arxiv: 2605.03760 · v1 · submitted 2026-05-05 · 🧮 math.OC

Recognition: unknown

Learning Dominant States in Elementary Resource Constrained Shortest Path Problems

Saverio Basso , Matteo Salani

Authors on Pith no claims yet

Pith reviewed 2026-05-07 15:21 UTC · model grok-4.3

classification 🧮 math.OC

keywords machine learningdynamic programmingshortest pathresource constraineddominating statessupervised learningnormalizationoptimization

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The pith

Machine learning can identify dominating states from dynamic programming labels in elementary resource constrained shortest path problems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests whether supervised learning can pick out promising states during dynamic programming for elementary resource constrained shortest path problems. The authors solved 41 single-resource instances, gathered hundreds of millions of states, and trained models on simple features that can be computed in constant time. This matters because these problems require exploring large numbers of states, and learning to discard non-promising ones early could shrink the search space dramatically. The models perform strongly on the full collection of generated states but show more mixed results when applied to states from new instances.

Core claim

Solving 41 ERCSPP instances with iterative relaxation produced two large datasets of labels. After designing constant-time features and applying a normalization step, supervised learning techniques were used to classify states as dominating. The resulting models reliably separate dominating states within the same problem but show declining accuracy when tested on previously unseen instances.

What carries the argument

Supervised classifiers trained on constant-time ad-hoc features of labels to predict dominance after a normalization step.

If this is right

Data-driven filtering of states can reduce the number of labels processed in dynamic programming for these problems.
Normalization reveals consistent patterns across successive relaxations within a single instance.
The same pipeline supports hybrid solvers that combine traditional relaxation with learned dominance checks.
Performance is stronger when the model stays within the distribution of training instances.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If more diverse training instances are added, generalization to unseen problems may improve enough for routine use in solvers.
Constant-time features alone may prove sufficient for dominance decisions in other combinatorial optimization settings that rely on label-based dynamic programming.
Integrating the classifier directly into the relaxation loop could yield measurable speedups on large instances without changing the underlying algorithm.

Load-bearing premise

The constant-time ad-hoc features capture enough patterns to distinguish dominating states and those patterns generalize beyond the 41 training instances.

What would settle it

Apply the trained models to a fresh set of ERCSPP instances outside the original 41 and measure whether they still correctly flag dominating states at high accuracy.

Figures

Figures reproduced from arXiv: 2605.03760 by Matteo Salani, Saverio Basso.

**Figure 1.** Figure 1: Average number of labels per iteration, for each instance. view at source ↗

**Figure 2.** Figure 2: Dataset G. Scatter plot of Label objective (y-axis) against critical view at source ↗

**Figure 3.** Figure 3: Dataset I. Scatter plot of Label objective (y-axis) against critical view at source ↗

**Figure 4.** Figure 4: Dataset G. Boxplot of objective for Pareto and dominated labels. view at source ↗

**Figure 5.** Figure 5: Dataset I. Boxplot of objective for Pareto and dominated labels. view at source ↗

**Figure 6.** Figure 6: Boxplot of critical resource consumption for Pareto and dominated view at source ↗

**Figure 7.** Figure 7: Dataset G. Boxplot of the objective for each iteration of the algorithm. view at source ↗

**Figure 8.** Figure 8: Dataset I. Boxplot of the objective for each iteration of the algorithm. view at source ↗

**Figure 9.** Figure 9: Dataset G. Plotting objective normalized with previous data (Prev) view at source ↗

**Figure 10.** Figure 10: Dataset I. Plotting objective normalized with previous data (Prev) view at source ↗

read the original abstract

In this work, we investigate whether machine learning can be leveraged to identify promising states in dynamic programming algorithms, focusing on Elementary Resource Constrained Shortest Path Problems (ERCSPP). More in detail, we solved 41 single resource instances from SPPRCLIB using iterative relaxation techniques through the PathWyse library, systematically collecting all generated states (i.e. labels). We designed ad-hoc features computable in constant time and constructed two datasets: one containing all generated labels (G) and another with only those inserted into data pools (I), totaling several hundred million labels. Machine learning tools are then employed to explore these datasets, revealing significant patterns between successive relaxations. Leveraging these insights, we propose a normalization approach and apply supervised learning techniques to distinguish dominating states, both within subsequent relaxations of the same problem and in previously unseen instances. Our results demonstrate the effectiveness of this approach on Dataset G, while for Dataset I, performance varies, showing strong results within the same instance but declining for unseen ones. Overall, these findings open new perspectives for the development of data-driven dynamic programming algorithms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Supervised learning on ad-hoc features predicts state dominance reasonably inside the same ERCSPP instance but loses accuracy on unseen ones, based on labels from 41 SPPRCLIB problems.

read the letter

This paper checks if machine learning can help spot dominant states early in dynamic programming for elementary resource-constrained shortest path problems. The results show it works okay on the data they trained on but struggles to generalize to new instances. They solved 41 single-resource instances from the SPPRCLIB library using the PathWyse solver. From the relaxation process they pulled hundreds of millions of states and labeled them. They came up with a set of simple features that can be computed quickly and used those to train supervised models that predict whether a state will dominate others in later steps. They also tried a normalization step to make the data more usable. The work is straightforward and reports the outcomes without much spin. Dataset G, which has all generated labels, gives good results. Dataset I, limited to inserted states, does well when tested on the same instance but the accuracy falls when applied to instances the model has not seen before. That pattern is worth knowing if you are thinking about using this in practice. The main limitation is that limited generalization. If the classifier mostly picks up on patterns tied to the specific instance rather than universal dominance rules, then for a brand new problem you would still need to run the full solver first to gather training data. That reduces the practical speedup. The features are described as ad-hoc, and without more detail on exactly which ones were chosen or how the models were tuned, it is difficult to reproduce or improve the approach. Training on just 41 instances also leaves open the question of how well it scales to the wider range of problems that come up in logistics. This kind of work is aimed at people who build exact solvers for constrained shortest paths and want to inject data-driven shortcuts. Someone already familiar with the ERCSPP literature and the SPPRCLIB benchmark would find the datasets and the within-versus-across instance comparison useful. It deserves a serious referee because the experimental design is clear and the negative result on unseen instances is reported plainly, which helps the field avoid over-optimism about ML for this task.

Referee Report

3 major / 1 minor

Summary. The paper investigates whether supervised machine learning can identify dominating states within dynamic programming for Elementary Resource Constrained Shortest Path Problems (ERCSPP). The authors solve 41 single-resource instances from SPPRCLIB via iterative relaxation in the PathWyse library, collect all generated labels into Dataset G and the subset inserted into pools into Dataset I (hundreds of millions of labels total), design ad-hoc constant-time features, and train classifiers to distinguish dominating states both within the same instance and on unseen instances. They report strong effectiveness on G and variable results on I (strong within-instance but declining on unseen instances), suggesting potential for data-driven DP enhancements.

Significance. If the central empirical claims hold after addressing generalization and reproducibility issues, the work would demonstrate a viable path toward hybrid ML-DP algorithms that prune non-dominating states in ERCSPP, potentially yielding faster exact solutions on large instances. The scale of the collected data and the focus on constant-time features are positive aspects, but the observed drop in performance on unseen instances limits the immediate practical significance for previously unseen problems.

major comments (3)

[Abstract and results description] Abstract and results description: The claim of 'effectiveness' on Dataset G and 'strong results within the same instance' on Dataset I is not supported by any reported quantitative metrics (accuracy, precision, recall, F1, or AUC), baseline comparisons, or error analysis, making it impossible to assess whether the supervised learning actually outperforms simple dominance rules or random guessing.
[Methodology and experimental setup] Methodology and experimental setup: The ad-hoc features are described only as 'computable in constant time' without explicit definitions, formulas, or justification for why they capture transferable dominance patterns rather than instance-specific correlations; combined with training on only 41 instances, this leaves the generalization claim (declining performance on unseen instances) without a concrete test or ablation.
[Results on Dataset I] Results on Dataset I: The explicit note of 'declining for unseen ones' directly undermines the utility claim for data-driven DP on previously unseen ERCSPP instances; no analysis is provided of whether this stems from overfitting, feature insufficiency, or label noise, which is load-bearing for the paper's stated goal of opening perspectives for new algorithms.

minor comments (1)

[Abstract] The abstract and text use 'Dataset G' and 'Dataset I' without clarifying the exact construction criteria for 'inserted into data pools' in the first mention.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed review of our manuscript on applying supervised learning to identify dominating states in ERCSPP. We address each major comment point by point below, indicating where revisions will be made to improve clarity, rigor, and completeness while preserving the exploratory nature of the work.

read point-by-point responses

Referee: [Abstract and results description] Abstract and results description: The claim of 'effectiveness' on Dataset G and 'strong results within the same instance' on Dataset I is not supported by any reported quantitative metrics (accuracy, precision, recall, F1, or AUC), baseline comparisons, or error analysis, making it impossible to assess whether the supervised learning actually outperforms simple dominance rules or random guessing.

Authors: We agree that the abstract and results description would be strengthened by explicit quantitative metrics. The current manuscript emphasizes the discovery of patterns and the feasibility of the approach rather than exhaustive benchmarking, but this omission makes evaluation difficult. In the revised version, we will update the abstract and expand the results section to report accuracy, precision, recall, F1-score, and AUC for the classifiers on both Dataset G and Dataset I. We will also add baseline comparisons against simple dominance rules and random guessing, together with a basic error analysis. These additions will be presented in new tables and will allow readers to directly assess performance relative to non-ML alternatives. revision: yes
Referee: [Methodology and experimental setup] Methodology and experimental setup: The ad-hoc features are described only as 'computable in constant time' without explicit definitions, formulas, or justification for why they capture transferable dominance patterns rather than instance-specific correlations; combined with training on only 41 instances, this leaves the generalization claim (declining performance on unseen instances) without a concrete test or ablation.

Authors: The features were intentionally designed to be lightweight and instance-independent so they could be used inside a DP loop without overhead. We acknowledge that the manuscript provides insufficient detail on their exact form and rationale. In the revision, we will add a dedicated subsection with explicit definitions, mathematical formulas, and justification for each feature, explaining why they are expected to reflect dominance properties that may generalize. We will also include an ablation study on feature importance and clarify the train/test protocol used for within-instance versus cross-instance evaluation, along with a discussion of the diversity present in the 41 SPPRCLIB instances. revision: yes
Referee: [Results on Dataset I] Results on Dataset I: The explicit note of 'declining for unseen ones' directly undermines the utility claim for data-driven DP on previously unseen ERCSPP instances; no analysis is provided of whether this stems from overfitting, feature insufficiency, or label noise, which is load-bearing for the paper's stated goal of opening perspectives for new algorithms.

Authors: We recognize that the performance drop on unseen instances is a central and potentially limiting observation. The manuscript presents it as an empirical finding that opens perspectives rather than as a solved capability. In the revision, we will expand the discussion of Dataset I results with an analysis of contributing factors, including possible overfitting to instance-specific patterns, limitations of the current constant-time feature set, and label noise arising from the iterative relaxation procedure. This will be supported by additional experiments such as grouped cross-validation and feature-sensitivity checks, thereby clarifying the boundaries of the method and strengthening the forward-looking claims. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical ML results on collected DP states are self-contained

full rationale

The paper's chain is: solve 41 instances via PathWyse to generate and label states, extract ad-hoc constant-time features, train supervised classifiers on the resulting datasets G and I, then report direct test performance (strong on G; within-instance strong but cross-instance weak on I). Dominance labels originate from the external solver's actual relaxations, not from the ML model or any fitted quantity defined in terms of the target prediction. No equations redefine the output as the input, no self-citation chain bears the central claim, and the reported decline on unseen instances is an explicit empirical observation rather than a constructed result. The work is therefore a standard experimental study whose conclusions do not reduce to their own inputs by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim depends on the assumption that ad-hoc constant-time features plus standard supervised learning can capture dominance patterns. No free parameters or invented entities are explicitly introduced in the abstract. The approach inherits standard assumptions from dynamic programming and machine learning.

axioms (2)

domain assumption Ad-hoc features computable in constant time are sufficient to distinguish dominating states
The paper relies on these features for the ML models without proving or justifying their completeness.
domain assumption Patterns learned from generated states in 41 instances generalize to dominance prediction
The reported within-instance success and cross-instance decline both rest on this generalization assumption.

pith-pipeline@v0.9.0 · 5487 in / 1438 out tokens · 60333 ms · 2026-05-07T15:21:27.744002+00:00 · methodology

discussion (0)

Reference graph

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