Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

Unsupervised Learning for the Elementary Shortest Path Problem

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read A neural solver can find near-optimal elementary paths without any labeled solutions, and it comes with a high-probability guarantee.

desk verdict Novel unsupervised heuristic for ESPP, but the paper's central high-probability certificate is not valid as stated, and the trained loss is not the one being bounded. read the letter →

arxiv 2508.01557 v1 pith:T6ZZKNWI submitted 2025-08-03 cs.LG

Jingyi Chen , Xinyuan Zhang , Xinwu Qian This is my paper · ORCID
classification cs.LG
keywords elementaryshortestpathproblemnegative-costcyclesunsupervisedgraphneuralnetworkprobabilisticmethodalgorithmicalignmentvaluefunctionestimationBellman-Fordhigh-probabilitycertificate
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

The paper sets out to show that the Elementary Shortest Path Problem, which is NP-hard in the presence of negative-cost cycles, can be solved near-optimally without any labeled solutions. It trains an unsupervised graph neural network to output node value estimates and edge-selection probabilities, and proves that a particular surrogate loss gives a high-probability certificate that the decoded path's cost is within some tolerance of optimal. The key move is to avoid the exponentially many subtour-elimination constraints by driving the learned subgraph free of negative-cost cycles, so that standard shortest-path reasoning applies. Experiments on random, grid, and scale-free graphs up to 100 nodes support the claim that the method outperforms unsupervised baselines and classical heuristics and generalizes across sizes and topologies.

What carries the argument

The load-bearing device is the negative-cost-cycle surrogate Φ_θ(G), defined as the average over edges of the slack δ_uv = [d_θ(u) − w_uv − d_θ(v)]_+ plus a probability-weighted version of the same slack; when the learned value estimates satisfy d_θ(u) ≤ w_uv + d_θ(v) on every edge, no negative-cost cycle survives in the weighted subgraph. Coupled with the value-operator alignment term |$F_θ^{{(k*+1)}}$(G) − $F_θ^{{(k*)}}$(G)|, the loss L = λ1 times that alignment term plus λ2 times Φ supplies the high-probability near-optimality certificate. In the implemented model, a message-passing network with a min-aggregation mirroring the Bellman operator outputs d_θ(v), edge probabilities p_uv = σ(d_θ(v) − d_θ(u)), and the full loss adds flow conservation, distributional alignment to a softmax target, softmin dynamic-programming alignment, unrolled Bellman-Ford consistency, and an advantage baseline from a fractional LP relaxation.

What would settle it

Train ESPP-NNAA while logging the certified loss L = λ1|$F_θ^{{(k*+1)}}$ − $F_θ^{{(k*)}}$| + λ2Φ_θ; if L_Full decreases while L stays large or grows on graphs that still contain negative-cost cycles, the theorem does not apply to the trained model. Conversely, on any graph family where a model with small L decodes paths with large optimality gap, the Lemma 2 bound would be violated.

Watch

Extended reading notes

Core claim

The paper claims that the Elementary Shortest Path Problem becomes tractable in practice through an unsupervised graph neural network that jointly learns a node value function and edge probabilities from a surrogate loss. The central assertion is Theorem 1: with enough independent training graphs, if the empirical surrogate loss is small, then with probability at least 1−β over the training data, the decoded elementary path has optimality gap larger than γ with probability at most α. The proof decomposes the objective into a value-operator alignment term and a negative-cost-cycle surrogate, showing that a small surrogate forces the absence of negative cycles with high probability, after which an elementary shortest path exists and the optimality gap is bounded.

Load-bearing premise

The practical training loss L_Full includes terms beyond the two-term loss that Theorem 1 certifies, and the paper gives no proof that minimizing L_Full keeps the certified loss small.

Editorial extensions

If this is right

  • Elementary paths can be decoded in polynomial time from learned probabilities, and increasing the number of decoding samples reduces the optimality gap, with 200 samples nearly matching exact solutions on 30- and 50-node graphs.
  • Training on small Erdős–Rényi graphs transfers to larger graphs and to grid and Barabási–Albert topologies, so the learned value functions appear to capture routing structure rather than memorizing instances.
  • The certificate is directly relevant to branch-and-price frameworks, where repeatedly solving ESPP as a pricing subproblem requires provable performance guarantees rather than heuristic-only solutions.
  • Ablations show that the distributional alignment term is the most critical component; removing it collapses the gap from roughly −76% to −3% at 50 nodes, indicating that aligning edge probabilities with Bellman-optimal transitions carries most of the solution quality.

Reading between the lines

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

  • The certificate theorem bounds only the two-term loss L, while training minimizes L_Full, which includes a negative advantage term and three additional inductive-bias terms; closing that gap would require proving that minimizing L_Full controls L, or re-proving the theorem directly for L_Full.
  • The same negative-cycle surrogate could be applied to other combinatorial problems where feasibility is enforced by eliminating forbidden structures rather than by adding exponential constraint sets, such as resource-constrained or prize-collecting path variants.
  • A testable extension is to measure the empirical distribution of Φ and the Bellman residual on held-out graphs during training; the certificate predicts that small Φ concentrates probability on negative-cycle-free subgraphs, which could be checked directly.
  • Because the decoder is sequential and elementary by construction, the method could be plugged into column-generation pricing loops where current LP reduced costs define edge weights, an application the paper names but does not implement.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript proposes ESPP-NNAA, an unsupervised graph-neural-network framework for the elementary shortest path problem (ESPP) on graphs with negative-cost cycles. The method learns node value estimates and edge-selection probabilities, and it minimizes a surrogate loss intended to reduce negative-cost cycles while embedding algorithmic alignment with Bellman-Ford. The paper claims a high-probability certificate of near-optimality for the decoded elementary path, and it reports experiments on Erdős–Rényi, grid, and Barabási–Albert graphs of up to 100 nodes, including comparisons with unsupervised baselines, an exact labeling algorithm, LP heuristics, beam search, and randomized decoding.

Significance. If the theoretical certificate were valid, the paper would be a substantial contribution: it would provide a label-free, finite-sample performance guarantee for an NP-hard combinatorial problem, and the experimental study is a genuine strength, with multiple graph families, cross-topology generalization tests, and component ablations. However, the central theoretical apparatus is not sound. The main probabilistic lemma is false as stated, and the guarantee is proved for a simplified loss that is not the objective actually minimized during training. The empirical comparisons may still be of interest, but the advertised theoretical contribution does not hold as written.

major comments (3)
  1. [Section 3, Proposition 1] The high-probability non-negative-cycle guarantee is false as stated. Consider G with vertices s, a, b, t and edges s→a (cost 0), a→b (cost −η), b→a (cost 0), b→t (cost 0). Set dθ(s)=dθ(t)=0 and dθ(a)=dθ(b)=−η/2; the printed condition dθ(u) ≤ w_uv − dθ(v) holds on every edge. Set p_ab=p_ba=1/2, p_sa=0.1, and p_bt=1. Then δ_sa=η/2, δ_ab=η, and δ_ba=δ_bt=0, so Φθ(G)=0.5125η. For η=0.0002 and ε=0.001, we have Φθ(G)=1.025×10^{-4} ≤ ε/(M|E|)=1.25×10^{-4}, but the negative cycle a→b→a survives with probability p_ab p_ba = 1/4, so Pr(Z̃=0)=0.75, contradicting the claimed Pr(Z̃=0) ≥ 0.999. The union-bound proof lacks any lower bound on the positive slack of surviving negative cycles; a negative cycle with tiny edge violations can survive with high probability while keeping Φ small. Proposition 1 is therefore not a valid certificate, and Theorem 1, which depends on it, does not establish the advertised guarantee.
  2. [Section 4, Integrated Surrogate Loss Function] Theorem 1 bounds the simplified loss L(G,θ)=λ1|F^{(k*+1)}_θ(G)−F^{(k*)}_θ(G)|+λ2Φθ(G), but the objective actually trained is LFull(θ)=E[LADV+λ1LFlow+λ2Φθ+λ3(LDA+LDPA+LAB)]. No inequality in the paper relates LFull to L: LADV=Σ_uv w_uv p_uv − c_LP may be negative, and LFlow is a flow-conservation penalty, not the value-operator difference appearing in Theorem 1. Consequently, even if Proposition 1 were correct, minimizing LFull on training graphs does not imply the small empirical L(G_i,θ̂) required for the high-probability near-optimality conclusion. The certificate is established only for a loss that the manuscript does not train.
  3. [Section 3, Lemma 2] The deterministic bound Δ(G,θ) ≤ K1ε1 + K2ε2 is asserted without proof in the main text, and the constants K1, K2 are not shown to be uniform over G and θ. The optimality gap Δ belongs to a decoded path and therefore depends on the sampled edge probabilities, whereas the right-hand side depends on puv only through one term of Φθ(G). Without an explicit construction of K1 and K2 that is uniform in the quantities entering the concentration argument, the stated inequality is vacuous: for a fixed finite graph and parameters, some K1,K2 always exist regardless of whether the bound carries any information. The subsequent use of K=max(K1/λ2, K2/λ1) in Theorem 1 requires such uniformity, and the paper provides no argument for it.
minor comments (4)
  1. [Section 3, Definition 3] The displayed condition dθ(u) ≤ w_uv − dθ(v) is inconsistent with the definition δ_uv=[dθ(u)−w_uv−dθ(v)]_+, which measures violation of dθ(u) ≤ w_uv + dθ(v). The sign error propagates into Proposition 1 and Lemma 1; the correct Bellman-style condition should be dθ(u) ≤ w_uv + dθ(v).
  2. [Section 5, Table 1] The caption of Table 1 does not define the reported 'gap'; because the values are negative and Definition 1 defines Δ(G,θ)≥0, the gap appears to be computed relative to beam search rather than relative to the optimal path. Please state the reference solver and the exact formula used for the percentage gap.
  3. [Section 4 and Section 5] The hyperparameters λ1, λ2, λ3, the temperature τ, the number of unrolled Bellman-Ford steps T, and the number of decoding samples N are introduced but no values or ranges are reported in the experiments. Reporting these values would substantially improve reproducibility.
  4. [Section 3, Proposition 1] The random variable Z̃ is described as the random number of negative-cost cycles, but the underlying probability space is not formalized in the main text. The proof appears to assume independent edge survival with probabilities p_uv; this should be stated explicitly before the proposition.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's derivation is not equivalent to its inputs by construction; the theorem/loss mismatch and Proposition 1 gap are soundness concerns, not circular reductions.

full rationale

Walking the derivation chain, I find no step where a predicted quantity is defined as, or fitted to, the quantity it is said to predict. The value estimates d_theta and edge probabilities p_uv are learned from surrogate losses, not from optimal labels or from the reported optimality gaps, and the Gurobi-derived heuristic cost cLP is an external baseline rather than a fitted parameter renamed as a prediction. The main load-bearing issue is a soundness gap, not circularity: Theorem 1 certifies L = lambda1 |F(k*+1)-F(k*)| + lambda2 Phi, while the implemented loss LFull adds LADV, LFlow, and inductive-bias terms, with no supplied inequality reducing LFull to L; similarly, Proposition 1 appears false as stated for a small-cost two-edge negative cycle. These are invalid-derivation problems, not input-output equivalence. Cited prior work (Karalias-Loukas, Xu et al., Wang et al.) is used as methodological context, not to import the central ESPP guarantee, and there is no load-bearing self-citation. Hence no significant circularity; score 0.

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

The central certificate rests on inherited probabilistic-method assumptions, a sign-corrected Bellman inequality that is misprinted, and an unproven relationship between the simplified bounded loss and the full trained loss. Several hyperparameters that materially affect performance are not reported as specific values.

free parameters (4)
  • loss weights λ1, λ2, λ3 = not reported
    Penalty weights balance value-operator alignment, NCC penalty, and inductive biases; the paper says they are tuned but does not list values.
  • temperature τ = not reported
    Controls softmax and softmin sharpness in distributional alignment, DP alignment, and Bellman-Ford unrolling; path quality depends on it.
  • Bellman-Ford unroll steps T = not reported
    Number of message-passing and BF iterations; convergence and training cost depend on T, which is not specified.
  • number of decoding samples N = 50, 100, or 200 studied
    The decoder repeats walks and keeps the best path; more samples reduce gap but increase inference cost. The default used in the main tables is not stated.
assumptions (5)
  • standard math No negative-cost cycles in G implies an elementary shortest path exists (Proposition 2).
    Used to justify reducing ESPP to ordinary SPP once the NCC penalty is small. This is standard only if the intended value inequality is dθ(u) ≤ wuv + dθ(v).
  • domain assumption A distribution over edge subsets with low expected cost and high feasibility certifies good solutions under the Karalias-Loukas probabilistic method.
    Adopted from prior work and not re-proven here; it requires the surrogate feasibility penalty to be tractable and to control sampling probability.
  • domain assumption The softmin relaxation and Bellman-Ford unrolling preserve the optimality structure of the hard min operator.
    LDPA and LAB use softmin as a differentiable surrogate for the non-differentiable min; no bound on the softmin-versus-min gap is provided.
  • ad hoc to paper The printed inequality dθ(u) ≤ wuv − dθ(v) in Definition 3 is a typo for dθ(u) ≤ wuv + dθ(v).
    With the printed minus sign, δuv is always zero and Lemma 1 cannot hold; the subsequent proofs require the plus sign.
  • ad hoc to paper Extra loss terms (LADV, LFlow, LDA, LDPA, LAB) can be absorbed without breaking the optimality-gap bound.
    Theorem 1 bounds only λ1|ΔF| + λ2Φ, and no inequality relates the full L_Full to that quantity. LADV can be negative, so the trained loss is not automatically bounded below by the theorem's loss.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Unsupervised Learning for the Elementary Shortest Path Problem." pith.science (2026). https://pith.science/paper/T6ZZKNWI

@misc{pith2026250801557,
  author       = {Pith},
  title        = {Pith review of: Unsupervised Learning for the Elementary Shortest Path Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T6ZZKNWI}},
  note         = {Machine review of arXiv:2508.01557}
}
read the original abstract

The Elementary Shortest-Path Problem(ESPP) seeks a minimum cost path from s to t that visits each vertex at most once. The presence of negative-cost cycles renders the problem NP-hard. We present a probabilistic method for finding near-optimal ESPP, enabled by an unsupervised graph neural network that jointly learns node value estimates and edge-selection probabilities via a surrogate loss function. The loss provides a high probability certificate of finding near-optimal ESPP solutions by simultaneously reducing negative-cost cycles and embedding the desired algorithmic alignment. At inference time, a decoding algorithm transforms the learned edge probabilities into an elementary path. Experiments on graphs of up to 100 nodes show that the proposed method surpasses both unsupervised baselines and classical heuristics, while exhibiting high performance in cross-size and cross-topology generalization on unseen synthetic graphs.

Figures

Figures reproduced from arXiv: 2508.01557 by the authors.

Figure 1
Figure 1. Unsupervised Learning for PM-based ESPP. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Studies on Sampling, Density, and Distribution. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. (a) Original graph with negative weights (in red). [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Discrete Flow-Based Generative Models for Measurement Optimization in Quantum Computing

    quant-ph 2025-09 conditional novelty 4.0 of 10

    GFlowNet-based graph coloring finds Hamiltonian groupings with lower estimated measurement costs than sorted insertion on small molecules, subject to selection bias and a missing abstract claim.

Reference graph

Works this paper leans on

34 extracted references · 29 canonical work pages · cited by 1 Pith paper

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address archivePrefix author booktitle chapter edition editor eid eprint howpublished institution isbn journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.a...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

  3. [3]

    Graph Reduction with Unsupervised Learning in Column Generation: A Routing Application

    Abouelrous, A.; Bliek, L.; Gabor, A. F.; Wu, Y.; and Zhang, Y. 2025. Graph Reduction with Unsupervised Learning in Column Generation: A Routing Application. arXiv preprint arXiv:2504.08401

    work page Pith review arXiv 2025

  4. [4]

    Albert, R.; and Barab \'a si, A.-L. 2002. Statistical mechanics of complex networks. Reviews of modern physics, 74(1): 47

    work page 2002

  5. [5]

    Alon, N.; and Spencer, J. H. 2016. The probabilistic method. John Wiley & Sons

    work page 2016

  6. [6]

    Amaldi, E.; Coniglio, S.; and Taccari, L. 2014. Maximum throughput network routing subject to fair flow allocation. In International Symposium on Combinatorial Optimization, 1--12. Springer

    work page 2014

  7. [7]

    Bellman, R. 1958. On a routing problem. Quarterly of applied mathematics, 16(1): 87--90

    work page 1958

  8. [8]

    Bengio, Y.; Lodi, A.; and Prouvost, A. 2021. Machine learning for combinatorial optimization: a methodological tour d’horizon. European Journal of Operational Research, 290(2): 405--421

    2021

Show all 34 references
  1. [9]

    B.; Lodi, A.; Morris, C.; and Veli c kovi \'c , P

    Cappart, Q.; Ch \'e telat, D.; Khalil, E. B.; Lodi, A.; Morris, C.; and Veli c kovi \'c , P. 2023. Combinatorial optimization and reasoning with graph neural networks. Journal of Machine Learning Research, 24(130): 1--61

    2023

  2. [10]

    Chabrier, A. 2006. Vehicle routing problem with elementary shortest path based column generation. Computers & Operations Research, 33(10): 2972--2990

    2006

  3. [11]

    Cheng, H.; Zheng, H.; Cong, Y.; Jiang, W.; and Pu, S. 2023. Select and optimize: Learning to solve large-scale tsp instances. In International conference on artificial intelligence and statistics, 1219--1231. PMLR

    2023

  4. [12]

    Drexl, M.; and Irnich, S. 2014. Solving elementary shortest-path problems as mixed-integer programs. OR spectrum, 36(2): 281--296

    2014

  5. [13]

    Erd o s, P.; and R \'e nyi, A. 1959. On random graphs I. Publ. math. debrecen, 6(290-297): 18

    1959

  6. [14]

    Feillet, D.; Dejax, P.; and Gendreau, M. 2005. Traveling salesman problems with profits. Transportation science, 39(2): 188--205

    2005

  7. [15]

    Feillet, D.; Dejax, P.; Gendreau, M.; and Gueguen, C. 2004. An exact algorithm for the elementary shortest path problem with resource constraints: Application to some vehicle routing problems. Networks: An International Journal, 44(3): 216--229

    2004

  8. [16]

    Ford Jr, L. R. 1956. Network flow theory. Technical report, RAND Corporation

    1956

  9. [17]

    Fu, Z.-H.; Qiu, K.-B.; and Zha, H. 2021. Generalize a small pre-trained model to arbitrarily large tsp instances. In Proceedings of the AAAI conference on artificial intelligence, volume 35, 7474--7482

    2021

  10. [18]

    M.; et al

    Gasse, M.; Bowly, S.; Cappart, Q.; Charfreitag, J.; Charlin, L.; Ch \'e telat, D.; Chmiela, A.; Dumouchelle, J.; Gleixner, A.; Kazachkov, A. M.; et al. 2022. The machine learning for combinatorial optimization competition (ml4co): Results and insights. In NeurIPS 2021 competit...

    2022

  11. [19]

    Gurobi Optimization, LLC . 2024. Gurobi Optimizer Reference Manual

    2024

  12. [20]

    Hu, C. 2024. Assessing and enhancing graph neural networks for combinatorial optimization: novel approaches and application in maximum independent set problems. arXiv preprint arXiv:2411.05834

    2024 arXiv

  13. [21]

    Hudson, B.; Li, Q.; Malencia, M.; and Prorok, A. 2021. Graph neural network guided local search for the traveling salesperson problem. ICLR

    2021

  14. [22]

    K.; Cappart, Q.; Rousseau, L.-M.; and Laurent, T

    Joshi, C. K.; Cappart, Q.; Rousseau, L.-M.; and Laurent, T. 2022. Learning the travelling salesperson problem requires rethinking generalization. Constraints, 27(1): 70--98

    2022

  15. [23]

    Karalias, N.; and Loukas, A. 2020. Erdos goes neural: an unsupervised learning framework for combinatorial optimization on graphs. Advances in Neural Information Processing Systems, 33: 6659--6672

    2020

  16. [24]

    B.; Morris, C.; and Lodi, A

    Khalil, E. B.; Morris, C.; and Lodi, A. 2022. Mip-gnn: A data-driven framework for guiding combinatorial solvers. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, 10219--10227

    2022

  17. [25]

    Kool, W.; Van Hoof, H.; and Welling, M. 2018. Attention, learn to solve routing problems! ICLR

    2018

  18. [26]

    Lowerre, B. T. 1976. The harpy speech recognition system. Carnegie Mellon University

    1976

  19. [27]

    Min, Y.; Bai, Y.; and Gomes, C. P. 2023. Unsupervised learning for solving the travelling salesman problem. Advances in neural information processing systems, 36: 47264--47278

    2023

  20. [28]

    Pan, X.; Jin, Y.; Ding, Y.; Feng, M.; Zhao, L.; Song, L.; and Bian, J. 2023. H-tsp: Hierarchically solving the large-scale traveling salesman problem. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 37, 9345--9353

    2023

  21. [29]

    Smith, K. A. 1999. Neural networks for combinatorial optimization: a review of more than a decade of research. Informs journal on Computing, 11(1): 15--34

    1999

  22. [30]

    Taccari, L. 2016. Integer programming formulations for the elementary shortest path problem. European Journal of Operational Research, 252(1): 122--130

    2016

  23. [31]

    Wang, Z.; Schaul, T.; Hessel, M.; Hasselt, H.; Lanctot, M.; and Freitas, N. 2016. Dueling network architectures for deep reinforcement learning. In International conference on machine learning, 1995--2003. PMLR

    2016

  24. [32]

    S.; Kawarabayashi, K.-i.; and Jegelka, S

    Xu, K.; Li, J.; Zhang, M.; Du, S. S.; Kawarabayashi, K.-i.; and Jegelka, S. 2019. What can neural networks reason about? ICLR

    2019

  25. [33]

    Yang, R.; Zhang, Y.; Chen, A.; Wang, X.; Yuan, S.; Chen, J.; Yang, D.; and Xiao, Y. 2025. ARIA: Training Language Agents with Intention-Driven Reward Aggregation. arXiv preprint arXiv:2506.00539

    2025 arXiv

  26. [34]

    Zheng, J.; He, K.; Zhou, J.; Jin, Y.; and Li, C.-M. 2021. Combining reinforcement learning with Lin-Kernighan-Helsgaun algorithm for the traveling salesman problem. In Proceedings of the AAAI conference on artificial intelligence, volume 35, 12445--12452

    2021

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.