GFPack++: Attention-Driven Gradient Fields for Optimizing 2D Irregular Packing

Tianyang Xue; Lin Lu; Yang Liu; Mingdong Wu; Hao Dong; Yanbin Zhang; Renmin Han; Baoquan Chen

cs.AIJun 9, 2024

GFPack++: Attention-Driven Gradient Fields for Optimizing 2D Irregular Packing

Authors: Tianyang Xue, Lin Lu, Yang Liu, Mingdong Wu, Hao Dong, Yanbin Zhang, Renmin Han, Baoquan Chen

Organizations: Shandong University, China · Microsoft Research Asia, China · Peking University, China

Abstract

2D irregular packing is a classic combinatorial optimization problem with various applications, such as material utilization and texture atlas generation. Due to its NP-hard nature, conventional numerical approaches typically encounter slow convergence and high computational costs. Previous research (GFPack) introduced a generative method for gradient-based packing, providing early evidence of its feasibility but faced limitations such as insufficient rotation support, poor boundary adaptability, and high overlap ratios. In this paper, we propose GFPack++, a deeply investigated framework that adopts attention-based geometry and relation encoding, enabling more comprehensive modeling of complex packing relationships. We further design a constrained gradient and a weighting function to enhance both the feasibility of the produced solutions and the learning effectiveness. Experimental results on multiple datasets demonstrate that GFPack++ achieves higher space utilization, supports continuous rotation, generalizes well to arbitrary boundaries, and infers orders of magnitude faster than previous approaches. Codes for this paper are at https://github.com/TimHsue/GFPack-pp.

Figures & tables

Figure 1 : The proposed attention-driven diffusion model for 2D irregular packing, GFPack++, supports continuous rotation and accommodating arbitrary boundaries.

Figure 2 : (a) Our approach is a diffusion-based generation method. (b) We use a sequence-to-sequence model to encode and decode the state and velocities at each time step and generate the next state accordingly. Our model consists of two components: an attention-based relationship encoder (c) and an attention-based geometry encoder (d). The former encodes the geometric feature, spatial, and boundary relationships among the input variables using the attention mechanism. The latter computes the geometric features $\varepsilon^{g}$ using a multi-level GCN feature aggregation network and Attention Pooling.

Figure 3 : Local Feature Representation. Our method leverages the information aggregation capability of GCNs combined with the multi-scale abilities of residual connections. It aggregates the geometric features of each point and its neighboring area with Eq. 1 .

Figure 4 : Attention pooling for local features. For the local features of each point, we aggregate them into the shape features of the polygon using an attention mechanism.

Dataset	XAtlas	NFP	SVGnest	GFPack++ ( $b=128$ )	GFPack++ ( $b=512$ )
Garment	62.26 \| 69.53 \| 74.70 \| 1.01 s	61.49 \| 65.49 \| 67.94 \| 6.13 s	69.38 \| 72.75 \| 75.13 \| 80.21 s	69.37 \| 74.22 \| 77.83 \| 3.25 s	74.15 \| 77.04 \| 80.62 \| 10.5 s
Dental	63.18 \| 70.86 \| 73.59 \| 1.25 s	60.42 \| 66.30 \| 68.13 \| 8.22 s	67.22 \| 73.64 \| 76.49 \| 96.73 s	70.57 \| 75.21 \| 78.67 \| 4.28 s	74.06 \| 77.53 \| 79.79 \| 14.5 s
Puzzle (square)	56.16 \| 66.48 \| 76.15 \| 1.21 s	56.90 \| 62.53 \| 68.13 \| 4.52 s	61.00 \| 67.22 \| 73.21 \| 42.55 s	82.22 \| 93.99 \| 98.64 \| 6.23 s	84.80 \| 95.12 \| 98.74 \| 32.1 s
Atlas (building)	50.56 \| 71.21 \| 87.62 \| 0.96 s	41.97 \| 67.32 \| 83.31 \| 2.76 s	44.97 \| 74.51 \| 90.54 \| 52.43 s	58.96 \| 78.78 \| 98.47 \| 8.12 s	66.03 \| 80.16 \| 98.88 \| 25.3 s
Atlas (object)	38.58 \| 60.98 \| 76.09 \| 1.43 s	32.18 \| 57.75 \| 83.12 \| 45.3 s	41.53 \| 63.87 \| 83.12 \| 402.2 s	31.76 \| 65.11 \| 86.08 \| 15.5 s	41.95 \| 67.47 \| 86.81 \| 65.4 s

Table 1 : Statistics of packing ratios (Min|Avg|Max) and time consumption for the average packing results. All of these results were optimized with enhancement algorithm.

Figure 5 : Samples from the teacher datasets.

Figure 6 : Utilization distribution shift for the dental data. The figure plots the histogram of utilization ratios of 1000 random data.

Figure 7 : Visualization of the attention intensity of the 7th-layer decoder at $t=0.2$ using data from the Puzzle (arbitrary) dataset. Attention intensity of each polygon is normalized by $\frac{a_{i}-a_{min}}{a_{max}-a_{min}}$ .

Dataset	Algo.	Util.(%)	Over.(%)	Time(s)
Garment	GFPack	69.82	0.85	81.2
	GFPack (E)	72.17	0.23	124.0
	GFPack++	74.25	0.08	7.9
	GFPack++ (E)	77.04	0.00	10.5
Dental	GFPack	66.59	1.43	80.4
	GFPack++	69.31	0.06	8.0

Table 2 : Comparison of GFPack and GFPack++ on Garment and Dental. ’E’ indicates enhancement of model outputs, ’R’ denotes training with arbitrary rotations. ’Util.’ refers to the average spatial utilization, ’Over.’ is the average overlap of generated outcomes as a percentage of total polygon area, and ’Time’ represents the average generation time.

Geometric Enc.	Relation Enc.	Over.	Util.
GFPack	GFPack	13.2%	-
GFPack++	GFPack	8.38%	-
GFPack	GFPack++	1.89%	72.84%
GFPack++ (AvgPool)	GFPack++	0.85%	73.11%
GFPack++	GFPack++	0.08%	74.25%

Table 3 : Performance comparison of GFPack++ and GFPack using different geometric encoders (first column) and relational encoders (second column) on the dental dataset. The results do not include enhancement.

Figure 8 : Polygon number scalability of GFPack++ (16 to 128).

Batch Size	Valid Solutions	Time
GFPack++ ( $b=128$ )	38.53%	4.21 s
GFPack++ ( $b=512$ )	67.54%	7.95 s

Table 4 : Generalization test on Dental (alphabet).

Batch Size	IoU (Min $\|$ Avg $\|$ Max)	Time
GFPack++ ( $b=128$ )	80.39 % $\|$ 93.75 % $\|$ 96.80 %	3.27 s
GFPack++ ( $b=512$ )	82.43 % $\|$ 94.91 % $\|$ 98.86 %	4.23 s

Table 5 : Generalization test on Puzzle (arbitrary).

Algo.	Building	Object
[ 40 ]	68.3 $\|$ 82.7 $\|$ 98.0	37.7 $\|$ 68.7 $\|$ 86.2
GFPack++	66.0 $\|$ 80.2 $\|$ 98.9	42.0 $\|$ 67.5 $\|$ 86.8
GFPack++*	61.5 $\|$ 78.4 $\|$ 97.4	36.2 $\|$ 66.2 $\|$ 83.9

Table 6 : Comparison of packing ratios (Min—Avg—Max) between GFPack++ and [ 40 ] . GFPack++* means our method trained on the general dataset.

Teacher dataset	Min	Avg	Max
Garment	63.02%	72.39%	78.90%
Dental	67.33%	74.16%	77.36%
Atlas (building)	39.92%	74.55%	95.74%
Atlas (object)	24.69%	63.25%	85.18%
Atlas (general)	40.76%	67.54%	92.41%

Table S1 : Teacher dataset utilization distributions.

Figure S1 : The statistics of polygon vertex counts. The horizontal axis represents the number of vertices in a polygon, while the vertical axis indicates the frequency of these vertex counts. Due to the wide distribution of vertex counts in the Atlas dataset, we have divided it into two parts: the main distribution and the outliers.

Dataset	Min	Avg	Max	Std
Garment	5	81.57	284	76.83
Dental	17	34.47	123	14.50
Puzzle	4	8.25	29	2.86
Atlas (building)	4	6.52	70	3.87
Atlas (object)	4	67.41	1120	102.38

Table S2 : Polygon vertex counts.

Dataset	Polygon Count
Garment	313
Dental	440
Atlas (building)	3262
Atlas (object)	6764

Table S3 : Polygon counts in each dataset.

Dataset	GFPack++		XAtlas		SVGnest
Dataset	Bef.	Aft.	Bef.	Aft.	Bef.	Aft.
Garment	74.25%	77.04%	68.90%	69.53%	70.98%	72.75%
Dental	75.11%	77.53%	69.88%	70.86%	72.55%	73.64%

Table S4 : Average improvements achieved through enhancement.

Figure S2 : Some results from GFPack++ across different datasets. Note that the utilization enhancement only applies to regular boundaries, so small overlaps may be observed in the puzzle (arbitrary) results.

Figure S3 : Comparative results from GFPack++, SVGnest, and XAtlas applied to Garment, Dental, Puzzle, and Atlas datasets. In the lower left corner of the Garment dataset, a shared similar local layout pattern between GFPack++ and SVGnest can be observed. For the Puzzle dataset, GFPack++ outperforms the others, nearly achieving a global optimum due to its continuous rotation solving.

Figure S4 : Comparative results from GFPack++, [ 40 ] , SVGnest, and XAtlas applied to Atlas datasets.

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Open paper AskPDF arXiv

The two-dimensional irregular knapsack problem in a fixed circular container is an important combinatorial optimization problem for maximizing material utilization in manufacturing. Conventional geometric packing solvers can produce tightly packed layouts, yet they often partition the residual space into isolated small pockets that cannot fit valuable unplaced polygons. To overcome this late-stage packing bottleneck, we propose a failure-aware large neighborhood search framework named GeoNest, driven by a graph policy trained via reinforcement learning. Specifically, we first construct neighborhoods by pairing failed target polygons with residual pockets. We then use explanatory poses to identify the placed polygons that block candidate insertions. These diagnosed blocking relations define bounded, fixed-item repair subproblems for the underlying geometric solver. Finally, the graph policy selects the most promising subproblem for execution. For evaluation, we introduce CircleNest-Bench, a benchmark comprising 2,391 load-controlled instances from four contour sources, including a held-out industrial CAD source. Experimental results demonstrate that, under the same total time budget, GeoNest improves mean utilization over a state-of-the-art standalone packing solver by about 0.9% on average across the three main test sets and by about 0.6% on the held-out industrial set.

Dataset	XAtlas	NFP	SVGnest	GFPack++ ( $b=128$ )	GFPack++ ( $b=512$ )
Garment	62.26 \| 69.53 \| 74.70 \| 1.01 s	61.49 \| 65.49 \| 67.94 \| 6.13 s	69.38 \| 72.75 \| 75.13 \| 80.21 s	69.37 \| 74.22 \| 77.83 \| 3.25 s	74.15 \| 77.04 \| 80.62 \| 10.5 s
Dental	63.18 \| 70.86 \| 73.59 \| 1.25 s	60.42 \| 66.30 \| 68.13 \| 8.22 s	67.22 \| 73.64 \| 76.49 \| 96.73 s	70.57 \| 75.21 \| 78.67 \| 4.28 s	74.06 \| 77.53 \| 79.79 \| 14.5 s
Puzzle (square)	56.16 \| 66.48 \| 76.15 \| 1.21 s	56.90 \| 62.53 \| 68.13 \| 4.52 s	61.00 \| 67.22 \| 73.21 \| 42.55 s	82.22 \| 93.99 \| 98.64 \| 6.23 s	84.80 \| 95.12 \| 98.74 \| 32.1 s
Atlas (building)	50.56 \| 71.21 \| 87.62 \| 0.96 s	41.97 \| 67.32 \| 83.31 \| 2.76 s	44.97 \| 74.51 \| 90.54 \| 52.43 s	58.96 \| 78.78 \| 98.47 \| 8.12 s	66.03 \| 80.16 \| 98.88 \| 25.3 s
Atlas (object)	38.58 \| 60.98 \| 76.09 \| 1.43 s	32.18 \| 57.75 \| 83.12 \| 45.3 s	41.53 \| 63.87 \| 83.12 \| 402.2 s	31.76 \| 65.11 \| 86.08 \| 15.5 s	41.95 \| 67.47 \| 86.81 \| 65.4 s

Abstract

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