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REVIEW 4 major objections 4 minor 1 cited by

Free-Energy Machine for Combinatorial Optimization

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

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Pith's one-line read The Free-Energy Machine claims a single annealed mean-field solver outperforms problem-specific solvers on MaxCut, balanced MinCut, and Max k-SAT without Ising formulations.

desk verdict A solid, honest extension of mean-field annealing to Potts and higher-order problems, with correct math and public code, but the 'surpasses specialized solvers' claim rests on per-instance tuning that a reviewer should push on. read the letter →

arxiv 2412.09285 v1 pith:AMWBVHWC submitted 2024-12-12 cond-mat.stat-mech cond-mat.dis-nnmath.OCphysics.comp-ph

Zi-Song Shen , Feng Pan , Yao Wang , Yi-Ding Men , Wen-Biao Xu , Man-Hong Yung , Pan Zhang This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.dis-nnmath.OCphysics.comp-ph MSC 90C2782B2068T20 PACS 05.10.-a89.20.Ff75.10.Nr
keywords combinatorialoptimizationfree-energyminimizationmean-fieldannealingautomaticdifferentiationmaximumcutbalancedminimumMaxk-SATGPUparallelization
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 sets out to establish that one algorithm, the Free-Energy Machine (FEM), can serve as a general and fast heuristic for combinatorial optimization by replacing the original discrete search with the minimization of an annealed variational mean-field free energy. The cost function of a problem is treated as the energy of a spin system, and the ground state is recovered from the zero-temperature limit of the Boltzmann distribution, approximated by softmax-parameterized per-variable marginals and updated with gradient optimizers such as Adam. Because the gradients come from automatic differentiation, the same framework applies to binary, multi-state, and higher-order interacting variables without converting the problem into an Ising formulation. The paper reports best-known cuts on most G-set MaxCut instances, lower cut values than METIS on large balanced minimum cut problems, a roughly 22-27% reduction in communication volume in a chip-verification task, and optimal answers on 448 of 454 MaxSAT 2016 instances in about 0.07 seconds on a GPU.

What carries the argument

The load-bearing object is the variational mean-field free energy $F_{\rm MF}(\{P_i(\sigma_i)\},\beta)=U_{\rm MF}-S_{\rm MF}/\beta$, where $U_{\rm MF}$ is the mean-field expectation of the problem's cost and $S_{\rm MF}$ is the entropy of the product distribution over marginals. Each marginal $P_i(\sigma_i)$ is the softmax of a local field $h_i(\sigma_i)$, so the discrete search becomes a smooth optimization with exact gradients. Annealing $\beta$ from small to large values first lets the entropy dominate, then drives the marginals toward one-hot states; reading out the argmax of the marginals and selecting the lowest-energy replica yields the proposed solution. The machinery is general because the only problem-specific ingredient is the energy expectation, which can be fed to automatic differentiation or to explicit gradient formulas.

What would settle it

Run FEM on a held-out set of new and previously published instances using a single fixed default hyperparameter configuration (for example, the most common settings in the supplementary tables) and compare the resulting cuts and MaxSAT gaps with the per-instance-tuned numbers reported in the paper; if the fixed configuration systematically misses best-known values, the general-unified-solver claim would not survive.

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Extended reading notes

Core claim

The central claim is that minimizing replicas of the variational mean-field free energy $F_{\rm MF}=\sum_\sigma P_{\rm MF}(\sigma)E(\sigma)+(1/\beta)\sum_\sigma P_{\rm MF}(\sigma)\ln P_{\rm MF}(\sigma)$, annealed from high to low temperature, is a unified and competitive way to solve NP-hard combinatorial optimization. The marginals $P_i(\sigma_i)$ are parameterized by local fields through a softmax, the gradients are computed by automatic differentiation or explicit formulas, and many replicas are minimized in parallel on GPUs. The paper argues that this single recipe, without Ising formulations, produces the best-known cut on the 2000-node complete graph K2000, reaches the best-known value on most G-set instances with shorter time-to-solution than the discrete simulated bifurcation machine on 33 of 54 instances, beats METIS on q-way balanced minimum cut across real-world and million-node random graphs, and finds the best-known answer on 448 of 454 random MaxSAT 2016 instances, with an average GPU time of 0.074 seconds.

Load-bearing premise

The reported superiority assumes the results come from one fixed general method, whereas the supplementary tables give different optimizers, learning rates, temperature ranges, gradient scales, replica counts, and penalty schedules to individual instances, so the published numbers may not be reproducible with a single configuration on new instances.

Editorial extensions

If this is right

  • If the benchmarks hold, a single FEM implementation, differing by one line between MaxCut and balanced MinCut, could replace several problem-specific solvers for QUBO, Potts, and higher-order problems.
  • For a new combinatorial problem, a user only needs to write down the mean-field expectation of its cost; automatic differentiation then supplies the gradients, removing the need for Ising conversion.
  • The parallel-replica design means runtimes can shrink almost linearly with GPU or FPGA resources, making million-variable instances solvable in seconds or less.
  • In balanced graph partitioning, FEM's reported lower edge cuts on million-node graphs would translate directly into reduced communication overhead in parallel computing and chip verification.
  • On MaxSAT-style problems, sub-second average solution times would give practitioners a fast alternative to dedicated incomplete MaxSAT solvers on competition-size instances.

Reading between the lines

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

  • The generality claim could be tested directly by freezing one hyperparameter configuration and running FEM across all G-set, Walshaw, and MaxSAT 2016 instances without per-instance tuning; the supplementary tables currently vary those settings, so that experiment would separate the method from its tuning.
  • Because the paper writes energy expectations for graph coloring, community detection, and quantum error correction in the supplement, the same annealing scheme is immediately testable on those problems without further methodological work.
  • The mean-field product ansatz is the least expressive part of the loop; upgrading it to TAP or Bethe free energies, as the paper itself mentions for future work, could improve accuracy on hard glassy instances while keeping the optimization machinery intact.
  • The converged marginals could serve as a warm start for exact or local-search solvers, a hybrid use the paper does not explore.
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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 Free-Energy Machine (FEM), a variational mean-field annealing method for combinatorial optimization. The authors represent discrete variables by softmax-parameterized marginal probabilities, minimize a temperature-annealed variational free energy for several replicas in parallel using automatic differentiation and standard machine-learning optimizers, and read out solutions by argmax. The method is applied to MaxCut (Ising-type binary variables), q-way balanced minimum cut (Potts-type multi-state variables), and Max k-SAT (higher-order interactions), with additional chip-verification and random-graph scalability experiments. The paper claims that FEM solves these problems in a unified framework, without Ising conversion, and markedly surpasses state-of-the-art specialized solvers in both solution quality and speed.

Significance. If the empirical claims survive scrutiny, FEM would be a valuable contribution: a single GPU-friendly mean-field annealing framework covering binary, multi-state, and higher-order cost functions, with explicit gradient formulas (Eqs. S21-S28) that are derived correctly and reduce computation time. The variational free-energy expressions in Eqs. (9)-(11) are sound, and the paper provides reproducible code and detailed supplementary tables. The core physics idea is not entirely new - references to earlier mean-field annealing are included - but the combination with massive replica parallelism, automatic differentiation, and modern optimizers is a useful engineering advance. The main weakness is that the headline claim of being a general solver that surpasses specialized algorithms is not established by the reported protocol, because per-instance hyperparameter selection is used throughout the benchmarks. The mathematical framework itself is not circular: the free energies and gradients follow from the problem definitions with no fitted constants. The empirical general-solver claim, however, requires a fixed-configuration or properly separated tuning protocol.

major comments (4)
  1. [Methods, 'Hyperparameter tunning'; Tables S2, S3, S5] The central claim that FEM is a general method that markedly surpasses specialized solvers is weakened by the benchmark protocol. Tables S2, S3, and S5 assign different optimizers, learning rates, temperature ranges, gradient scales, replica counts, and penalty schedules to individual instances, and the chip-verification experiment chooses the Louvain coarsening level separately for q=4,8 versus q=16. Because the same benchmark instances were used to select these settings, the reported cuts, TTS values, and solution times do not demonstrate that a single FEM configuration, or even a fixed per-problem protocol, generalizes to unseen instances. Please report held-out results with a fixed configuration, or explicitly reposition FEM as requiring per-instance tuning and report the tuning cost as part of the time-to-solution.
  2. [Fig. 3(c) and Table S1] The MaxCut TTS comparison supports a more qualified claim than the text states. FEM achieves shorter TTS than dSBM on 33 of 54 G-set instances, but dSBM remains faster on many random, toroidal, and planar instances, including G6-G13, G22-G34, and G43-G47; for some instances FEM TTS is orders of magnitude larger (e.g., G33: 659,607 ms versus 57,766 ms, and G51-G54 are all slower). This is consistent with FEM being competitive rather than uniformly superior, and the Discussion should be revised to quantify the comparison and describe where FEM is and is not faster.
  3. [Table I and accompanying text] The bMinCut results do not uniformly support the statement that FEM 'significantly outperforms KaFFPaE with a large q value.' In Table I, FEM is sometimes worse than KaFFPaE (add20 q=16: 2057 versus 2054; data q=8: 669 versus 668; bcsstk33 q=8: 34440 versus 34437) and sometimes better. FEM is consistently better than METIS, which is a solid result, but the KaFFPaE comparison should be reported with run-to-run variability and the claim should be restricted to the instances and settings where the advantage actually holds.
  4. [Fig. 6 and Table S4] The MaxSAT speed comparison measures 'RC time' (replica computation time) and reports an average of 0.074 seconds per instance, but Table S5 shows per-instance hyperparameter tuning was performed in advance. The time spent selecting Nstep, Nreplica, Tmax, gamma_grad, learning rate, and momentum is not included in the comparison, while the incomplete-solver times from the competition represent full end-to-end runs. For a fair speed claim, the paper should either report end-to-end time including hyperparameter selection or state explicitly that the quoted time excludes tuning; as written, the speed advantage is overstated.
minor comments (4)
  1. [Methods, Eq. (9)] The sentence 'The expression of FMF can be explicitly formulated' appears in several places; for the MaxCut free energy in Eq. (9), the summation over σ_i and σ_j and the factor 1/2 conventions should be stated clearly, since the code in the Methods divides the cut by 2 during inference but the free energy does not.
  2. [Supplementary, Eq. (S33)] The phrase 'a slight modification on the constraint term in the formula of mean-field free energy should be made' has a grammatical issue and should be rewritten as 'we modify the constraint term in the mean-field free energy as follows.'
  3. [Supplementary, Table S5] The caption states 'the inverse-proportional scheduling is used for the annealing,' but Table S5 lists Tmin and Tmax rather than βmin and βmax; please define the relationship to the β schedule used in the main text to avoid confusion.
  4. [Discussion, paragraph 3] The claim that FEM 'markedly surpasses contemporary algorithms tailored specifically for each problem' should be tempered to reflect the mixed TTS results on G-set, the mixed KaFFPaE comparison, and the per-instance tuning used in the benchmarks.

Circularity Check

0 steps flagged · score 2.0 of 10

No definitional circularity: the free-energy formulas are derived from the problem energies, and benchmarks are external; the only caveat is per-instance hyperparameter tuning, which affects the generality claim but is not a circular derivation.

full rationale

The derivation chain is self-contained. The mean-field free energy in Eq. (3) is defined from the Boltzmann distribution and the KL divergence, and the problem-specific free energies in Eqs. (9), (10), and (11) are obtained by substituting the stated energy functions into the mean-field expectation; no fitted constants are folded into these formulas, and the reported solutions are obtained by minimizing those free energies. The benchmarks use external references as ground truth: best-known G-set cuts, METIS and KaFFPaE on Walshaw graphs, and MaxSAT 2016 competition best-known energies. There is no uniqueness theorem imported from the authors' prior work, no ansatz smuggled via self-citation, and no renaming of a known result as a new derivation. The circularity-adjacent concern is per-instance hyperparameter selection: Tables S2, S3, and S5 list instance-specific optimizers, learning rates, temperature ranges, gradient scales, replica counts, and penalty schedules, and the Methods section states that hyperparameters were 'carefully tuned.' This means the reported performance is not evidence for a single fixed configuration, and the general-solver claim would be stronger with a held-out configuration protocol. However, this is a benchmark-fairness and reproducibility issue, not a circularity in the mathematical derivation, because the reported cut and energy values are not algebraically forced by the hyperparameter values; they are genuine optimization outcomes. The central physics and machine-learning content remains independent of the benchmark-specific tuning.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The free-parameter list is dominated by optimizer and annealing choices that are tuned per instance; the axioms are the standard mean-field annealing assumption plus heuristic gradient modifications and the coarsening assumption. No new physical entities are introduced.

free parameters (6)
  • Annealing temperature range and schedule type (Tmax, Tmin, exponential/inverse-proportional) = e.g., G-set: Tmax from 0.19 to 22.94, Tmin from 3.12e-6 to 1.1e-3 (Table S2)
    Controls exploration-exploitation balance; tuned per instance and directly affects solution quality.
  • Optimizer and hyperparameters (SGD/RMSprop/Adam: lr, momentum, weight decay, alpha, beta1, beta2) = e.g., K2000 uses RMSprop lr=0.03, alpha=0.56, momentum=0.63, weight decay=0.013; per-instance values in Tables S2, S3…
    Chosen separately for each instance to obtain reported results.
  • Gradient scale gamma_grad = e.g., 0.65 to 10.8 on G-set (Table S2)
    Hyperparameter in Eq. (S20) scaling field gradients; tuned per instance.
  • Balance penalty schedule (lambda_max, linear ramp) = e.g., 0.2 for 3elt q=4; up to 2.07 for add20 q=2 (Table S3)
    Soft constraint weight for balanced partitions; tuned per graph and q.
  • Replica count Nreplica/Nbatch = e.g., 50 to 1000 across benchmarks
    More replicas improve best-found solution; computational budget chosen per benchmark.
  • Louvain coarsening level for chip verification = level 3 for q=4,8; level 2 for q=16
    Chosen after inspecting the community graph; affects whether the balanced constraint is feasible.
assumptions (4)
  • domain assumption Minimizing the annealed variational mean-field free energy over product distributions yields marginals whose per-variable argmax approximates the ground state of E(sigma).
    Assumed throughout the Results and Eq. (3); no proof that the annealing path avoids bad local minima for arbitrary COPs.
  • ad hoc to paper One-hot discretization and L1-norm gradient normalization preserve the optimizer's ability to reach global optima.
    Heuristic modifications in Eqs. (S25) and (S27); justified only by observed performance.
  • domain assumption Louvain coarsening preserves near-optimal balanced minimum cut structure of the original graph.
    Used in the chip-verification application; no approximation guarantee given.
  • ad hoc to paper Per-instance hyperparameter selection is a valid protocol for benchmarking a general method.
    Underlies all comparative claims; Tables S2, S3, S5 show instance-specific settings.

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Pith. "Pith review of Free-Energy Machine for Combinatorial Optimization." pith.science (2026). https://pith.science/paper/AMWBVHWC

@misc{pith2026241209285,
  author       = {Pith},
  title        = {Pith review of: Free-Energy Machine for Combinatorial Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AMWBVHWC}},
  note         = {Machine review of arXiv:2412.09285}
}
abstract

Finding optimal solutions to combinatorial optimization problems is pivotal in both scientific and technological domains, within academic research and industrial applications. A considerable amount of effort has been invested in the development of accelerated methods that leverage sophisticated models and harness the power of advanced computational hardware. Despite the advancements, a critical challenge persists, the dual demand for both high efficiency and broad generality in solving problems. In this work, we propose a general method, Free-Energy Machine (FEM), based on the ideas of free-energy minimization in statistical physics, combined with automatic differentiation and gradient-based optimization in machine learning. The algorithm is flexible, solving various combinatorial optimization problems using a unified framework, and is efficient, naturally utilizing massive parallel computational devices such as graph processing units (GPUs) and field-programmable gate arrays (FPGAs). We benchmark our algorithm on various problems including the maximum cut problems, balanced minimum cut problems, and maximum $k$-satisfiability problems, scaled to millions of variables, across both synthetic, real-world, and competition problem instances. The findings indicate that our algorithm not only exhibits exceptional speed but also surpasses the performance of state-of-the-art algorithms tailored for individual problems. This highlights that the interdisciplinary fusion of statistical physics and machine learning opens the door to delivering cutting-edge methodologies that will have broad implications across various scientific and industrial landscapes.

Figures

Figures reproduced from arXiv: 2412.09285 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. (b) shows a schematic of the typical evolutions of in￾ternal energy UMF = P σ PMF(σ)E(σ), and entropy S MF = − P σ PMF(σ) ln PMF(σ) as a function of β. The correspond￾ing evolutions of {Pi(σi)} of the replicas with the annealing are depicted in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

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Reference graph

Works this paper leans on

70 extracted references · 30 canonical work pages · cited by 1 Pith paper

  1. [1]

    Du \ and\ author P

    author author D. Du \ and\ author P. M. \ Pardalos ,\ @noop title Handbook of combinatorial optimization ,\ Vol. volume 4 \ ( publisher Springer Science & Business Media ,\ year 1998 ) NoStop

    work page 1998

  2. [2]

    Arora \ and\ author B

    author author S. Arora \ and\ author B. Barak ,\ @noop title Computational complexity: a modern approach \ ( publisher Cambridge University Press ,\ year 2009 ) NoStop

    work page 2009

  3. [3]

    Kirkpatrick , author C

    author author S. Kirkpatrick , author C. D. \ Gelatt Jr ,\ and\ author M. P. \ Vecchi ,\ title title Optimization by simulated annealing ,\ https://doi.org/10.1126/science.220.4598.671 journal journal Science \ volume 220 ,\ pages 671 ( year 1983 ) NoStop

    doi:10.1126/science.220.4598.671

  4. [4]

    Selman , author H

    author author B. Selman , author H. A. \ Kautz , author B. Cohen , et al. ,\ title title Noise strategies for improving local search ,\ https://cdn.aaai.org/AAAI/1994/AAAI94-051.pdf journal journal AAAI \ volume 94 ,\ pages 337 ( year 1994 ) NoStop

    work page 1994

  5. [5]

    Glover \ and\ author M

    author author F. Glover \ and\ author M. Laguna ,\ @noop title Tabu search \ ( publisher Springer ,\ year 1998 ) NoStop

    work page 1998

  6. [6]

    Boettcher \ and\ author A

    author author S. Boettcher \ and\ author A. G. \ Percus ,\ title title Optimization with extremal dynamics ,\ https://doi.org/10.1103/PhysRevLett.86.5211 journal journal Phys. Rev. Lett. \ volume 86 ,\ pages 5211 ( year 2001 ) NoStop

    work page doi:10.1103/physrevlett.86.5211 2001

  7. [7]

    author author F. Barahona ,\ title title On the computational complexity of ising spin glass models ,\ https://doi.org/10.1088/0305-4470/15/10/028 journal journal Journal of Physics A: Mathematical and General \ volume 15 ,\ pages 3241 ( year 1982 ) NoStop

    doi:10.1088/0305-4470/15/10/028

  8. [8]

    author author E. S. \ Tiunov , author A. E. \ Ulanov ,\ and\ author A. Lvovsky ,\ title title Annealing by simulating the coherent Ising machine ,\ https://doi.org/10.1364/OE.27.010288 journal journal Optics Express \ volume 27 ,\ pages 10288 ( year 2019 ) NoStop

    doi:10.1364/oe.27.010288

Show all 70 references
  1. [9]

    author author A. D. \ King , author W. Bernoudy , author J. King , author A. J. \ Berkley ,\ and\ author T. Lanting ,\ title title Emulating the coherent Ising machine with a mean-field algorithm ,\ https://doi.org/10.48550/arXiv.1806.08422 journal journal arXiv preprint arXiv...

    doi arXiv

  2. [10]

    Goto , author K

    author author H. Goto , author K. Tatsumura ,\ and\ author A. R. \ Dixon ,\ title title Combinatorial optimization by simulating adiabatic bifurcations in nonlinear Hamiltonian systems ,\ https://doi.org/10.1126/sciadv.aav2372 journal journal Science Advances \ volume 5 ,\ pag...

    doi

  3. [11]

    Goto , author K

    author author H. Goto , author K. Endo , author M. Suzuki , author Y. Sakai , author T. Kanao , author Y. Hamakawa , author R. Hidaka , author M. Yamasaki ,\ and\ author K. Tatsumura ,\ title title High-performance combinatorial optimization based on classical mechanics ,\ htt...

    doi

  4. [12]

    author author M. W. \ Johnson , author M. H. \ Amin , author S. Gildert , author T. Lanting , author F. Hamze , author N. Dickson , author R. Harris , author A. J. \ Berkley , author J. Johansson , author P. Bunyk , et al. ,\ title title Quantum annealing with manufactured spi...

    doi

  5. [13]

    Inagaki , author K

    author author T. Inagaki , author K. Inaba , author R. Hamerly , author K. Inoue , author Y. Yamamoto ,\ and\ author H. Takesue ,\ title title Large-scale Ising spin network based on degenerate optical parametric oscillators ,\ https://doi.org/10.1038/nphoton.2016.68 journal j...

    doi

  6. [14]

    Honjo , author T

    author author T. Honjo , author T. Sonobe , author K. Inaba , author T. Inagaki , author T. Ikuta , author Y. Yamada , author T. Kazama , author K. Enbutsu , author T. Umeki , author R. Kasahara , et al. ,\ title title 100,000-spin coherent Ising machine ,\ https://doi.org/10....

    doi

  7. [15]

    Pierangeli , author G

    author author D. Pierangeli , author G. Marcucci ,\ and\ author C. Conti ,\ title title Large-scale photonic Ising machine by spatial light modulation ,\ https://doi.org/10.1103/PhysRevLett.122.213902 journal journal Phys. Rev. Lett. \ volume 122 ,\ pages 213902 ( year 2019 ) NoStop

    doi

  8. [16]

    Mallick , author M

    author author A. Mallick , author M. K. \ Bashar , author D. S. \ Truesdell , author B. H. \ Calhoun , author S. Joshi ,\ and\ author N. Shukla ,\ title title Using synchronized oscillators to compute the maximum independent set ,\ https://doi.org/10.1038/s41467-020-18445-1 jo...

    doi

  9. [17]

    Cai , author S

    author author F. Cai , author S. Kumar , author T. Van Vaerenbergh , author X. Sheng , author R. Liu , author C. Li , author Z. Liu , author M. Foltin , author S. Yu , author Q. Xia , et al. ,\ title title Power-efficient combinatorial optimization using intrinsic noise in mem...

    doi

  10. [18]

    author author N. A. \ Aadit , author A. Grimaldi , author M. Carpentieri , author L. Theogarajan , author J. M. \ Martinis , author G. Finocchio ,\ and\ author K. Y. \ Camsari ,\ title title Massively parallel probabilistic computing with sparse Ising machines ,\ https://doi.o...

    doi

  11. [19]

    Mohseni , author P

    author author N. Mohseni , author P. L. \ McMahon ,\ and\ author T. Byrnes ,\ title title Ising machines as hardware solvers of combinatorial optimization problems ,\ https://doi.org/10.1038/s42254-022-00440-8 journal journal Nature Reviews Physics \ volume 4 ,\ pages 363 ( ye...

    doi

  12. [20]

    Kochenberger , author J.-K

    author author G. Kochenberger , author J.-K. \ Hao , author F. Glover , author M. Lewis , author Z. L \"u , author H. Wang ,\ and\ author Y. Wang ,\ title title The unconstrained binary quadratic programming problem: a survey ,\ https://doi.org/10.1007/s10878-014-9734-0 journa...

    doi

  13. [21]

    author author A. Lucas ,\ title title Ising formulations of many NP problems ,\ https://doi.org/10.3389/fphy.2014.00005 journal journal Frontiers in Physics \ volume 2 ,\ pages 5 ( year 2014 ) NoStop

    2014

  14. [22]

    author author E. Gardner ,\ title title Spin glasses with p -spin interactions ,\ https://doi.org/10.1016/0550-3213(85)90374-8 journal journal Nuclear Physics B \ volume 257 ,\ pages 747 ( year 1985 ) NoStop

    1985 doi

  15. [23]

    author author R. M. \ Karp ,\ https://doi.org/10.1007/978-3-540-68279-0_8 title Reducibility among combinatorial problems \ ( publisher Springer ,\ year 2010 ) NoStop

    2010 doi

  16. [24]

    \ Wu ,\ title title The Potts model ,\ https://doi.org/10.1103/RevModPhys.54.235 journal journal Rev

    author author F.-Y. \ Wu ,\ title title The Potts model ,\ https://doi.org/10.1103/RevModPhys.54.235 journal journal Rev. Mod. Phys. \ volume 54 ,\ pages 235 ( year 1982 ) NoStop

    1982 doi

  17. [25]

    author author T. R. \ Jensen \ and\ author B. Toft ,\ @noop title Graph coloring problems \ ( publisher John Wiley & Sons ,\ year 2011 ) NoStop

    2011

  18. [26]

    author author M. E. \ Newman ,\ title title Modularity and community structure in networks ,\ https://doi.org/https://doi.org/10.1073/pnas.0601602103 journal journal Proceedings of the National Academy of Sciences \ volume 103 ,\ pages 8577 ( year 2006 ) NoStop

    doi

  19. [27]

    Papadimitriou \ and\ author K

    author author C. Papadimitriou \ and\ author K. Steiglitz ,\ https://books.google.com.sg/books?id=cDY-joeCGoIC title Combinatorial Optimization: Algorithms and Complexity ,\ Dover Books on Computer Science\ ( publisher Dover Publications ,\ year 1998 ) NoStop

    1998

  20. [28]

    M \'e zard , author G

    author author M. M \'e zard , author G. Parisi ,\ and\ author R. Zecchina ,\ title title Analytic and algorithmic solution of random satisfiability problems ,\ https://doi.org/https://doi.org/10.1126/science.1073287 journal journal Science \ volume 297 ,\ pages 812 ( year 2002...

    doi

  21. [29]

    author author D. P. \ Kingma \ and\ author J. Ba ,\ title title Adam: A method for stochastic optimization ,\ https://doi.org/10.48550/arXiv.1412.6980 journal journal arXiv preprint arXiv:1412.6980 \ ( year 2014 ) NoStop

    2014 doi arXiv

  22. [30]

    author author D. A. \ Chermoshentsev , author A. O. \ Malyshev , author M. Esencan , author E. S. \ Tiunov , author D. Mendoza , author A. Aspuru-Guzik , author A. K. \ Fedorov ,\ and\ author A. I. \ Lvovsky ,\ title title Polynomial unconstrained binary optimisation inspired ...

    doi arXiv

  23. [31]

    Bybee , author D

    author author C. Bybee , author D. Kleyko , author D. E. \ Nikonov , author A. Khosrowshahi , author B. A. \ Olshausen ,\ and\ author F. T. \ Sommer ,\ title title Efficient optimization with higher-order Ising machines ,\ https://doi.org/10.1038/s41467-023-41214-9 journal jou...

    doi

  24. [32]

    Kanao \ and\ author H

    author author T. Kanao \ and\ author H. Goto ,\ title title Simulated bifurcation for higher-order cost functions ,\ https://doi.org/10.35848/1882-0786/acaba9 journal journal Applied Physics Express \ volume 16 ,\ pages 014501 ( year 2022 ) NoStop

    doi

  25. [33]

    Reifenstein , author T

    author author S. Reifenstein , author T. Leleu , author T. McKenna , author M. Jankowski , author M.-G. \ Suh , author E. Ng , author F. Khoyratee , author Z. Toroczkai ,\ and\ author Y. Yamamoto ,\ title title Coherent SAT solvers: a tutorial ,\ https://doi.org/10.1364/AOP.47...

    doi

  26. [34]

    M \'e zard , author G

    author author M. M \'e zard , author G. Parisi , author N. Sourlas , author G. Toulouse ,\ and\ author M. Virasoro ,\ title title Replica symmetry breaking and the nature of the spin glass phase ,\ https://hal.science/jpa-00209816/file/ajp-jphys_1984_45_5_843_0.pdf journal jou...

    1984

  27. [35]

    M \'e zard , author G

    author author M. M \'e zard , author G. Parisi ,\ and\ author M. A. \ Virasoro ,\ https://doi.org/https://doi.org/10.1142/0271 title Spin glass theory and beyond: An Introduction to the Replica Method and Its Applications ,\ Vol. volume 9 \ ( publisher World Scientific Publish...

    doi

  28. [36]

    Wu , author L

    author author D. Wu , author L. Wang ,\ and\ author P. Zhang ,\ title title Solving statistical mechanics using variational autoregressive networks ,\ https://doi.org/10.1103/PhysRevLett.122.080602 journal journal Phys. Rev. Lett. \ volume 122 ,\ pages 080602 ( year 2019 ) NoStop

    doi

  29. [37]

    Hibat-Allah , author E

    author author M. Hibat-Allah , author E. M. \ Inack , author R. Wiersema , author R. G. \ Melko ,\ and\ author J. Carrasquilla ,\ title title Variational neural annealing ,\ https://doi.org/10.1038/s42256-021-00401-3 journal journal Nature Machine Intelligence \ volume 3 ,\ pa...

    doi

  30. [38]

    Paszke , author S

    author author A. Paszke , author S. Gross , author F. Massa , author A. Lerer , author J. Bradbury , author G. Chanan , author T. Killeen , author Z. Lin , author N. Gimelshein , author L. Antiga , et al. ,\ title title Pytorch: An imperative style, high-performance deep learn...

    2019

  31. [39]

    Abadi , author A

    author author M. Abadi , author A. Agarwal , author P. Barham , author E. Brevdo , author Z. Chen , author C. Citro , author G. S. \ Corrado , author A. Davis , author J. Dean , author M. Devin , et al. ,\ title title Tensorflow: Large-scale machine learning on heterogeneous d...

    doi arXiv

  32. [40]

    author author Y. Y. \ Boykov \ and\ author M.-P. \ Jolly ,\ title title Interactive graph cuts for optimal boundary & region segmentation of objects in N-D images ,\ in\ https://doi.org/10.1109/ICCV.2001.937505 booktitle Proceedings eighth IEEE international conference on comp...

    2001

  33. [41]

    o tschel , author M. J \

    author author F. Barahona , author M. Gr \"o tschel , author M. J \"u nger ,\ and\ author G. Reinelt ,\ title title An application of combinatorial optimization to statistical physics and circuit layout design ,\ https://doi.org/https://doi.org/10.1287/opre.36.3.493 journal jo...

    doi

  34. [42]

    Facchetti , author G

    author author G. Facchetti , author G. Iacono ,\ and\ author C. Altafini ,\ title title Computing global structural balance in large-scale signed social networks ,\ https://doi.org/https://doi.org/10.1073/pnas.1109521108 journal journal Proceedings of the National Academy of S...

    doi

  35. [43]

    B \"o hm , author T

    author author F. B \"o hm , author T. V. \ Vaerenbergh , author G. Verschaffelt ,\ and\ author G. Van der Sande ,\ title title Order-of-magnitude differences in computational performance of analog Ising machines induced by the choice of nonlinearity ,\ https://doi.org/10.1038/...

    doi

  36. [44]

    Rinaldy ,\ @noop title rudy graph generator ,\ howpublished http://www-user.tu-chemnitz.de/ helmberg/rudy.tar.gz NoStop

    author author G. Rinaldy ,\ @noop title rudy graph generator ,\ howpublished http://www-user.tu-chemnitz.de/ helmberg/rudy.tar.gz NoStop

  37. [45]

    Inagaki , author Y

    author author T. Inagaki , author Y. Haribara , author K. Igarashi , author T. Sonobe , author S. Tamate , author T. Honjo , author A. Marandi , author P. L. \ McMahon , author T. Umeki , author K. Enbutsu , et al. ,\ title title A coherent Ising machine for 2000-node optimiza...

    doi

  38. [46]

    Ye ,\ @noop title G-set test problems ,\ howpublished https://web.stanford.edu/ yyye/yyye/Gset/ NoStop

    author author Y. Ye ,\ @noop title G-set test problems ,\ howpublished https://web.stanford.edu/ yyye/yyye/Gset/ NoStop

  39. [47]

    Walshaw ,\ @noop title The graph partitioning archive ,\ howpublished https://chriswalshaw.co.uk/partition/ NoStop

    author author C. Walshaw ,\ @noop title The graph partitioning archive ,\ howpublished https://chriswalshaw.co.uk/partition/ NoStop

  40. [48]

    author author M. J. \ Schuetz , author J. K. \ Brubaker ,\ and\ author H. G. \ Katzgraber ,\ title title Combinatorial optimization with physics-inspired graph neural networks ,\ https://doi.org/10.1038/s42256-022-00468-6 journal journal Nature Machine Intelligence \ volume 4 ...

    doi

  41. [49]

    Ushijima-Mwesigwa , author C

    author author H. Ushijima-Mwesigwa , author C. F. \ Negre ,\ and\ author S. M. \ Mniszewski ,\ title title Graph partitioning using quantum annealing on the D-Wave system ,\ in\ https://doi.org/10.1145/3149526.3149531 booktitle Proceedings of the Second International Workshop ...

  42. [50]

    author author P. W. \ Holland , author K. B. \ Laskey ,\ and\ author S. Leinhardt ,\ title title Stochastic blockmodels: First steps ,\ https://doi.org/https://doi.org/10.1016/0378-8733(83)90021-7 journal journal Social Networks \ volume 5 ,\ pages 109 ( year 1983 ) NoStop

    doi

  43. [51]

    Karrer \ and\ author M

    author author B. Karrer \ and\ author M. E. \ Newman ,\ title title Stochastic blockmodels and community structure in networks ,\ https://doi.org/http://dx.doi.org/10.1103/PhysRevE.83.016107 journal journal Phys. Rev. E \ volume 83 ,\ pages 016107 ( year 2011 ) NoStop

    doi

  44. [52]

    Karypis \ and\ author V

    author author G. Karypis \ and\ author V. Kumar ,\ title title A fast and high quality multilevel scheme for partitioning irregular graphs ,\ https://doi.org/10.1137/S1064827595287997 journal journal SIAM Journal on Scientific Computing \ volume 20 ,\ pages 359 ( year 1998 ) NoStop

    doi

  45. [53]

    Sanders \ and\ author C

    author author P. Sanders \ and\ author C. Schulz ,\ title title Think locally, act globally: Highly balanced graph partitioning ,\ in\ https://link.springer.com/chapter/10.1007/978-3-642-38527-8_16 booktitle International Symposium on Experimental Algorithms \ ( organization S...

    doi

  46. [54]

    Acer , author E

    author author S. Acer , author E. G. \ Boman , author C. A. \ Glusa ,\ and\ author S. Rajamanickam ,\ title title Sphynx: A parallel multi-gpu graph partitioner for distributed-memory systems ,\ https://doi.org/https://doi.org/10.1016/j.parco.2021.102769 journal journal Parall...

  47. [55]

    Chuzhoy , author Y

    author author J. Chuzhoy , author Y. Gao , author J. Li , author D. Nanongkai , author R. Peng ,\ and\ author T. Saranurak ,\ title title A deterministic algorithm for balanced cut with applications to dynamic connectivity, flows, and beyond ,\ in\ https://doi.org/https://doi....

  48. [56]

    author author W. K. \ Lam ,\ @noop title Hardware design verification: simulation and formal method-based approaches (Prentice Hall Modern semiconductor design series) \ ( publisher Prentice Hall PTR ,\ year 2005 ) NoStop

    2005

  49. [57]

    Patil \ and\ author D

    author author S. Patil \ and\ author D. Kulkarni ,\ title title K-way spectral graph partitioning for load balancing in parallel computing ,\ https://doi.org/10.1007/s41870-021-00777-w journal journal International Journal of Information Technology \ volume 13 ,\ pages 1893 ( ...

    doi

  50. [58]

    author author V. D. \ Blondel , author J.-L. \ Guillaume , author R. Lambiotte ,\ and\ author E. Lefebvre ,\ title title Fast unfolding of communities in large networks ,\ https://doi.org/10.1088/1742-5468/2008/10/P10008 journal journal Journal of statistical mechanics: theory...

    2008 doi

  51. [59]

    @noop title The graph partitioning framework KaHIP ,\ howpublished https://github.com/KaHIP/KaHIP NoStop

  52. [60]

    Moln \'a r , author F

    author author B. Moln \'a r , author F. Moln \'a r , author M. Varga , author Z. Toroczkai ,\ and\ author M. Ercsey-Ravasz ,\ title title A continuous-time MaxSAT solver with high analog performance ,\ https://doi.org/10.1038/s41467-018-07327-2 journal journal Nature Communica...

    doi

  53. [61]

    @noop title Eleventh Max-SAT Evaluation ,\ howpublished http://www.maxsat.udl.cat/16/benchmarks/index.html NoStop

  54. [62]

    author author S. A. \ Cook ,\ title title The complexity of theorem-proving procedures ,\ in\ https://doi.org/10.1145/800157.805047 booktitle Proceedings of the Third Annual ACM Symposium on Theory of Computing ,\ series and number STOC '71 \ ( publisher Association for Comput...

  55. [63]

    author author S. Cook ,\ title title The P versus NP problem ,\ https://www.cs.utoronto.ca/ toni/Courses/Complexity/handouts/cook-clay.pdf journal journal Clay Mathematics Institute \ volume 2 ,\ pages 6 ( year 2000 ) NoStop

    2000

  56. [64]

    Tieleman , author G

    author author T. Tieleman , author G. Hinton , et al. ,\ title title Lecture 6.5-rmsprop: Divide the gradient by a running average of its recent magnitude ,\ @noop journal journal COURSERA: Neural networks for machine learning \ volume 4 ,\ pages 26 ( year 2012 ) NoStop

    2012

  57. [65]

    @noop title Optim tools of pytorch ,\ howpublished https://pytorch.org/docs/stable/optim.html NoStop

  58. [66]

    Bilbro , author R

    author author G. Bilbro , author R. Mann , author T. Miller , author W. Snyder , author D. van den Bout ,\ and\ author M. White ,\ title title Optimization by mean field annealing ,\ https://papers.nips.cc/paper_files/paper/1988/file/ec5decca5ed3d6b8079e2e7e7bacc9f2-Paper.pdf ...

    1988

  59. [67]

    @noop title Metis software package: version 5.1.0 ,\ howpublished http://glaros.dtc.umn.edu/gkhome/metis/metis/download NoStop

  60. [68]

    Sanders \ and\ author C

    author author P. Sanders \ and\ author C. Schulz ,\ title title Kahip v3. 00--karlsruhe high quality partitioning--user guide ,\ https://doi.org/10.48550/arXiv.1311.1714 journal journal arXiv preprint arXiv:1311.1714 \ ( year 2013 b ) NoStop

    doi arXiv

  61. [69]

    Fujisaki , author H

    author author J. Fujisaki , author H. Oshima , author S. Sato ,\ and\ author K. Fujii ,\ title title Practical and scalable decoder for topological quantum error correction with an Ising machine ,\ https://doi.org/10.1103/PhysRevResearch.4.043086 journal journal Phys. Rev. Res...

    doi

  62. [70]

    Fujisaki , author K

    author author J. Fujisaki , author K. Maruyama , author H. Oshima , author S. Sato , author T. Sakashita , author Y. Takeuchi ,\ and\ author K. Fujii ,\ title title Quantum error correction with an ising machine under circuit-level noise ,\ https://doi.org/10.1103/PhysRevResea...

    doi

Pith tools

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