ATLAS: Scaffold-Free Algorithm Synthesis by LLMs via Embedding-Guided Quality-Diversity Search

Danial Yazdani; Mohammad Nabi Omidvar; Yuan Sun; Maksud Ibrahimov; Xiaodong Li

Source-linked AI summary

ATLAS: Scaffold-Free Algorithm Synthesis by LLMs via Embedding-Guided Quality-Diversity Search

Danial Yazdani, Mohammad Nabi Omidvar, Yuan Sun, Maksud Ibrahimov, Xiaodong Li

arXiv:2608.15546v2Aug 16, 2026cs.AIcs.NE

TL;DR

LLM-based algorithm design usually searches within human-prescribed scaffolds, limiting full-algorithm exploration in combinatorial optimization. ATLAS removes that restriction with embedding-guided quality-diversity search and outperforms component-synthesis methods and a matched full-synthesis baseline across four NP-hard benchmarks while retaining multiple competitive designs.

Problem
Existing LLM-based methods synthesize components within predefined scaffolds, while scaffold-free full-algorithm synthesis explores a larger, harder-to-navigate design space.
Method
ATLAS uses an embedding-organized coverage archive and three-layer search to refine, diversify, and recombine scaffold-free algorithms.
Results
Across four NP-hard benchmarks, ATLAS outperforms component-synthesis methods and a matched full-synthesis baseline, remains competitive with strong human-designed algorithms, and retains multiple competitive designs.
Takeaways & Limitations
Embedding-guided quality-diversity search can make the enlarged full-algorithm design space practically searchable while preserving multiple competitive algorithmic designs.

Abstract

from arXiv · show

Most LLM-based automated algorithm design methods optimize a designated component within a human-specified scaffold, fixing overall organization and component interactions. We present ATLAS, an embedding-guided quality-diversity framework for scaffold-free full-algorithm synthesis in combinatorial optimization. The problem specification supplies objectives and constraints; a minimal I/O interface fixes only instance and solution formats; the LLM chooses and restructures components, interactions, and control flow. This freedom enlarges the search space, risking invalid candidates and premature convergence to one design region. ATLAS independently detects execution, interface, and feasibility failures, recomputes objectives, and applies error-conditioned repair; similarity-based archive management preserves algorithms across embedding-space regions to counter premature convergence. Its three-layer search refines the best design, gives other regions dedicated refinement opportunities, and performs cross-region synthesis to recombine components and their interactions. Across four NP-hard problems, ATLAS outperforms several state-of-the-art component-synthesis methods and a matched full-synthesis baseline while remaining competitive with strong human-designed algorithms. One ATLAS run retains several algorithms with comparable performance from distinct embedding-space regions rather than a single design. Code inspection finds that these multi-component designs differ in their primary construction or global-search backbone. Our results suggest that embedding-guided quality-diversity search can make the enlarged full-algorithm design space practically searchable. Source code and exact executable prompts are available at https://github.com/Danial-Yazdani/ATLAS .

I. INTRODUCTION

ATLAS introduces scaffold-free, embedding-guided quality-diversity search for complete algorithm synthesis, allowing LLMs to choose components, interactions, and control flow under only problem and I/O requirements. Its archive-based three-layer search preserves multiple regions, supports cross-region recombination, and evaluates failures independently across four NP-hard benchmarks.

Scaffold-Free Full-Algorithm Synthesis: ATLAS synthesizes complete optimization algorithms from problem objectives, constraints, and minimal I/O formats without prescribing internal components, roles, or control flow.The LLM controls end-to-end solution construction and constraint handling.
Embedding-Guided Quality-Diversity and Multimodal Search: ATLAS organizes executable algorithms in a pretrained embedding space, using coverage-preserving retrieval, clustering, and redundancy control to mitigate premature convergence.The representation is based on algorithm names, descriptions, and preprocessed source code rather than execution-defined program semantics.
Hierarchical Three-Layer Search: Its three search layers refine the best region, give non-elite regions dedicated opportunities, and recombine components and interactions across regions.Preserved alternatives serve as varied references for cross-region synthesis, enabling the LLM to adapt or combine mechanisms from different algorithms.
Independent Evaluation and Failure-Conditioned Recovery: ATLAS independently detects execution, interface, malformed-output, and feasibility failures, verifies solutions, recomputes objectives, and routes classified evidence for recovery.It applies an all-or-nothing validity rule that records failures instead of converting them into finite penalties.
Empirical Validation: Across four NP-hard benchmarks, ATLAS outperforms several state-of-the-art component-synthesis methods and a matched full-synthesis baseline, while remaining competitive with strong human-designed algorithms.A run retains multiple competitive algorithms in separate archive regions, consistent with multimodal search.

II. RELATED WORK … 1) ATLAS Setting:

The paper situates ATLAS among component-synthesis, full-algorithm-synthesis, and meta-optimization methods, then defines a scaffold-free setting where complete executable algorithms are generated within minimal external boundaries. ATLAS evaluates and archives whole algorithms while permitting multi-component structures and coordinated revisions without requiring an explicit function-role decomposition.

II. RELATED WORK: LLM-driven component-synthesis methods generate functions for predefined roles within user-specified algorithmic control flow.Examples include FUNSEARCH, EOH, REEVO, HSEvo, MCTS-AHD, CALM, HEURAGENIX, MEOH, and EOH-S.
II. RELATED WORK: Full-algorithm methods instead generate or evolve complete algorithms, including LLAMEA, ALPHAEVOLVE, and concurrently developed A2DEPT.These methods differ in domain, representation, and refinement or evolution mechanisms.
II. RELATED WORK: Meta-optimization over generated algorithms must balance exploiting strong designs with exploring and preserving useful alternatives.Prior LLM-based methods use mechanisms such as island models and UCT-guided tree search to retain or explore alternatives.
A. Overview: ATLAS navigates scaffold-free full-algorithm synthesis through an embedding-guided archive, semantic operators, and a three-layer search that preserves and refines multiple algorithmic regions.The method assigns generated algorithms and an external evaluator distinct responsibilities while combining initialization, archiving, and search procedures.
1) ATLAS Setting:: ATLAS generates complete executable optimization algorithms that independently handle construction, constraints, internal state, and stopping logic within fixed external boundaries.Each algorithm receives an instance through a fixed input interface and returns a structured solution; no user-written algorithm skeleton is supplied.
1) ATLAS Setting:: The setting fixes problem semantics, a minimal I/O interface, and an execution protocol without prescribing internal algorithm structure.The protocol uniformly specifies execution conditions and resource limits within each comparison.
1) ATLAS Setting:: The external evaluator executes each algorithm, verifies returned-solution feasibility, and independently recomputes its objective rather than completing or repairing the solution.Thus, scaffold freedom means no prescribed algorithm decomposition, not no evaluation requirements.
1) ATLAS Setting:: ATLAS represents and revises multi-component designs without explicit function-role decomposition, using operators with distinct edit scopes over complete source code.TUNE preserves structure, IMPROVE preserves the core approach, while COMBINE and DIVERGE coordinate changes across functions or structures.

2) Problem Formulation: … C. Coverage-Preserving Archive

ATLAS maintains a bounded archive of valid algorithms and uses embedding-based similarity to remove redundancy while preserving coverage across distinct semantic regions. Its archive cycle validates and repairs candidates, clusters regions, allocates layered refinement and synthesis, and prunes near-duplicates under capacity constraints.

3) ATLAS Approach:: The framework organizes search with embeddings, clustering, and a three-layer allocation strategy to counter concentration around early designs and under-refinement of promising weaker regions.Its broader architecture includes similarity, coverage-preserving archiving, semantic operators, initialization, and three-layer search.
C. Coverage-Preserving Archive: ATLAS iteratively evaluates and inserts valid algorithms, organizes and clusters the archive in embedding space, and generates candidates through three-layer search until the synthesis budget is exhausted.The high-level procedure returns the training-best algorithm, the finalized archive, and cluster representatives.
C. Coverage-Preserving Archive: ATLAS maintains a bounded archive of valid algorithms as a semantic repertoire, adding validated candidates while preserving competitive algorithms across distinct embedding-space regions.Multimodality is defined operationally by retention across representation-space regions, not by verified mechanistic families.
C. Coverage-Preserving Archive: Candidate processing executes each algorithm on all training instances, verifies outputs and feasibility, independently recomputes objectives, and repairs failures using aggregated evidence before reevaluation.Only candidates succeeding on every training instance are marked valid and retained with their training objective and lineage.
C. Coverage-Preserving Archive: The archive cycle embeds retained algorithms, removes embedding-near duplicates, clusters the full archive, excludes the elite region, and selects each surviving non-elite region’s training-best representative.The best algorithm is the training-best member of the archive, while final cluster representatives are likewise selected by training objective.
C. Coverage-Preserving Archive: Three-layer search intensifies the current best region, matures each represented non-elite region, and searches across regions through refinement, within-region combination, and cross-region synthesis.Cross-region operations combine representatives, anchor hybrids to the best algorithm, and diverge from references to seek separated approaches.
C. Coverage-Preserving Archive: ATLAS forms hybrids by combining representatives from different regions, combining the best algorithm with non-elite representatives, and applying Diverge to references from separated regions.The LLM synthesizes candidates from the complete source code of their references, and candidates enter the valid set only after processing.
C. Coverage-Preserving Archive: Archive management removes embedding-near redundancy and, when capacity is exceeded, prunes the most similar pairs first while protecting the updated best neighborhood.Two deduplication tiers use distance thresholds, with performance-aware removal additionally requiring a small objective difference; ties are broken uniformly at random.

1) Similarity-Based Deduplication:

ATLAS controls archive size by deduplicating the most similar algorithms while protecting the best design and its nearest neighbors as refinement anchors.

Similarity-Based Deduplication:: When the archive exceeds Nmax, ATLAS repeatedly removes the lower-performing algorithm from the most similar pair.The pair is selected by minimizing embedding distance Dij.
Similarity-Based Deduplication:: The elite region G* = {a*} ∪ k-NN(a*, k) is excluded from removal to preserve refinement anchors.This protects the best algorithm and its k nearest neighbors.

2) Capacity-Constrained Pruning: … F. Three-Layer Search Strategy

ATLAS combines semantic operators, failure-conditioned repair, diversity-preserving archive management, and a three-layer embedding-space search to synthesize and retain complete algorithms. Its workflow initializes diverse regions, refines promising and alternative designs, and recombines components across regions while rejecting infeasible or malformed candidates.

2) Capacity-Constrained Pruning: … F. Three-Layer Search Strategy: ATLAS prunes algorithms by marginal coverage contribution rather than global fitness, preserving nonredundant representation across embedding-space regions without unbounded archive growth.Similarity-first management retains candidates that expand archive coverage even when they are not currently optimal.
D. Semantic Search Operators: CREATE generates complete algorithms without references; IMPROVE, TUNE, and SIMPLIFY refine one archived algorithm, while COMBINE and DIVERGE synthesize from multiple references.The operators are prompted with the problem specification and executable interface requirements, and REPAIR regenerates rejected candidates from classified evaluation evidence.
1) Zero-Reference Generation:: CREATE is used only for initialization and supplies numerical instance–solution examples to clarify output structure and constraint satisfaction without exposing implementation logic.It generates candidates directly from the problem specification and I/O requirements, with no reference algorithm.
2) Single-Reference Refinement:: IMPROVE changes logic and efficiency, TUNE adjusts numerical parameters, and SIMPLIFY removes redundant code while each preserves the reference algorithm’s fundamental paradigm.These single-reference operators provide refinement with progressively different targets, including parsimony pressure against code bloat.
3) Multi-Reference Synthesis:: COMBINE integrates complementary ideas from multiple references, whereas DIVERGE explicitly seeks differences, using nearby or distant embedding regions to enable local hybrids or large search jumps.Embedding distance determines whether multi-reference synthesis receives local context or cross-region context.
4) Repair:: Full-algorithm candidates must construct and validate feasible solutions themselves, so ATLAS rejects any candidate with malformed, infeasible, or independently mis-scored output rather than applying a finite penalty.The evaluator never completes or modifies returned solutions, and failure on any evaluated instance invalidates the candidate; REPAIR then uses aggregate failure evidence.
E. Initialization: ATLAS bootstraps diversity by sampling CREATE at temperature T=1.0, then repeatedly applying DIVERGE to uniformly sampled archive references to expand coverage beyond clustered textbook approaches.The two phases are bootstrap generation and reference-conditioned diversification.
F. Three-Layer Search Strategy: The three-layer strategy concentrates refinement near the best algorithm, distributes refinement across other represented regions, and synthesizes across regions using components and interactions from multiple inputs.Embedding-space clustering enables local operations within regions and global operations across regions; the elite region contains the best algorithm and its k-nearest neighbors.

1) Archive Clustering: … 3) Evaluation Protocol:

ATLAS maintains a globally archived, embedding-clustered search that allocates refinement and synthesis across elite and non-elite regions. Its experiments evaluate this scaffold-free framework across four NP-hard problems against human-designed, component-based, and full-synthesis baselines under controlled multi-run protocols.

1) Archive Clustering:: Embedding-induced clusters are archive regions only; clustering does not infer mechanistic algorithm families.The clusters depend on the selected embedding and should not be interpreted as mechanistic categories.
1) Archive Clustering:: Empty clusters are discarded, representatives are replaced by the best surviving members, and Layers 2 and 3 operate on the resulting non-elite archive regions.Layer 3 additionally includes the quality anchor a∗ in its COMBINE and DIVERGE streams.
2) Layer Roles:: Layer 1 refines the elite region, while Layer 2 gives each surviving non-elite representative dedicated within-region refinement and, when possible, local recombination.Layer 2 avoids requiring global-fitness parent selection for inferior regions.
3) Archive-Conditional Allocation:: The nominal work of Layers 2 and 3 scales with surviving cluster count Rt, while clustering, elite membership, and representatives are recomputed each iteration.Each Layer 3 stream schedules Rt candidates subject to the remaining global budget.
3) Archive-Conditional Allocation:: ATLAS uses one global archive with transient embedding-induced clusters, unlike island models with persistent migrating demes or MAP-Elites with fixed descriptor grids.The comparison is motivational: ATLAS preserves alternatives without adopting either baseline mechanism.
1) Benchmark Problems:: The evaluation covers CVRP, CVRPTW, FSS, and QAP, representing routing, scheduling, and assignment objectives with their associated constraints.All four are NP-hard combinatorial optimization problems.
2) Baselines:: Comparisons include domain-specific human-designed methods, REEVO, EOH, MCTS-AHD, and the adapted scaffold-free full-synthesis baseline EOH-FULL.LLM methods share a backend, matched token-budget termination, and a common greedy-construction scaffold for component-based comparisons.
3) Evaluation Protocol:: Each LLM method performs R = 5 independent synthesis runs, selects the best training algorithm, and evaluates it on 31 independently generated test instances.Search and archive construction access only the 31-instance training split; any invalid or infeasible test outcome causes the run to fail.

4) ATLAS Configuration: … 2) Generalization Beyond Default Benchmark Settings:

ATLAS is configured and implemented with documented hyperparameters, executable prompts, controlled compute settings, and per-run resource estimates. Across default and scaled benchmarks, it outperforms LLM-based baselines, remains close to human-designed references, and faces increasing hyperparameter sensitivity at larger scales.

5) ATLAS Implementation:: Compared methods use single-core, single-threaded candidate–instance evaluations, while different instances run in parallel.This execution model also applies to human-designed domain-specific baselines when applicable.
5) ATLAS Implementation:: 500 operator calls and approximately 5M–7M total tokens yield an estimated GPT-5-MINI (LOW) API cost of approximately $4.2–$5.8 per synthesis run.The token total includes reasoning tokens and is roughly two-thirds input and one-third output.
1) Comparison with Baselines:: ATLAS statistically outperforms every LLM-based baseline across all four default benchmark settings.EOH performs best among component-synthesis methods in three settings, while MCTS-AHD performs best on FSS; EOH-FULL substantially improves over component-based implementations.
B. Main Experimental Results: ATLAS remains within 0.46% of the best human-designed reference on all four default settings, with gaps of 0.347% on FSS, 0.094% on CVRP, 0.000% on CVRPTW, and 0.460% on QAP.On CVRP, ATLAS is not significantly different from PyVRP; on CVRPTW, it matches the lowest reported mean at displayed precision.
2) Generalization Beyond Default Benchmark Settings:: ATLAS achieves the best results among LLM-based methods in all four additional half- and double-scale settings re-run from scratch on newly generated instances.Its FSS gaps are 0.029% and 1.651% at smaller and larger scales; on CVRP, it matches the best domain-specific mean at n25 and has a 1.820% gap at n100.
2) Generalization Beyond Default Benchmark Settings:: Performance becomes more sensitive to numerical hyperparameters as problem scale increases, potentially contributing to wider gaps at larger scales.The TUNE operator proposes numerical changes from source code and evaluates them through standard selection, but lacks dedicated feedback-driven hyperparameter optimization; source-code-only proposals are unlikely to robustly identify settings.

3) Training-Selected Archive Representatives: · C. Component Ablations and Design Analyses · V. CONCLUSION

ATLAS retains multiple competitive, structurally different algorithms across embedding-space regions, and its complete three-layer search configuration outperforms ablated variants. The framework enables scaffold-free full-algorithm synthesis while leaving hyperparameter configuration as a remaining performance opportunity.

3) Training-Selected Archive Representatives:: The final archive retains representatives from multiple embedding-space regions, each fixed as the training-best member of a final cluster.The reported set comes from the best-training-objective run among R=5 synthesis runs.
3) Training-Selected Archive Representatives:: Code inspection finds structural diversity in primary construction and global-search components, alongside shared refinement mechanisms and component reuse.Examples include regret insertion, savings-based construction, clustering, ruin-and-recreate, and label- or beam-inspired seeding.
3) Training-Selected Archive Representatives:: The archive and Layer 3 preserve several competitive, structurally different multi-component algorithms rather than only one best design.Layer 3 can recombine components and interactions from different input algorithms into new hybrids.
C. Component Ablations and Design Analyses: ATLAS improves over EOH-FULL under shared full-algorithm operators, repair, evaluator, LLM, and budgets, supporting the value of archive-based semantic quality-diversity search.EOH-FULL itself improves over shared-scaffold component implementations in the controlled comparison.
C. Component Ablations and Design Analyses: The complete three-layer configuration performs best, while ablations support intensive elite-region refinement, representative-level refinement, and the superiority of Layer 3 alone over Layer 1 alone.Variant A versus C isolates Layer 1, Variant C versus D isolates Layer 2, and Variant D outperforms Variant B.
C. Component Ablations and Design Analyses: MGTE-LARGE-EN-V1.5 with concatenated name, description, and preprocessed code is the strongest single-encoder embedding strategy.Dual-phase initialization with DIVERGE yields greater initial embedding-space dispersion than prompt variation alone while maintaining high validity.
C. Component Ablations and Design Analyses: ATLAS remains effective across several LLM backends; stronger models perform better, but the differences remain moderate.The comparison is reported as an underlying-LLM analysis.
V. CONCLUSION: ATLAS performs scaffold-free full-algorithm synthesis using only problem objectives, constraints, and a minimal external I/O interface, without prescribing internal structure.Generated algorithms must satisfy problem requirements, while full-algorithm operators leave component inventory and control flow to synthesis.

APPENDIX A EXTENDED RELATED WORK AND COMPARATIVE ANALYSIS … 1) Permutation Flow Shop Scheduling (FSS):

ATLAS is positioned as a scaffold-free, full-algorithm synthesis method within established LLM-based design taxonomies and contrasted with component-based, multi-algorithm, and other full-synthesis approaches. The appendices also specify implementation details and formal benchmark definitions, including permutation flow shop scheduling.

A. Positioning ATLAS in Existing Taxonomies: ATLAS is classified as LLM-as-Designer, high-level algorithm generation, and MetaBBO algorithm generation via in-context learning.These taxonomies respectively describe the model’s role, intervention level, and combination of learning mechanism with meta-level task.
Synthesis: Scaffold-Free Full-Algorithm Synthesis under a Fixed I/O Interface: Unlike methods that optimize designated components inside prescribed scaffolds, ATLAS synthesizes complete algorithms from a problem specification under a minimal I/O interface.Its internal decomposition, component interactions, and control flow are not prescribed.
1) Component-Based Single-Algorithm Synthesis: Component-based single-algorithm methods retain a predefined framework or structured hierarchy and refine targets such as priority functions, scoring rules, operators, or key functions.The reviewed examples include FUNSEARCH, EOH, REEVO, MCTS-AHD, CALM, HEURAGENIX, and MTHS.
2) Component-Based Multi-Algorithm Synthesis: Designed-multiplicity component methods make multiple heuristics an explicit output objective, whereas ATLAS preserves candidates by embedding-space coverage rather than optimizing portfolio or Pareto criteria.MEOH targets quality–runtime Pareto fronts, while EOH-S targets complementary portfolios across instance regimes.
3) Full-Synthesis: Full-synthesis methods differ in domain, external interface, and structural prior: LLAMEA evolves complete metaheuristics, ALPHAEVOLVE edits marked regions, and A2DEPT uses function-level representations and hierarchical edits.LLAMEA’s published setting uses a fixed callable interface and a single active evolutionary trajectory; ALPHAEVOLVE retains supplied code structure; A2DEPT imposes function-level inductive bias.
3) Full-Synthesis: ATLAS’s undecomposed representation can produce coordinated multi-function workflows, illustrated by a CVRP algorithm combining construction, repair, local-search, large-neighborhood, and acceptance mechanisms.The example connects alternative construction procedures, savings merging, regret repair, relocation, 2-opt, large-neighborhood search, and simulated-annealing acceptance.
APPENDIX B IMPLEMENTATION DETAILS: The released implementation maintains exact executable prompts that combine objectives, problem specifications, I/O requirements, response structure, and selected reference information.Prompt assembly is described as occurring in framework/prompt_system.py, with source code and prompts maintained in the public repository.
1) Permutation Flow Shop Scheduling (FSS): For permutation flow shop scheduling, a job permutation is applied across all machines to minimize makespan under sequential machine processing and capacity constraints.Completion times follow C_l,k = max(C_l−1,k, C_l,k−1) + p_πl,k, with non-preemption and precedence enforced by the formulation.

2) Capacitated Vehicle Routing Problem (CVRP): … 4) Quadratic Assignment Problem (QAP):

The paper formalizes CVRP, CVRPTW, and QAP, then generates synthetic benchmark instances with fixed train–test splits and problem-specific parameters. The formulations specify routing feasibility, time-window dynamics, quadratic facility-location costs, and implementation conventions.

2) Capacitated Vehicle Routing Problem (CVRP):: CVRP represents solutions as an unlimited-fleet set of customer routes, with positive customer demands constrained by vehicle capacity Q.Each route is an ordered sequence of distinct customers, and route feasibility includes customer coverage and capacity constraints.
3) CVRP with Time Windows (CVRPTW):: CVRPTW extends CVRP with customer and depot time windows, service times, waiting, and return-to-depot deadlines.Service starts at σi = max(τi, ei), and routes become infeasible when service starts after a customer’s latest time li or returns after depot closure.
4) Quadratic Assignment Problem (QAP):: QAP assigns each facility to a distinct location through a permutation π, minimizing quadratic flow–distance interaction costs.The flow and distance matrices are symmetric with zero diagonals, and the evaluator sums contributions over facility pairs.
B. Benchmark Instance Generation: Synthetic instances follow LLM4AD benchmark generators for CVRP, CVRPTW, and QAP, while FSS uses a newly described synthetic generator instead of COBench instances.This replacement makes FSS instance generation consistent with the other benchmark problems.
B. Benchmark Instance Generation: Each benchmark setting contains Ntrain = 31 training instances and Ntest = 31 independently generated test instances, using split seeds 2024 and 42.Only training instances are accessible during synthesis and archive construction; the test split is accessed after the algorithm is frozen.
1) CVRP:: CVRP instances use n = 50 customers, capacity Q = 40, coordinates uniformly sampled in [0, 1]^2, integer demands in {1, . . . , 9}, and Euclidean travel distances.The depot is sampled alongside customers, while its demand is fixed to 0.
2) CVRPTW:: CVRPTW instances retain n = 50 and Q = 40, with service times and window lengths sampled from U(0.15, 0.20) and depot closing time Tmax = 4.6.Customer early times depend on depot-to-customer distance, and late times equal early times plus the sampled window length.
3) Permutation Flow Shop Scheduling (FSS): / 4) Quadratic Assignment Problem (QAP):: FSS uses n = 50 jobs and m = 10 machines with processing times sampled as integers in {10, . . . , 99}, whereas QAP uses n = 50 and symmetric zero-diagonal matrices with entries in {1, . . . , 100}.FSS solutions are job permutations applied across machines; QAP matrices are generated by sampling, symmetrizing, and zeroing diagonals.

C. Evaluation Protocol … 4) Three-Layer Search Strategy:

The paper evaluates scaffolded components and full executable algorithms under matched benchmark conditions, then details ATLAS’s archive-based workflow, from initialization and maintenance through three-layer search and final reporting. Its protocol separates training-based selection from locked test evaluation and uses diversity-preserving search to refine elites, explore regions, and synthesize across clusters.

1) Evaluators:: Component baselines execute synthesized heuristics inside fixed scaffolds, whereas ATLAS and EOH-FULL execute complete algorithms end-to-end with feasibility and output validation.ATLAS and EOH-FULL use identical full-algorithm evaluators; objective computation is identical for identical feasible solutions.
2) Synthesis Runs and LLM Configuration:: LLM-based methods use GPT-5-MINI at low reasoning effort and T = 1.0, with R = 5 independent synthesis runs using different random seeds.Each run yields one final optimization algorithm, while human-designed baselines are evaluated once.
3) Budgets and Runtime Caps:: ATLAS ablations and sensitivity analyses use B = 500 evaluated operator executions, while cross-method comparisons match total token consumption per synthesis run.Per-instance runtime caps are 210 s for FSS, 30 s for CVRP, 120 s for CVRPTW, and 240 s for QAP.
5) Reporting and Statistical Testing:: ATLAS selects the best-training algorithm from each run, then evaluates it on all Ntest = 31 test instances without resuming search or modifying the archive.Success requires all 31 executions to meet runtime, memory, execution, formatting, finite-objective, and feasibility checks.
D. ATLAS: Algorithms and Pseudocode: The detailed workflow expands from initialization and candidate generation to distance updates, deduplication, archive-size management, clustering, and final archive representatives.Algorithm B.1 returns the best algorithm a∗, archive A, and cluster representatives T.
2) Initialization:: Initialization combines diverse CREATE generation with paradigm-diversifying DIVERGE, while deduplication removes semantic or negligible-gain near-duplicates before capacity pruning.Similarity-first pruning removes the worse member of the closest pair, preserving embedding-space coverage; the protected elite region G∗ retains refinement capacity.
4) Three-Layer Search Strategy:: The three-layer strategy intensifies refinement around a∗, distributes refinement across cluster regions, and performs cross-cluster or best-distant COMBINE for global synthesis.DIVERGE includes a∗ as a reference to encourage alternatives separated from the already sampled best region.

E. ATLAS: Configuration and Hyperparameters · 1) Embedding Strategy: · 2) Clustering Details:

ATLAS represents algorithms with normalized early-fusion MGTE embeddings and organizes the archive through distance-based k-medoids clustering. Clustering uses repeated FasterPAM runs, automatic Silhouette-based K selection, singleton handling, and best-performing cluster representatives for downstream search.

1) Embedding Strategy:: ATLAS embeds each algorithm by applying MGTE-LARGE-EN-V1.5 to its name, description, and preprocessed code under early fusion.Comments, docstrings, and blank lines are removed before concatenation; cosine similarity is computed in the resulting embedding space.
1) Embedding Strategy:: Early-fusion + MGTE achieved the highest Class Cohesion Index and second-best accuracy, motivating its practical default use.The representation is described as simpler, more practical, and based on an open-source model than the top-accuracy variant.
1) Embedding Strategy:: All embeddings are ℓ2-normalized, making cosine similarity equivalent to the dot product between normalized vectors.The normalization is applied before similarity computation.
2) Clustering Details:: ATLAS clusters candidates with k-medoids on a precomputed cosine-distance matrix using FasterPAM, whose medoids are actual archive algorithms.This supports representative-based downstream search while operating directly on arbitrary pairwise distances.
2) Clustering Details:: Each candidate K receives 10 independent FasterPAM runs with max_iter = 300, retaining the run with the lowest k-medoids objective.PAM BUILD initializes every run, while multiple seeds reduce initialization sensitivity and poor local minima.
2) Clustering Details:: K is selected by maximizing average Silhouette score over candidate values, with Kmin = 3 and ties broken toward smaller K.Kmin = 3 ensures at least two clusters remain after removing the cluster containing the current best algorithm for Layer 3 operations.
2) Clustering Details:: After clustering, singleton clusters are merged into the nearest non-singleton cluster using minimum average distance.For each cluster, the best-performing member, defined by the lowest objective value, becomes its representative for refinement and cross-cluster synthesis.

3) Hyperparameter Configuration:

ATLAS fixes its main hyperparameters across experiments to balance archive coverage, deduplication, initialization diversity, and refinement across embedding-space regions. The configuration uses a 100-algorithm archive, 50 initial candidates, and a three-layer search centered on a five-member elite set.

Archive and similarity configuration: Nmax = 100 bounds the archive, enabling coverage across multiple embedding-space regions while keeping pairwise-distance and clustering overhead manageable.The square-root clustering heuristic yields at most Kmax ≈10 candidate clusters.
Archive and similarity configuration: τstrict = 0.95 removes predominantly cloned, functionally equivalent, trivially edited, or near-identical algorithms.The threshold was selected from exploratory analysis of 8 pilot runs on FSS and CVRP.
Initialization: 50 initial algorithms are generated, with 20 Phase 1 CREATE calls and 30 Phase 2 DIVERGE calls, allocating 10% of the default B = 500 operator-call budget to initialization.Deduplication can reduce the retained initial archive below 50 before the main loop.
Layer 1 configuration: |G∗| = 5 defines the elite set as a∗ plus four nearest neighbors, balancing local refinement against resources reserved for Layers 2–3.The small neighborhood supplies multiple refinement anchors while bounding Layer 1 cost.
Layer 1 configuration: nintensify = 3 applies three intensification rounds to a∗, with IMPROVE, TUNE, and SIMPLIFY always applied while neighbors receive probabilistic refinement.This configuration maintains strong exploitation pressure while compensating for strict deduplication.

4) LLM Configuration: • … G. Baselines

ATLAS uses GPT-5-MINI with fixed temperature, embedding-guided search budgets, and resource-contained evaluation. Its validation and repair pipeline rejects invalid candidates, conditionally regenerates failed algorithms, and compares against human-designed and LLM-based baselines.

4) LLM Configuration: •: ATLAS uses GPT-5-MINI with low reasoning effort for practical full-algorithm synthesis over hundreds of operator calls.The configuration prioritizes capability alongside cost and latency.
4) LLM Configuration: •: T = 1.0 is fixed across synthesis operators and compared LLM-based methods to balance generation diversity and stability.The same temperature is used for all compared LLM-based synthesis methods.
5) Embedding configuration:: The embedding configuration uses MGTE-LARGE-EN-V1.5 with early fusion of algorithm name, description, and preprocessed code.Both choices were based on the representation analysis in Appendix C-A.
6) Search Budget: •: B = 500 evaluated operator executions is the default ATLAS budget, while baseline comparisons use problem-specific token-matched budgets.Token budgets are calibrated from average usage under 500 full-synthesis executions because synthesis types consume different numbers of tokens.
1) Sequential LLM Calls, Parallel Instance Evaluation:: ATLAS sequentially executes LLM calls and parallelizes instance evaluation with a CPU-aware process pool.Worker count is the smallest of instance count, available CPU cores, and the user-specified limit.
2) Entry-Point Checking and Worker-Process Execution:: Entry-point checks, worker processes, timeouts, and RLIMIT_AS contain execution failures and excessive allocation, but do not provide a security sandbox.Forward experiments run under Linux/WSL2, and native Windows execution is refused.
4) Independent Constraint and Objective Validation:: The authoritative validator independently checks output structure, constraints, and objectives; candidates must succeed on every training instance to enter the archive.Failures trigger one REPAIR attempt, and repaired candidates are admitted only after complete successful re-evaluation.
6) Failure-Conditioned Repair:: REPAIR applies only after evaluation-stage execution or validation failures, classifies diagnostics deterministically, and regenerates source using failure-specific correction objectives.Malformed responses and entry-point failures bypass repair; regenerated candidates undergo the same checks and all-or-nothing acceptance policy.

1) Human-Designed Domain-Specific Baseline Implementations: … 5) Embedding Strategies:

The paper establishes controlled human-designed, component-synthesis, and full-synthesis baselines, then analyzes embedding representations using curated executable algorithms, varied pretrained encoders, preprocessing, and four text/code fusion strategies.

1) Human-Designed Domain-Specific Baseline Implementations:: Human-designed baselines use documented defaults or fixed upstream configurations without test-result tuning, while runtime caps are common upper bounds and may allow continued baseline improvement.Implementations include PyVRP, OR-Tools, VROOM, NEH, IG-TB, Iterative Beam Search, RoTS, Simulated Annealing, BLS, and BMA.
2) Component-Synthesis Baseline Implementations:: REEVO, EOH, and MCTS-AHD are evaluated through LLM4AD with shared scaffolds, prompts, interfaces, evaluators, benchmark instances, seeds, budgets, runtime caps, and GPT-5-MINI settings.The shared greedy-construction scaffold and infrastructure reduce confounding from prompt wording, scaffold design, evaluators, instances, and stopping criteria.
3) EOH-FULL: Full-Algorithm Synthesis Baseline:: EOH-FULL preserves EOH’s fitness-driven evolutionary structure while replacing component generation with ATLAS’s full-synthesis operators under matched prompts, repair, evaluation, instances, configuration, caps, and budgets.Its population size remains 20, rather than ATLAS’s archive cap Nmax = 100, and it retains EOH’s original initialization scheme.
3) EOH-FULL: Full-Algorithm Synthesis Baseline:: ATLAS organizes search semantically across embedding neighborhoods, cluster representatives, and cross-cluster combinations, whereas EOH-FULL uses fitness-driven selection and maintains an active evolutionary population.The comparison isolates the effect of replacing EOH’s evolutionary search organization with ATLAS’s semantic quality-diversity search.
APPENDIX C COMPONENT ABLATIONS AND DESIGN ANALYSES: The appendix distinguishes component ablations from design analyses, with embedding, initialization, reasoning-effort, and underlying-LLM studies examining supporting design choices.Semantic search-guidance and search-layer studies are classified as component ablations.
A. Embedding Representation Analysis Against Curated Family Labels; 1) Aim and Research Questions:: Embedding analysis evaluates whether pretrained representations support nearest-neighbor retrieval, clustering, deduplication, and similarity pruning while distinguishing author-labeled algorithm families.The research questions address family discrimination, text-versus-code modality contribution, fusion strategy, and encoder characteristics.
2) Dataset Construction and Family Annotation:: The curated benchmark contains 308 algorithms across FSS, CVRP, CVRPTW, and QAP, selected from more than 2,000 validated implementations to control inter-label separation and intra-label diversity.The problems span routing, scheduling, and assignment, while candidates were generated with multiple LLM backends and varied prompting strategies.

6) Evaluation Metrics: … 3) Results and Analysis:

The paper evaluates embedding strategies using local agreement and label-based cohesion, selects MGTE-LARGE-EN-V1.5 with concatenation for ATLAS, and finds that divergence-based initialization produces substantially more diverse valid algorithm archives than stochastic or persona-based variation.

6) Evaluation Metrics:: NNA measures nearest-neighbor agreement with curated family labels, while CCI measures silhouette-based cohesion and separation relative to those labels.Both metrics are computed for each embedding strategy; CCI ranges from −1 to +1, with larger values indicating tighter grouping and clearer separation.
7) Results:: Text-only strategies achieve 89–95% NNA across tested general-purpose models, outperforming code-only strategies at 50–78% NNA.The results indicate that algorithm names and descriptions strongly reproduce the curated broad-family labels.
8) Analysis and Interpretation:: Embedding scale is not decisive: MGTE-LARGE-EN-V1.5 achieves 95.38% NNA with concatenation, compared with 95.33% for QWEN3-EMBEDDING-4B.Within the OpenAI family, TEXT-EMBEDDING-3-SMALL exceeds TEXT-EMBEDDING-3-LARGE across all three single-encoder strategies.
8) Analysis and Interpretation:: Concatenation is model-dependent, improving NNA over text-only by +3.94pp for MGTE, +5.80pp for Qwen3-4B, and +3.03pp for Qwen3-0.6B.It instead degrades performance for OpenAI-Small by -3.29pp and OpenAI-Large by -4.82pp.
8) Analysis and Interpretation:: Late fusion reaches 97.40% NNA at α = 0.6, while MGTE-LARGE-EN-V1.5 with concatenation is selected for its 95.38% NNA, highest CCI of 0.437, simplicity, and accessibility.Late fusion requires separate text and code embeddings plus a tunable mixing weight; the selected single-encoder configuration requires one embedding call per algorithm.
1) Aim and Research Question:: The initialization study asks whether ATLAS’s dual-phase initialization yields more diverse valid initial algorithms than prompt variation alone.It compares three strategies across GPT-4.1-MINI, GPT-5 MINI, CLAUDE HAIKU 4.5, and LLAMA 4 MAVERICK on CVRP and FSS.
2) Setup:: The study generates 50 algorithms per trial at T = 1.0, repeats five independent trials, and measures validity, MNND local sparsity, and MPD global spread in MGTE concatenated embedding space.The strategies are stochastic-only baseline, +Random Persona, and +Divergence; higher values are better for Valid, MNND, and MPD.
3) Results and Analysis:: +Divergence outperforms baseline and random persona variation in all 16 diversity comparisons, with CVRP gains of 68–287% for MNND and 29–331% for MPD.On FSS, its gains range from 94–210% for MNND and 42–227% for MPD, while validity remains 84–98% on CVRP and 89–98% on FSS.

C. Semantic Search-Guidance Ablation … 2) Impact on Archive Diversity:

Across the ablations, ATLAS performs best when semantic guidance and all three search layers are retained, while Low reasoning offers the best practical quality–cost trade-off and stronger underlying LLMs improve final performance. The archive-diversity analysis further shows that GPT-5-MINI and GPT-5 retain structurally varied competitive representatives, although representative diversity is descriptive rather than a headline performance measure.

C. Semantic Search-Guidance Ablation — 3) Results and Analysis:: Semantic guidance using embeddings for both Layer 1 neighbor selection and archive grouping achieves the best performance on CVRP and FSS versus fitness-based and random guidance.All variants retain the same archive mechanisms, elite-set size, adaptive cluster count, and representative-selection rule; they differ only in guidance-specific neighbor selection and grouping.
C. Semantic Search-Guidance Ablation — 1) Aim and Research Question:: The semantic representation supports targeted local refinement, coherent representative-level refinement, and separated cross-region synthesis across ATLAS’s three search layers.Embedding-based neighbors guide Layer 1, embedding groups provide local context in Layer 2, and semantically separated representatives support Layer 3 synthesis.
D. Search-Layer Ablation — 3) Analysis:: The full three-layer configuration achieves the best performance on both benchmarks, followed by Layers 2+3, Layer 3 only, and Layer 1 only.The ordering is consistent with complementary roles for local refinement, distributed representative search, and cross-cluster synthesis.
D. Search-Layer Ablation — 3) Analysis:: Removing Layer 1 lowers performance, while Layer 1 alone is insufficient; distributed refinement and cross-cluster synthesis maintain competitive performance across multiple promising archive regions.The comparison between the full configuration and Layers 2+3 indicates a contribution from intensive local refinement, while the Layer 1-only result shows that representative-level and cross-cluster operations are also needed.
E. Reasoning-Effort Sensitivity and Cost Analysis — 3) Results and Analysis:: Increasing reasoning effort from Minimal to Low improves both benchmarks, with total tokens increasing approximately 3% on CVRP and 5% on FSS.The passage characterizes Low as a stronger modification setting at relatively minor additional token cost.
E. Reasoning-Effort Sensitivity and Cost Analysis — 3) Results and Analysis:: Beyond Low, additional gains are limited: CVRP shows no statistically significant difference among Low, Medium, and High, while FSS gains less than 0.06 percentage points despite higher cost.Compared with Low, Medium increases token usage approximately 28% on CVRP and 31% on FSS, while High increases it approximately 73% on CVRP and 84% on FSS.
F. Effect of the Underlying LLM on ATLAS — 1) Performance Comparison:: On FSS, GPT-5 achieves the best mean performance, GPT-5-MINI is close behind, and GPT-5-NANO and CLAUDE HAIKU 4.5 are weaker but acceptable under the fixed ATLAS framework.The GPT-5 family uses the same low reasoning level, while CLAUDE HAIKU 4.5 uses its default setting without extended thinking.
F. Effect of the Underlying LLM on ATLAS — 2) Impact on Archive Diversity:: GPT-5-MINI and GPT-5 retain strong, structurally varied competitive archive representatives, but this descriptive repertoire view does not measure diversity as a headline performance outcome.Representative membership is fixed from training objectives, while test gaps only order and interpret the displayed representatives.