Graph Learning GRASP (GL-GRASP)

• The paper introduces a novel hybrid method that integrates graph representation learning with GRASP to efficiently minimize arc crossings in constrained incremental graph drawings.
• It employs PCA-reduced node embeddings to form a semi-greedy candidate list, guiding vertex selection and placement under strict hierarchical and displacement constraints.
• Empirical results show that GL-GRASP, particularly with HOPE and SDNE variants, outperforms classic GRASP heuristics on denser instances with lower gaps and faster convergence.

Searching arXiv for GL-GRASP and closely related papers to ground the article. Graph Learning GRASP (GL-GRASP) is a hybrid metaheuristic for the Constrained Incremental Graph Drawing Problem (C-IGDP) that incorporates Graph Representation Learning (GRL) into the construction phase of a Greedy Randomized Search Procedure (GRASP). In the formulation presented in “An Efficient Hybridization of Graph Representation Learning and Metaheuristics for the Constrained Incremental Graph Drawing Problem” (Charytitsch et al., 21 Aug 2025), GL-GRASP uses node embeddings, reduced to two dimensions by PCA, to guide incremental vertex selection and placement under hierarchical, precedence, and displacement constraints. The method is designed for layered graph drawings in which original vertices must preserve their relative order and may move only within a bounded displacement, while incremental vertices and arcs are inserted so as to minimize total arc crossings. The paper positions GL-GRASP as a learning-augmented variant of prior GRASP heuristics for C-IGDP, and explicitly distinguishes it from other unrelated methods whose acronyms also include “GRASP” (Charytitsch et al., 21 Aug 2025).

1. Problem setting and optimization model

GL-GRASP is defined for the Constrained Incremental Graph Drawing Problem on a hierarchical graph $G = (V, A, \Lambda)$ partitioned into $\Lambda$ layers, with $V = V^1 \cup \dots \cup V^\Lambda$ and $A = A^1 \cup \dots \cup A^{(\Lambda-1)}$ connecting successive layers (Charytitsch et al., 21 Aug 2025). An initial drawing $D = (G, \Pi_0)$ provides layerwise positions $\Pi_0 = \{\pi_0^1, \dots, \pi_0^\Lambda\}$, where $$\pi_0^\lambda: V^\lambda \to \{1, \dots, |V^\lambda|\}$$. Incremental graph drawing augments the instance with new vertices and arcs, denoted by $IV^\lambda = V^\lambda \cup \hat{V}^\lambda$ and $IA^\lambda = A^\lambda \cup \hat{A}^\lambda$, without changing the number of layers (Charytitsch et al., 21 Aug 2025).

The output is a new drawing $I = (IG, \Pi)$ for $\Lambda$0, with updated position functions $\Lambda$1 and $\Lambda$2 (Charytitsch et al., 21 Aug 2025). The objective is to minimize the total number of arc crossings across adjacent layers. The model uses binary precedence variables $\Lambda$3 to encode whether vertex $\Lambda$4 precedes vertex $\Lambda$5 in layer $\Lambda$6, and binary crossing variables $\Lambda$7 to encode whether arcs $\Lambda$8 and $\Lambda$9 cross between layers $V = V^1 \cup \dots \cup V^\Lambda$0 and $V = V^1 \cup \dots \cup V^\Lambda$1 (Charytitsch et al., 21 Aug 2025).

The optimization objective is

$V = V^1 \cup \dots \cup V^\Lambda$2

subject to crossing-consistency constraints, transitivity of precedence, mutual exclusivity of pairwise precedence, preservation of the original relative order of non-incremental vertices, and an absolute displacement bound $V = V^1 \cup \dots \cup V^\Lambda$3 for those original vertices (Charytitsch et al., 21 Aug 2025). Position encoding is linked to precedence by

$V = V^1 \cup \dots \cup V^\Lambda$4

for all $V = V^1 \cup \dots \cup V^\Lambda$5 (Charytitsch et al., 21 Aug 2025). The incremental-stability constraints are central: original vertices retain their pairwise order from $V = V^1 \cup \dots \cup V^\Lambda$6, and each may move at most $V = V^1 \cup \dots \cup V^\Lambda$7 positions from its initial slot (Charytitsch et al., 21 Aug 2025).

This formulation places GL-GRASP within constrained layered drawing rather than graph learning in the sense of Laplacian estimation or graph similarity prediction. A common source of confusion is nomenclature: the 2025 GL-GRASP method is distinct from “GraSP: Simple yet Effective Graph Similarity Predictions” (Zheng et al., 2024), from “GRASPEL: Graph Spectral Learning at Scale” (Wang et al., 2019), and from the digital pathology method “GRASP: GRAph-Structured Pyramidal Whole Slide Image Representation” (Mirabadi et al., 2024).

2. Baseline GRASP heuristics and their role

GL-GRASP builds on baseline GRASP heuristics introduced for C-IGDP and retains the standard two-phase metaheuristic structure: a construction phase followed by local search (Charytitsch et al., 21 Aug 2025). Each GRASP iteration constructs a feasible solution using a semi-greedy Restricted Candidate List (RCL), then improves it with neighborhood moves consisting of swaps and insertions of incremental vertices (Charytitsch et al., 21 Aug 2025). Termination is controlled by a maximum number of iterations $V = V^1 \cup \dots \cup V^\Lambda$8 and a stall limit $V = V^1 \cup \dots \cup V^\Lambda$9 (Charytitsch et al., 21 Aug 2025).

The baseline construction heuristics are denoted C1, C2, and C3 (Charytitsch et al., 21 Aug 2025). C1 is degree-driven barycenter: it starts by choosing a high-degree vertex, placing it randomly within allowable positions, maintaining an RCL based on a degree threshold, and using the barycenter of neighbors in adjacent layers to choose a feasible location (Charytitsch et al., 21 Aug 2025). C2 is incremental barycenter: it starts from the original drawing and adds incremental vertices using the same semi-greedy rule as C1 (Charytitsch et al., 21 Aug 2025). C3 uses a minimum-crossing insertion-cost function $A = A^1 \cup \dots \cup A^{(\Lambda-1)}$0, defined as the number of additional crossings created if vertex $A = A^1 \cup \dots \cup A^{(\Lambda-1)}$1 is inserted in position $A = A^1 \cup \dots \cup A^{(\Lambda-1)}$2, and selects vertices by the minimum achievable insertion cost $A = A^1 \cup \dots \cup A^{(\Lambda-1)}$3 (Charytitsch et al., 21 Aug 2025).

For C3, the RCL threshold is

$A = A^1 \cup \dots \cup A^{(\Lambda-1)}$4

with $A = A^1 \cup \dots \cup A^{(\Lambda-1)}$5 (Charytitsch et al., 21 Aug 2025). More generally, the RCL rule is written as

$A = A^1 \cup \dots \cup A^{(\Lambda-1)}$6

where $A = A^1 \cup \dots \cup A^{(\Lambda-1)}$7 is the candidate score (Charytitsch et al., 21 Aug 2025).

The baseline pseudocode comprises: initialize the partial solution; score candidates; form the RCL; choose a candidate uniformly at random from the RCL; place it using barycenter or insertion-cost rules; repeat until all incremental vertices are inserted; then apply best-improvement swap and insert local search until a local minimum is reached (Charytitsch et al., 21 Aug 2025). The paper notes that C3 has higher construction complexity because it evaluates $A = A^1 \cup \dots \cup A^{(\Lambda-1)}$8 over all candidates and feasible positions, whereas C1 and C2 are cheaper and depend mainly on degrees, barycenters, and feasibility checks (Charytitsch et al., 21 Aug 2025).

GL-GRASP preserves this overall GRASP structure and the same local-search operators, but replaces the purely combinatorial construction logic with embedding-informed selection and placement (Charytitsch et al., 21 Aug 2025). This suggests that the contribution of GL-GRASP is not a new local-search neighborhood or a reformulation of the optimization model, but a learning-guided bias in the early stages of solution generation.

3. Graph Representation Learning components

The defining feature of GL-GRASP is its use of node embeddings to inject latent structural information into GRASP’s construction phase (Charytitsch et al., 21 Aug 2025). The paper evaluates four embedding methods: HOPE, Node2Vec, SDNE, and a spectral embedding method denoted SPEC (Charytitsch et al., 21 Aug 2025).

HOPE is described as preserving high-order proximity and asymmetric transitivity using Katz similarity, with an objective of minimizing matrix reconstruction error,

$A = A^1 \cup \dots \cup A^{(\Lambda-1)}$9

where $D = (G, \Pi_0)$0 is a similarity matrix and $D = (G, \Pi_0)$1 contains the node embeddings (Charytitsch et al., 21 Aug 2025). Node2Vec is presented as a random-walk skip-gram method that maximizes neighborhood likelihood using negative sampling,

$D = (G, \Pi_0)$2

with biased walks controlled by $D = (G, \Pi_0)$3, though the specific hyperparameters are not given and CogDL defaults are used (Charytitsch et al., 21 Aug 2025). SDNE is described as a deep autoencoder preserving first-order and second-order proximity through

$D = (G, \Pi_0)$4

(Charytitsch et al., 21 Aug 2025). SPEC uses the eigenvectors of the normalized Laplacian $D = (G, \Pi_0)$5,

$D = (G, \Pi_0)$6

(Charytitsch et al., 21 Aug 2025).

The embedding dimensions are method-specific. HOPE uses $D = (G, \Pi_0)$7, with the CogDL implementation using left and right singular vectors; SPEC uses $D = (G, \Pi_0)$8; Node2Vec and SDNE use $D = (G, \Pi_0)$9 (Charytitsch et al., 21 Aug 2025). The paper states that CogDL defaults are used and that detailed parameters are provided only in the Supplementary Material (Charytitsch et al., 21 Aug 2025).

After embedding computation, GL-GRASP applies PCA to project the embeddings into two dimensions, producing coordinates $\Pi_0 = \{\pi_0^1, \dots, \pi_0^\Lambda\}$0 for each node (Charytitsch et al., 21 Aug 2025). Euclidean distance in this 2D space becomes the main structural signal used during construction:

$\Pi_0 = \{\pi_0^1, \dots, \pi_0^\Lambda\}$1

(Charytitsch et al., 21 Aug 2025). The paper characterizes these distances as encoding latent structural patterns such as communities, structural roles, and proximities across layers, which can then inform drawing decisions (Charytitsch et al., 21 Aug 2025).

In this sense, GL-GRASP uses GRL neither as an end-to-end learned optimizer nor as a predictor of crossings. Instead, embeddings act as a geometric prior for semi-greedy construction. A plausible implication is that GL-GRASP belongs to the class of learning-augmented heuristics in which learned representations bias classical combinatorial search without replacing it.

4. Embedding-informed construction in GL-GRASP

The construction phase of GL-GRASP begins with a partial solution $\Pi_0 = \{\pi_0^1, \dots, \pi_0^\Lambda\}$2 that initially contains only the original drawing and its fixed-order vertices (Charytitsch et al., 21 Aug 2025). For each candidate incremental vertex $\Pi_0 = \{\pi_0^1, \dots, \pi_0^\Lambda\}$3, the method computes the minimum embedding-space distance to any already placed adjacent neighbor:

$\Pi_0 = \{\pi_0^1, \dots, \pi_0^\Lambda\}$4

(Charytitsch et al., 21 Aug 2025). Lower $\Pi_0 = \{\pi_0^1, \dots, \pi_0^\Lambda\}$5 indicates that $\Pi_0 = \{\pi_0^1, \dots, \pi_0^\Lambda\}$6 lies close, in the learned representation, to some already positioned neighbor and is therefore a more promising candidate for insertion (Charytitsch et al., 21 Aug 2025).

The RCL is then formed by thresholding these $\Pi_0 = \{\pi_0^1, \dots, \pi_0^\Lambda\}$7 values using a random $\Pi_0 = \{\pi_0^1, \dots, \pi_0^\Lambda\}$8 sampled at each GRASP iteration:

$\Pi_0 = \{\pi_0^1, \dots, \pi_0^\Lambda\}$9

and

$$\pi_0^\lambda: V^\lambda \to \{1, \dots, |V^\lambda|\}$$0

(Charytitsch et al., 21 Aug 2025). A candidate $$\pi_0^\lambda: V^\lambda \to \{1, \dots, |V^\lambda|\}$$1 is then selected uniformly at random from the RCL (Charytitsch et al., 21 Aug 2025).

Placement is also embedding-informed. For each neighboring layer $$\pi_0^\lambda: V^\lambda \to \{1, \dots, |V^\lambda|\}$$2 in which $$\pi_0^\lambda: V^\lambda \to \{1, \dots, |V^\lambda|\}$$3 has already placed neighbors, GL-GRASP considers the closest available position to the nearest neighbor in $$\pi_0^\lambda: V^\lambda \to \{1, \dots, |V^\lambda|\}$$4 and the closest available position to the farthest neighbor in $$\pi_0^\lambda: V^\lambda \to \{1, \dots, |V^\lambda|\}$$5 (Charytitsch et al., 21 Aug 2025). It also computes an intermediate option as the closest available position to the average of the positions accumulated in the candidate set $$\pi_0^\lambda: V^\lambda \to \{1, \dots, |V^\lambda|\}$$6 (Charytitsch et al., 21 Aug 2025). One of these positions is then selected at random, with feasibility-preserving shifts applied when necessary so that the relative-order and absolute-displacement constraints remain satisfied (Charytitsch et al., 21 Aug 2025).

The construction procedure can therefore be summarized as follows: compute embeddings; project them to 2D; derive Euclidean distances for adjacent vertices; iteratively choose an incremental vertex with small minimum distance to a placed neighbor; place it near either its nearest neighbor, its farthest neighbor, or an average position; then continue until all incremental vertices are inserted (Charytitsch et al., 21 Aug 2025). After construction, the method applies the same local search as baseline GRASP: best-improvement swaps, then best-improvement inserts (Charytitsch et al., 21 Aug 2025).

The paper defines several variants. G_SPEC, G_HOPE, G_N2V, and G_SDNE compute embeddings once per run, while G_N2V* and G_SDNE* recompute embeddings at every GRASP iteration to increase diversification (Charytitsch et al., 21 Aug 2025). The latter are explicitly characterized as stochastic variants (Charytitsch et al., 21 Aug 2025).

5. Evaluation methodology and empirical results

Evaluation is based primarily on the Normalized Primal Integral (NPI), which captures the quality-time trade-off of the solution trajectory (Charytitsch et al., 21 Aug 2025). Let $$\pi_0^\lambda: V^\lambda \to \{1, \dots, |V^\lambda|\}$$7, where $$\pi_0^\lambda: V^\lambda \to \{1, \dots, |V^\lambda|\}$$8 is the best solution value among all GL-GRASP heuristics for the instance, let $$\pi_0^\lambda: V^\lambda \to \{1, \dots, |V^\lambda|\}$$9 be the improving solutions observed over time, and let $IV^\lambda = V^\lambda \cup \hat{V}^\lambda$0 be their timestamps. Then

$IV^\lambda = V^\lambda \cup \hat{V}^\lambda$1

with lower NPI indicating faster convergence to high-quality solutions (Charytitsch et al., 21 Aug 2025).

The experiments use hardware comprising four high-memory nodes, each with two Intel Xeon E5-2667v4 3.2 GHz CPUs and 512 GB memory, for a total of 64 cores and 2 TB RAM (Charytitsch et al., 21 Aug 2025). The benchmark set contains 609 test cases derived from 240 base instances with $IV^\lambda = V^\lambda \cup \hat{V}^\lambda$2, densities $IV^\lambda = V^\lambda \cup \hat{V}^\lambda$3, non-incremental vertices per layer in $IV^\lambda = V^\lambda \cup \hat{V}^\lambda$4, incremental ratios $IV^\lambda = V^\lambda \cup \hat{V}^\lambda$5, and displacement bounds $IV^\lambda = V^\lambda \cup \hat{V}^\lambda$6 (Charytitsch et al., 21 Aug 2025). A second set of newly generated dense instances uses $IV^\lambda = V^\lambda \cup \hat{V}^\lambda$7, $IV^\lambda = V^\lambda \cup \hat{V}^\lambda$8, $IV^\lambda = V^\lambda \cup \hat{V}^\lambda$9, and non-incremental nodes per layer in $IA^\lambda = A^\lambda \cup \hat{A}^\lambda$0 (Charytitsch et al., 21 Aug 2025).

The methods compared are the GL-GRASP variants, literature baselines GRASP2 and GRASP3, and Gurobi 10.0.2 on the benchmark set (Charytitsch et al., 21 Aug 2025). The GRASP parameters are fixed to $IA^\lambda = A^\lambda \cup \hat{A}^\lambda$1 and $IA^\lambda = A^\lambda \cup \hat{A}^\lambda$2, with $IA^\lambda = A^\lambda \cup \hat{A}^\lambda$3 chosen randomly in $IA^\lambda = A^\lambda \cup \hat{A}^\lambda$4 per iteration (Charytitsch et al., 21 Aug 2025).

The main findings are structured across three experiments (Charytitsch et al., 21 Aug 2025). In the comparison among GL-GRASP variants, G_SPEC achieved the lowest mean gap overall, G_SDNE was the fastest on average, and G_HOPE provided a strong balance and was second-best overall when considering both gap and time (Charytitsch et al., 21 Aug 2025). On mean NPI, G_HOPE was best and G_SDNE second-best, while the stochastic variants G_N2V* and G_SDNE* had significantly higher runtimes and worse NPI (Charytitsch et al., 21 Aug 2025). Performance-profile results show G_SDNE and G_HOPE dominating NPI; at $IA^\lambda = A^\lambda \cup \hat{A}^\lambda$5, G_SDNE has the best NPI in about 42% of instances and G_HOPE in about 30% (Charytitsch et al., 21 Aug 2025).

In the comparison with literature baselines, GRASP3 yields the best average gaps on small $IA^\lambda = A^\lambda \cup \hat{A}^\lambda$6 and low-density instances, but at significantly higher runtimes because of the expensive C3 construction (Charytitsch et al., 21 Aug 2025). GL-GRASP variants, especially G_HOPE and G_SDNE, outperform GRASP3 on larger $IA^\lambda = A^\lambda \cup \hat{A}^\lambda$7 and/or higher densities, with much lower runtimes (Charytitsch et al., 21 Aug 2025). GRASP2 is the fastest of the baselines on benchmark instances but has the worst gap on most classes (Charytitsch et al., 21 Aug 2025). Wilcoxon signed-rank tests at $IA^\lambda = A^\lambda \cup \hat{A}^\lambda$8 indicate that GRASP2 is significantly worse in gap than GRASP3, G_HOPE, and G_SDNE; there is no significant difference in gap between GRASP3, G_HOPE, and G_SDNE; and GRASP3 is significantly slower than G_HOPE and G_SDNE (Charytitsch et al., 21 Aug 2025).

On the newly generated denser instances with a time limit of 10,800 seconds, G_HOPE and G_SDNE each attain mean gap $IA^\lambda = A^\lambda \cup \hat{A}^\lambda$9, with mean runtimes of about 8,371 s and 8,074 s respectively (Charytitsch et al., 21 Aug 2025). GRASP2 attains mean gap $I = (IG, \Pi)$0 and mean time $I = (IG, \Pi)$1 s, while GRASP3 attains mean gap $I = (IG, \Pi)$2 and mean time $I = (IG, \Pi)$3 s (Charytitsch et al., 21 Aug 2025). Gurobi attains mean gap $I = (IG, \Pi)$4 within the same time limit (Charytitsch et al., 21 Aug 2025). G_HOPE and G_SDNE each achieve best gaps on 8 of the 12 dense instances (Charytitsch et al., 21 Aug 2025).

6. Interpretation, strengths, and limitations

The paper’s central interpretation is that GRL improves the construction phase by encoding global structure that barycenter-style local averaging does not capture (Charytitsch et al., 21 Aug 2025). Embedding-guided selection and placement near or away from already placed neighbors can produce better initial layouts, accelerate convergence, and reduce crossings early in the search (Charytitsch et al., 21 Aug 2025). This is supported empirically by the fact that GL-GRASP, especially with HOPE and SDNE, matches or exceeds the solution quality of GRASP3 while being significantly faster on many classes of instances (Charytitsch et al., 21 Aug 2025).

Several limitations are also explicit. Stochastic embedding recomputation in G_N2V* and G_SDNE* increases runtime and can degrade NPI, so diversification does not consistently improve the overall quality-time trade-off (Charytitsch et al., 21 Aug 2025). On small-layer, low-density instances, the exhaustive insertion-cost strategy of GRASP3 often remains superior in gap, which indicates that GL-GRASP’s advantage is most pronounced on larger or denser instances rather than universally across all regimes (Charytitsch et al., 21 Aug 2025). The embedding and PCA steps add preprocessing overhead, so in very small graphs that overhead may outweigh the benefit (Charytitsch et al., 21 Aug 2025).

The recommended practical choices are HOPE and SDNE, which offer the strongest balance of quality and runtime (Charytitsch et al., 21 Aug 2025). The paper recommends computing embeddings once per run unless additional diversification is needed and the time budget permits recomputation (Charytitsch et al., 21 Aug 2025). It also recommends the standard GRASP settings $I = (IG, \Pi)$5, $I = (IG, \Pi)$6, random $I = (IG, \Pi)$7, and Euclidean distance in 2D PCA space as the score-defining geometry (Charytitsch et al., 21 Aug 2025).

A plausible implication is that GL-GRASP is particularly effective when the graph’s latent structure aligns with the crossing-minimization geometry sufficiently well for embedding proximity to become a useful surrogate during construction. Where that alignment is weak, more expensive combinatorial scoring such as C3 may remain competitive.

7. Naming, reproducibility, and relation to adjacent work

The 2025 paper explicitly uses “Graph Learning GRASP (GL-GRASP)” to denote the incorporation of Graph Representation Learning into GRASP for C-IGDP (Charytitsch et al., 21 Aug 2025). This naming should not be conflated with other “GRASP”-like acronyms in the graph and machine learning literature. “GraSP: Simple yet Effective Graph Similarity Predictions” addresses graph similarity computation for GED and MCS prediction using RWPE-enhanced graph embeddings and a graph-level interaction module (Zheng et al., 2024). “GRASPEL: Graph Spectral Learning at Scale” is a spectral graph-learning method for estimating ultra-sparse Laplacian graphs from data (Wang et al., 2019). “GRASP: GRAph-Structured Pyramidal Whole Slide Image Representation” is a multi-magnification graph model for whole-slide image classification in digital pathology (Mirabadi et al., 2024). These methods are unrelated to GL-GRASP beyond acronym overlap.

Reproducibility is supported by public code and data links reported in the paper (Charytitsch et al., 21 Aug 2025). All algorithms are implemented in Python, with benchmark instances and newly generated dense instances made available through the project repository (Charytitsch et al., 21 Aug 2025). Embeddings are drawn from CogDL implementations, and Gurobi 10.0.2 is used for exact benchmark experiments (Charytitsch et al., 21 Aug 2025). The paper notes that seeds are not specified; randomization arises from the per-iteration sampling of $I = (IG, \Pi)$8 and from stochastic embedding methods where applicable (Charytitsch et al., 21 Aug 2025).

Within the broader literature, GL-GRASP exemplifies a learning-augmented metaheuristic rather than a replacement of metaheuristics by supervised learning or reinforcement learning (Charytitsch et al., 21 Aug 2025). Its reported contribution is a comparatively inexpensive use of representation learning to extract latent graph structure and inject it into a classical semi-greedy constructor. The empirical record in the paper suggests that this hybridization is especially effective for dense and large-scale C-IGDP instances, where exhaustive construction heuristics become prohibitively costly and simpler baselines lose solution quality (Charytitsch et al., 21 Aug 2025).