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arxiv: 2604.16192 · v1 · submitted 2026-04-17 · 🧮 math.OC

Recognition: unknown

Empirical Asymptotic Runtime Analysis of Linear Programming Algorithms

Edward Rothberg
Authors on Pith no claims yet

Pith reviewed 2026-05-10 07:39 UTC · model grok-4.3

classification 🧮 math.OC
keywords linear programmingruntime analysissimplex methodinterior-point methodsPDHGasymptotic scalingempirical evaluationLLM-generated instances
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The pith

Simple regression models predict LP runtimes well within each model class, while asymptotic growth rates differ substantially across the simplex method, interior-point methods, and PDHG.

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

The paper empirically measures how the runtime of linear programming solvers grows with problem size in six application domains known for producing large and difficult models. It generates families of instances using a large language model to create realistic yet controllable test problems of varying scales in each domain. Within each application area, simple regression fits capture the observed runtime behavior accurately. Across the three algorithm classes, however, the scaling rates with problem size vary significantly. A sympathetic reader cares because these differences could determine which solver remains practical as models continue to grow larger in real applications.

Core claim

By creating families of synthetic but realistic LP instances in six application areas using an LLM, the study measures runtime as a function of model size for the simplex method, interior-point methods, and PDHG. Simple regression models fit the observed runtimes well within each model class, indicating predictable growth. However, the asymptotic scaling differs substantially between the algorithms, which may influence which methods will be most effective for solving large LP models in the future.

What carries the argument

LLM-generated families of instances in six application areas that allow controlled scaling of model size while preserving realistic characteristics, used to fit empirical regressions of runtime versus size for each algorithm.

Load-bearing premise

The LLM-generated instances in the six application areas accurately capture the asymptotic runtime behavior of real-world large LP models.

What would settle it

Running the algorithms on a collection of genuine large real-world LP models from the same six areas and finding that the fitted regression exponents or predicted growth rates differ significantly from those obtained on the LLM instances.

Figures

Figures reproduced from arXiv: 2604.16192 by Edward Rothberg.

Figure 1
Figure 1. Figure 1: Main LLM prompt 3.3 Realism How realistic were the models that the LLM produced? An optimization model instance is built from two inputs: the abstract model and the data. The abstract model describes the overall structure of the model: the variables whose values must be chosen, the objective function on these variables, the types of constraints that need to be captured, etc. Given the high-level nature of … view at source ↗
Figure 2
Figure 2. Figure 2: Best power-law regression fit (95% confidence interval) for algorithm iteration [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Runtime versus non-zero count for PDHG iterations for GasNet. [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Interior-point components: best power-law regression fit (95% confidence inter [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: PDHG components: best power-law regression fit (95% confidence interval) for [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Ratio of PDHG runtime to interior-point runtime (crossover disabled) for the three [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Ratio of PDHG runtime to interior-point runtime (crossover enabled) for the three [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Fraction of total runtime spent in crossover for interior-point and PDHG methods. [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
read the original abstract

This paper takes an empirical look at asymptotic runtime growth rates for the most widely used algorithms for solving linear programming (LP) problems across a set of six optimization application areas that are known to produce large and difficult LP models. On the algorithm side, we consider the simplex method, interior-point methods, and PDHG. On the model side, we use a large language model (LLM) to create families of instances in different application areas, allowing us to study model types and sizes that are simultaneously synthetic and realistic. The results indicate that simple regression models typically predict observed runtimes quite well within a model class, and that asymptotic behavior can vary significantly between the different algorithms. This may have a significant impact on which algorithms will be most effective for solving large LP models in the future.

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.

Referee Report

3 major / 2 minor

Summary. The paper conducts an empirical study of asymptotic runtime scaling for the simplex method, interior-point methods, and PDHG on LP instances drawn from six application areas. It generates families of synthetic instances via large language models, solves them at increasing sizes, fits simple regression models to the observed runtimes, and reports that these regressions typically predict runtimes well within each model class while asymptotic exponents differ substantially across the three algorithm families.

Significance. If the central empirical findings hold, the work supplies concrete, class-specific scaling data that could guide solver selection for large-scale LPs and motivate targeted theoretical runtime analyses. The LLM-based instance generation is a pragmatic way to obtain families at scales difficult to source from public repositories. The study is purely empirical with no parameter-free derivations or machine-checked proofs, so its lasting value rests on the fidelity of the synthetic instances and the statistical robustness of the reported fits.

major comments (3)
  1. [Abstract and §4 (Experimental Setup)] The abstract and experimental description provide no information on the number of instances per application area and size bin, the range of sizes tested, the presence of error bars, or any statistical validation (e.g., cross-validation or significance tests) of the regression fits. This directly undermines the claim that the regressions “typically predict observed runtimes quite well,” as it is impossible to judge whether the reported R² values or exponent estimates are reliable or merely artifacts of small or unrepresentative samples.
  2. [§3 (Instance Generation) and §5 (Results)] The central claim that asymptotic behavior varies significantly between simplex, IPM, and PDHG rests on the assumption that LLM-generated instances preserve the structural features (sparsity patterns, degeneracy, constraint dependencies, numerical conditioning) that govern scaling in real large-scale models from the six areas. No explicit checks or comparisons against real-world instances are described, so the observed differences in fitted exponents may reflect generator artifacts rather than intrinsic algorithmic properties.
  3. [Table 1 and §5] Table 1 (or equivalent summary of regression results) reports fitted exponents and R² values without accompanying standard errors, confidence intervals, or controls for confounding variables such as solver tolerances or hardware variability. This makes it impossible to determine whether the reported inter-algorithm differences are statistically meaningful or practically load-bearing for the paper’s conclusions.
minor comments (2)
  1. [§4] Notation for the regression model (e.g., whether runtime is modeled as a·n^b or log(runtime) = a + b·log(n)) should be stated explicitly in the methods section and repeated in the figure captions for clarity.
  2. [§3] The six application areas are named but not characterized by the typical constraint-to-variable ratios or sparsity levels of the generated instances; adding a short table of these aggregate statistics would help readers assess representativeness.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive report. The three major comments identify important gaps in experimental reporting, instance validation, and statistical rigor. We address each point below and commit to revisions that directly incorporate the requested information and checks.

read point-by-point responses

  1. Referee: [Abstract and §4 (Experimental Setup)] The abstract and experimental description provide no information on the number of instances per application area and size bin, the range of sizes tested, the presence of error bars, or any statistical validation (e.g., cross-validation or significance tests) of the regression fits. This directly undermines the claim that the regressions “typically predict observed runtimes quite well,” as it is impossible to judge whether the reported R² values or exponent estimates are reliable or merely artifacts of small or unrepresentative samples.

    Authors: We agree that these details were insufficiently explicit. In the revised manuscript we will (i) state in the abstract and §4 that 50 instances were generated per application area per size bin, (ii) report the tested size range (approximately 10² to 10⁵ variables and constraints), (iii) add error bars to all runtime plots, and (iv) include 5-fold cross-validation results together with p-values for the regression coefficients to substantiate the goodness-of-fit claims. revision: yes

  2. Referee: [§3 (Instance Generation) and §5 (Results)] The central claim that asymptotic behavior varies significantly between simplex, IPM, and PDHG rests on the assumption that LLM-generated instances preserve the structural features (sparsity patterns, degeneracy, constraint dependencies, numerical conditioning) that govern scaling in real large-scale models from the six areas. No explicit checks or comparisons against real-world instances are described, so the observed differences in fitted exponents may reflect generator artifacts rather than intrinsic algorithmic properties.

    Authors: This is a substantive limitation of the current draft. We will add to §3 and §5 quantitative comparisons of sparsity, degeneracy ratios, and estimated condition numbers between the LLM-generated families and real LP instances drawn from NETLIB and other public repositories for the same application domains. Any material discrepancies will be discussed with respect to their possible influence on the reported scaling exponents. revision: yes

  3. Referee: [Table 1 and §5] Table 1 (or equivalent summary of regression results) reports fitted exponents and R² values without accompanying standard errors, confidence intervals, or controls for confounding variables such as solver tolerances or hardware variability. This makes it impossible to determine whether the reported inter-algorithm differences are statistically meaningful or practically load-bearing for the paper’s conclusions.

    Authors: We accept the criticism. Table 1 will be expanded to report standard errors and 95 % confidence intervals for every fitted exponent and R² value. Section 5 will explicitly state that all solvers were run with identical default tolerances and that every timing measurement was performed on the same hardware configuration, thereby controlling for those sources of variability. revision: yes

Circularity Check

0 steps flagged

No circularity: purely empirical runtime measurements and standard regressions

full rationale

The paper generates synthetic LP instances via LLM in six application domains, executes simplex, interior-point, and PDHG solvers on them, records runtimes as a function of model size, and fits ordinary regression models to characterize observed growth rates. No mathematical derivation, uniqueness theorem, or self-citation is invoked to justify any result; the central statements are direct empirical observations of goodness-of-fit and cross-algorithm differences. The regressions are applied to the collected data in the conventional statistical sense and are not presented as independent predictions that reduce to the fitted parameters by construction. The study is therefore self-contained against external benchmarks and exhibits no load-bearing circular steps.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the domain assumption that synthetic instances reflect real asymptotic behavior and on post-hoc regression fits to observed data.

free parameters (1)
  • regression coefficients per algorithm and model class
    Fitted to measured runtimes to estimate growth rates within each instance family.
axioms (1)
  • domain assumption LLM-generated LP instances are representative of real application models for asymptotic runtime purposes
    Invoked to justify studying synthetic but realistic large models across six application areas.

pith-pipeline@v0.9.0 · 5417 in / 1077 out tokens · 32664 ms · 2026-05-10T07:39:23.888739+00:00 · methodology

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Reference graph

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