437]

2017 arXiv
[45]

Andriolo, D

S. Andriolo, D. Junghans, T. Noumi and G. Shiu, A Tower Weak Gravity Conjecture from Infrared Consistency, Fortsch. Phys. 66 (2018) 1800020 [ 1802.04287]. – 29 –

2018 arXiv
[46]

Fierro Cota, A

C. Fierro Cota, A. Mininno, T. Weigand and M. Wiesner, The minimal weak gravity conjecture, JHEP 05 (2024) 285 [ 2312.04619]

2024 arXiv
[47]

H. L. Bodlaender, P. Gritzmann, V. Klee and J. Van Leeuwen, Computational complexity of norm-maximization, Combinatorica 10 (1990) 203

1990
[48]

Calder´ on-Infante, A

J. Calder´ on-Infante, A. M. Uranga and I. Valenzuela,The Convex Hull Swampland Distance Conjecture and Bounds on Non-geodesics , JHEP 03 (2021) 299 [ 2012.00034]

2021 arXiv
[49]

Etheredge, B

M. Etheredge, B. Heidenreich, J. McNamara, T. Rudelius, I. Ruiz and I. Valenzuela, Running decompactification, sliding towers, and the distance conjecture , JHEP 12 (2023) 182 [2306.16440]

2023 arXiv
[50]

Etheredge, B

M. Etheredge, B. Heidenreich, T. Rudelius, I. Ruiz and I. Valenzuela, Taxonomy of infinite distance limits, JHEP 03 (2025) 213 [ 2405.20332]

2025 arXiv
[51]

Grieco, I

A. Grieco, I. Ruiz and I. Valenzuela, EFT strings and dualities in 4d N = 1, 2504.16984

arXiv
[52]

P. Lin, A. Mininno and G. Shiu, Formulating the Weak Gravity Conjecture in AdS Space , 2503.05862

arXiv
[53]

Lanza, Neural network learning and Quantum Gravity , JHEP 07 (2024) 105 [2403.03245]

S. Lanza, Neural network learning and Quantum Gravity , JHEP 07 (2024) 105 [2403.03245]

2024 arXiv
[54]

T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein, Introduction to Algorithms. The MIT Press, 2nd ed., 2001

2001
[55]

Gr¨ unbaum,Convex Polytopes

B. Gr¨ unbaum,Convex Polytopes. Springer New York, 2003. – 30 –

2003

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [41]

Ye and E

Y. Ye and E. Tse, An extension of karmarkar projective algorithm for convex quadratic programming, Math. Program. 44 (1989) 157–179

work page 1989

[2] [23]

Gritzmann, V

P. Gritzmann, V. Klee and D. Larman, Largest j-simplices in n-polytopes, Discrete & Computational Geometry 13 (1995) 477

work page 1995

[3] [1]

M. R. Douglas, The statistics of string/M-theory vacua , in 2nd String Phenomenology 2003 , pp. 102–113, 7, 2003

work page 2003

[4] [2]

Ashok and M

S. Ashok and M. R. Douglas, Counting flux vacua , JHEP 01 (2004) 060 [hep-th/0307049]

arXiv 2004

[5] [3]

M. R. Douglas, Basic results in vacuum statistics , Comptes Rendus Physique 5 (2004) 965 [hep-th/0409207]

arXiv 2004

[6] [4]

Taylor and Y.-N

W. Taylor and Y.-N. Wang, The F-theory geometry with most flux vacua , JHEP 12 (2015) 164 [1511.03209]

arXiv 2015

[7] [5]

Vafa, The String landscape and the swampland , hep-th/0509212

C. Vafa, The String landscape and the swampland , hep-th/0509212

arXiv

[8] [6]

Arkani-Hamed, L

N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, The String landscape, black holes and gravity as the weakest force , JHEP 06 (2007) 060 [ hep-th/0601001]

arXiv 2007