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arxiv: 2604.06973 · v1 · submitted 2026-04-08 · 💻 cs.NE

Block-Bench: A Framework for Controllable and Transparent Discrete Optimization Benchmarking

Furong Ye , Frank Neumann , Thomas B\"ack , Niki van Stein This is my paper

Pith reviewed 2026-05-10 17:54 UTC · model grok-4.3

classification 💻 cs.NE
keywords discrete optimizationbenchmark constructionevolutionary algorithmsalgorithm behavior analysisblock functionsmulti-modal problemsself-adaptation
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The pith

The Block-Bench framework constructs discrete optimization benchmarks from composable block functions and dependency graphs, enabling analysis of algorithm behavior through intermediate values rather than only the final objective.

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

The paper introduces a method to build benchmark problems for discrete optimization by combining multiple block functions, each handling a subset of variables, linked by weights and an adjacency graph. This setup gives researchers explicit control over problem features like modality and variable interactions. A key benefit is the ability to look inside the optimization process by checking the values produced by each block during a run. This detailed view helps explain how different algorithms handle large, complex problems that have many local optima. The authors show how this design can guide improvements in evolutionary algorithm techniques such as self-adaptation and maintaining diversity.

Core claim

We build benchmark problems based on a set of block functions, where each block function maps a subset of variables to a real value. Problems are instantiated through a set of block functions, weight factors, and an adjacency graph representing the dependency among the block functions. Through analyzing intermediate block values, our framework allows to analyze algorithm behavior not only in the objective space but also at the level of variable representations in the obtained solutions. This capacity is particularly useful for analyzing discrete heuristics in large-scale multi-modal problems.

What carries the argument

Block functions that each map a subset of decision variables to a scalar value, assembled via an adjacency graph for dependencies and weight factors for combining into the overall objective.

Load-bearing premise

That the intermediate values from the block functions can be recorded and interpreted to yield useful insights into how an algorithm explores the search space, beyond what the single scalar objective value shows.

What would settle it

Running multiple algorithms on the same set of block-based benchmarks and finding that the block value trajectories do not correlate with or explain differences in overall performance or solution quality.

Figures

Figures reproduced from arXiv: 2604.06973 by Frank Neumann, Furong Ye, Niki van Stein, Thomas B\"ack.

Figure 1
Figure 1. Figure 1: Correlations between block function values (in axes) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Diverse landscapes constructed by block functions [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Correlations between block function values (in axes) [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The minimal (Top) and maximal (Bottom) attainable [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Convergence process of EAs for F5 over the FEs. [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Convergence process of EAs for F10 over the FEs. [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The attainable objective values of a 40D DBP (F3) and GCP (F7) with only Jump3 block functions vs. the number of 1-bits in the solution. 𝑚 = 1, 2, 4 from left to right. (a) Jump blocks (b) Epistasis blocks (c) Distinct block functions [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The FEs of (𝜇 + 1) GAs to reach the optimum of 40D DBPs and GCPs with Jump3, Epistasis , and heterogeneous block functions. added into block functions (𝑚 ≥ 3). In addition, when compar￾ing the performance on DBP and GCP instances, DBP generally appears to be easier for both algorithms, except for a noticeable performance shift of the standard GA in [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The bi-objective search space of GCPs with the [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The HV convergence on the bi-objective optimiza [PITH_FULL_IMAGE:figures/full_fig_p008_11.png] view at source ↗
read the original abstract

We present a novel approach for constructing discrete optimization benchmarks that enables fine-grained control over problem properties, and such benchmarks can facilitate analyzing discrete algorithm behaviors. We build benchmark problems based on a set of block functions, where each block function maps a subset of variables to a real value. Problems are instantiated through a set of block functions, weight factors, and an adjacency graph representing the dependency among the block functions. Through analyzing intermediate block values, our framework allows to analyze algorithm behavior not only in the objective space but also at the level of variable representations in the obtained solutions. This capacity is particularly useful for analyzing discrete heuristics in large-scale multi-modal problems, thereby enhancing the practical relevance of benchmark studies. We demonstrate how the proposed approach can inspire the related work in self-adaptation and diversity control in evolutionary algorithms. Moreover, we explain that the proposed benchmark design enables explicit control over problem properties, supporting research in broader domains such as dynamic algorithm configuration and multi-objective optimization.

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

2 major / 2 minor

Summary. The paper introduces Block-Bench, a framework for constructing discrete optimization benchmark problems using a set of block functions (each mapping a subset of variables to a real value), weight factors, and an adjacency graph to represent dependencies among blocks. It claims this construction provides fine-grained control over problem properties such as modality and variable interactions, and allows for transparent analysis of discrete algorithms by examining intermediate block values in addition to the overall objective value. The authors illustrate the approach's potential to inspire research in self-adaptation and diversity control in evolutionary algorithms, as well as its applicability to dynamic algorithm configuration and multi-objective optimization.

Significance. If the framework delivers on its promises of controllability and transparency, it could provide a valuable structured method for generating customizable benchmarks in discrete optimization and evolutionary computation. This would address limitations in existing black-box benchmarks by enabling systematic variation of problem features and deeper behavioral analysis, potentially inspiring new algorithmic techniques. The definitional construction is internally consistent and offers a clear path to reproducible problem instances.

major comments (2)
  1. [Abstract and applications section] The central claim that analyzing intermediate block values enables insights into algorithm behavior not obtainable from the scalar objective alone is load-bearing for the paper's motivation, yet the manuscript provides no quantitative validation, baseline comparisons, or concrete case studies demonstrating actionable insights from this analysis (see abstract and applications discussion).
  2. [Construction method] The assertion of 'fine-grained control' over properties such as modality and variable interactions via the block decomposition and adjacency graph lacks a systematic demonstration or enumeration of achievable configurations, making the controllability claim informal rather than rigorously shown.
minor comments (2)
  1. [Methods] The notation for block functions, weight factors, and the adjacency graph would benefit from explicit mathematical definitions or pseudocode to improve clarity and reproducibility.
  2. [Related work] A more explicit comparison to existing modular or decomposable benchmarks (e.g., NK-landscapes) would better situate the novelty of the block-function approach.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript introducing the Block-Bench framework. The comments correctly identify areas where additional evidence would strengthen the presentation of our claims on transparency and controllability. We respond to each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract and applications section] The central claim that analyzing intermediate block values enables insights into algorithm behavior not obtainable from the scalar objective alone is load-bearing for the paper's motivation, yet the manuscript provides no quantitative validation, baseline comparisons, or concrete case studies demonstrating actionable insights from this analysis (see abstract and applications discussion).

    Authors: We acknowledge that the current manuscript motivates the value of intermediate block-value analysis in the abstract and applications section but does not supply a quantitative demonstration or baseline comparison. To address this, we will add a dedicated case-study subsection. It will present a controlled multi-modal instance, run a representative discrete evolutionary algorithm, and contrast the behavioral insights obtained from tracking per-block values against those available from the scalar objective alone, including a simple baseline that uses only the objective. This addition will make the transparency claim concrete and actionable. revision: yes

  2. Referee: [Construction method] The assertion of 'fine-grained control' over properties such as modality and variable interactions via the block decomposition and adjacency graph lacks a systematic demonstration or enumeration of achievable configurations, making the controllability claim informal rather than rigorously shown.

    Authors: We agree that the controllability claim is currently supported primarily by the definitional construction and selected examples rather than by an explicit enumeration of reachable configurations. In the revised manuscript we will expand the construction-method section with a systematic overview: a table that enumerates representative choices of block-function families, weight assignments, and adjacency-graph densities, together with the resulting problem properties (number of local optima, interaction strength, etc.). This will render the fine-grained control claim more rigorous while preserving the original framework. revision: yes

Circularity Check

0 steps flagged

No significant circularity: definitional benchmark construction framework

full rationale

The manuscript defines a benchmark construction method via block functions, weights, and adjacency graphs to enable controllable discrete optimization problems and intermediate-value analysis. No equations, derivations, or predictions are presented that reduce the central claims to fitted inputs or self-citations by construction. The contribution is the framework definition itself, which is self-contained and does not rely on load-bearing self-citations or ansatzes imported from prior author work. This matches the expected non-circular outcome for a novel definitional proposal.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The framework rests on the definitional choice to decompose problems into independent block functions whose outputs are combined via weights and a graph; no numerical fitting or external data is required for the construction itself.

axioms (2)
  • domain assumption Any discrete optimization problem can be usefully decomposed into a collection of block functions each acting on a variable subset.
    Invoked when the paper states that problems are instantiated through a set of block functions.
  • domain assumption An adjacency graph on blocks meaningfully captures variable dependencies for analysis purposes.
    Used to represent dependency among block functions.
invented entities (1)
  • block function no independent evidence
    purpose: Modular component that maps a variable subset to a scalar value for transparent intermediate analysis.
    Core new building block introduced by the framework.

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