SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Our model stems from learning to construct partial solutions, making it appropriate for iteratively reconstructing partial solutions from different directions and different starting and destination nodes. It naturally leads to our proposed flexible construction mechanism, Random Re-Construct(RRC). Specifically, being the same as generating the partial solution during training, RRC randomly samples a partial solution from the initial solution and restructures it to obtain a new partial solution. If the new partial solution is superior, it replaces the old one. This iterative process can enhance the solution quality within a stipulated time budget.
In this section, we empirically compare our proposed LEHD model with other learning-based and classical solvers on TSP and CVRP instances with various sizes and distributions.
Problem Setting We follow the standard data generation procedure of the previous work [27] to generate TSP and CVRP datasets. The training sets consist of one million instances for TSP100 and CVRP100, respectively. The TSP test set includes 10,000 instances with 100 nodes and 128 instances for each of 200, 500, and 1000 nodes, respectively. Similarly, the CVRP test set is constructed with the same number of instances. We obtain optimal solutions for the TSP training set with the Concorde solver [2] and obtain optimal solutions for the CVRP training set with HGS [50]. The training and test datasets can be downloaded from Google Drive or Baidu Cloud.
Model Setting For our LEHD model, the embedding dimension is set to 128, and the number of attention layers in the decoder is set to 6. In each attention layer, the head number of MHA is set to 8, and the dimension of the feed-forward layer is set to 512. In all experiments, we train and test our LEHD models using a single NVIDIA GeForce RTX 3090 GPU with 24GB memory.
Training Both the TSP model and CVRP model are trained on one million instances of size 100, respectively. The optimizer is Adam [26] with an initial learning rate of 1e-4. The value of the learning rate decay is set to 0.97 per epoch for the TSP model and 0.9 per epoch for the CVRP model. With a batch size of 1024, we train the TSP model for 150 epochs and the CVRP model for 40 epochs since it converges much faster.
Baselines We compare our method with (1) Classical Solvers: Concorde [2], LKH3 [16], HGS [50], and OR-Tools [41]; (2) Constructive NCO: POMO [29], MDAM [54], EAS [19], SGBS [8], and BQ [12]; (3) Heatmap-based Method: Att-GCN+MCTS [13].
Metrics and Inference We report the optimality gap and inference time for all methods. The optimality gap measures the difference between solutions achieved by different learning and nonlearning-based methods and the optimal solutions obtained using Concorde for TSP and LKH3 for CVRP. Since the classical solvers are run on a single CPU, their reported inference times are not directly comparable to those of the learning-based methods executed on GPU.
For MDAM, POMO, SGBS, and EAS, we directly run their source code with default settings on our test set. For Att-GCN+MCTS, we report their original results for the corresponding setting. For BQ, following the settings described in the original paper, we reproduce and train the BQ model by ourselves and report the obtained results on our test set. For our proposed LEHD model, we report both the results for greedy inference as well as for Random Re-Construct (RRC) with various computational budgets.
6.1 Experimental Results
The main experimental results on uniformly distributed TSP and CVRP instances are reported in Table 1. For TSP, our proposed LEHD method can obtain good greedy inference performance with a very fast inference time on instances of all sizes. With only 100 RRC iterations, LEHD can significantly outperform all other learning-based methods with a reasonable inference time. If we have a larger budget of more than 500 RRC iterations, LEHD can achieve very promising performance for all instances, of which the gap is nearly optimal for TSP100 and TSP200, less than  $0.2\%$  for TSP500, and around  $0.8\%$  for TSP1000.
For CVRP, LEHD can also achieve promising greedy inference performance for all instances. With 300 RRC iterations, LEHD can outperform most learning-based methods on all CVRP instances, except for the EAS method on CVRP100, which requires a much longer inference time. The EAS's performance significantly decreases for solving larger instances such as CVRP500, and we fail to obtain a result for EAS on CVRP1000 with a reasonable budget. SGBS struggles with poor generalization and long inference time for solving large-scale instances. LEHD can also outperform
<sup>3</sup>https://drive.google.com/drive/folders/1LptBUGVxQlCZeWVxmcZUOf9WPlsqOROR?usp=sharing  
<sup>4</sup>https://pan.baidu.com/s/12uxjol_5pAlnm0j4F6D_RQ?pwd=rzja
Table 1: Experimental results on TSP and CVRP with uniformly distributed instances. The results of methods with an asterisk (*) are directly obtained from the original paper.
TSP100	TSP200	TSP500	TSP1000
Concorde	0.000%	34m	0.000%	3m	0.000%	32m	0.000%	7.8h
LKH	0.000%	56m	0.000%	4m	0.000%	32m	0.000%	8.2h
OR-Tools	2.368%	11h	3.618%	17m	4.682%	50m	4.885%	10h
Att-GCN+MCTS*	0.037%	15m	0.884%	2m	2.536%	6m	3.223%	13m
MDAM bs50	0.388%	21m	1.996%	3m	10.065%	11m	20.375%	44m
POMO augx8	0.134%	1m	1.533%	5s	22.187%	1m	40.570%	8m
SGBS	0.060%	40m	0.562%	4m	11.550%	54m	26.035%	7.4h
EAS	0.057%	6h	0.496%	28m	17.08%	7.8h	-	-
BQ greedy	0.579%	0.6m	0.895%	3s	1.834%	0.4m	3.965%	2.4m
BQ bs16	0.046%	11m	0.224%	1m	0.896%	6m	2.605%	38m
LEHD greedy	0.577%	0.4m	0.859%	3s	1.560%	0.3m	3.168%	1.6m
LEHD RRC	50	0.0284%	7.4m	0.123%	0.6m	0.482%	3.4m	1.416%	22m
100	0.0114%	13.7m	0.0761%	1.2m	0.343%	8m	1.218%	43m
300	0.0044%	40m	0.0363%	3.3m	0.223%	22m	0.899%	2.1h
500	0.0025%	1.1h	0.0280%	5.3m	0.193%	37m	0.818%	3.5h
1000	0.0016%	2.2h	0.0182%	10.5m	0.167%	1.2h	0.719%	7h
CVRP100	CVRP200	CVRP500	CVRP1000
LKH3	0.000%	12h	0.000%	2.1h	0.000%	5.5h	0.000%	7.1h
HGS	-0.533%	4.5h	-1.126%	1.4h	-1.794%	4h	-2.162%	5.3h
OR-Tools	6.193%	2h	6.894%	1h	9.112%	2.2h	11.662%	3h
MDAM bs50	2.211%	25m	4.304%	3m	10.498%	12m	27.814%	47m
POMO augx8	0.689%	1m	4.866%	7s	19.901%	1m	128.885%	10m
SGBS	0.079%	40m	2.581%	1m	15.343%	16m	136.980%	2.3h
EAS	-0.234%	15h	0.640%	33m	11.042%	9.3h	-	-
BQ greedy	2.993%	0.7m	3.527%	4s	5.121%	0.4m	9.812%	2.4m
BQ bs16	0.611%	10m	1.141%	0.6m	2.991%	6m	7.784%	39m
LEHD greedy	3.648%	0.5m	3.312%	3s	3.178%	0.3m	4.912%	1.6m
LEHD RRC	50	0.535%	7.2m	0.515%	0.6m	0.930%	8m	2.814%	27m
100	0.272%	17m	0.217%	1.1m	0.546%	14m	2.370%	45m
300	0.029%	52m	-0.146%	3.6m	0.045%	36m	1.582%	2.3h
500	-0.044%	1.4h	-0.246%	6m	-0.107%	1h	1.270%	4h
1000	-0.112%	2.8h	-0.383%	11.3m	-0.347%	2h	0.921%	8h
Table 2: Experimental results on TSPLib and CVRPLib. "#" denotes the number of instances in the corresponding set.
Size	#	POMO aug×8	BQ greedy	bs16	LEHD greedy	RRC
TSPLib	<100	6	0.792%	1.076%	0.505%	0.976%	0.481%
100-200	21	2.423%	2.684%	1.318%	2.336%	0.158%
200-500	15	13.413%	3.177%	2.183%	2.742%	0.200%
500-1k	6	31.678%	8.311%	5.521%	4.049%	1.310%
>1k	22	63.810%	42.566%	36.730%	11.267%	4.088%
All	70	26.439%	15.668%	12.923%	5.260%	1.529%
POMO aug×8	BQ greedy	bs16	LEHD greedy	RRC
CVRPLib	A (31-79)	27	4.970%	6.310%	1.627%	5.871%	0.647%
B (30-77)	23	4.747%	6.859%	2.221%	6.049%	0.812%
E (12-100)	11	11.402%	5.884%	1.211%	4.809%	0.541%
F (44-134)	3	15.973%	12.568%	7.404%	9.051%	3.009%
M (100-199)	5	4.861%	8.407%	3.691%	7.094%	1.817%
P (15-100)	23	15.525%	5.902%	2.393%	6.611%	0.917%
X (100-1k)	100	21.684%	12.526%	9.774%	12.520%	3.511%
All	192	15.450%	9.692%	6.153%	9.465%	2.253%
LKH3 on CVRP100-CVRP500 with a budget of 500 RRC iterations and can achieve an around  $1\%$  gap to LKH3 on CVRP1000 with 1000 RCC iterations. To the best of our knowledge, our method is the first purely learning-based NCO method that can outperform LKH3 on CVRP200 and CVRP500.
Table 2 shows the test results on real-world TSPLib [43] and CVRPLib [45] instances with different sizes and distributions. We report the results with greedy inference and 1000 RRC iterations for our proposed LEHD method. It is clear that LEHD has a good greedy inference performance and can significantly outperform POMO and BQ on all instances with 1000 RRC iterations. These results demonstrate the robust generalization ability of LEHD.
Table 3: Comparison of POMO and LEHD with the same SL training method.
TSP100	TSP200	TSP500	TSP1000
POMO augx8 SL	0.571%	3.970%	20.418%	34.419%
LEHD greedy SL	0.577%	0.859%	1.560%	3.168%
Heavy Decoder vs. Heavy Encoder We compare the performance of models with the Heavy Encoder structure (POMO) and the Heavy Decoder structure (LEHD). We also train POMO with SL on TSP100 and report the result in Table 3. According to the results, POMO trained by SL can also reach a good training performance on TSP100 while it still performs poorly on large-scale problems. Therefore, the promising generalization performance of our proposed method does come from the heavy decoder structure rather than the SL training.
Table 4: Effect of reinforcement learning and supervised learning on training the LEHD model with 10K testing instances for TSP50/100 and 128 testing instances for TSP200/500/1000.
TSP50	TSP100	TSP200	TSP500	TSP1000
RL	5.372%	7.891%	12.581%	25.377%	46.441%
SL	0.375%	0.789%	1.545%	3.500%	6.566%
Supervised Learning vs. Reinforcement Learning In Table 4, we conduct a comparison with reinforcement learning and supervised learning for training our proposed LEHD model. Due to the high computational cost, the reinforcement learning method with the POMO strategy cannot converge in a reasonable time. Therefore, we use each method to train a LEHD model separately on TSP50 with the same computational budget. The results in Table 4 confirm that our proposed LEHD model is more suitable to be trained by the supervised learning method.
Table 5: Comparison of POMO and LEHD with random sampling and RRC methods.
(10k instances) TSP100	(TSP200	(1k instances) TSP500	TSP1000
Concorde	0.000%	34m	0.000%	23m	0.000%	4h	0.000%	61h
POMO augx8	0.134%	1m	1.459%	0.6m	21.948%	8m	40.551%	1.1h
POMO augx8-sample100	0.080%	1.7h	1.609%	1.2h	37.285%	16h	82.015%	117h
POMO augx8-RRC100	0.115%	2m	1.283%	1.6m	11.900%	11m	25.378%	1.2h
POMO augx8-RRC1000	0.075%	12m	0.820%	8m	6.753%	29m	16.261%	2.1h
LEHD greedy	0.577%	0.4m	0.849%	0.2m	1.585%	2.1m	3.089%	13m
LEHD RRC 100	0.0114%	13.7m	0.053%	6.5m	0.357%	58m	1.195%	5.3h
1000	0.0016%	2.2h	0.015%	1h	0.179%	9.5h	0.735%	57h
The Effect of RRC We also conduct comparisons with POMO with random sampling (POMO-Sample100) as well as POMO with RRC (POMO-RRC100) to clearly present the advantages of LEHD and RRC.
The experimental results are shown in Table 5, and we have the following two main observations:
- POMO-RRC > POMO-Sample: The random sampling strategy is inefficient in improving POMO's solution quality, especially for large-scale problems. In contrast, our proposed RRC can efficiently improve POMO's performance at inference time.
- LEHD-RRC > POMO-RRC: Our proposed LEHD model structure provides enough model capacity for learning to construct partial solutions for different sizes. It is crucial for the strong generalization ability to large-scale problems.
6.3 Comparison with Search-based/Improvement-based Methods
Table 6: Comparison with SO-mixed [7]. The results of the SO-mixed method are directly obtained from the original paper.
(128 instances)
TSP200	TSP500	TSP1000
Concorde	0.000%	3m	0.000%	32m	0.000%	7.8h
SO-mixed	0.636%	21m	2.401%	32m	2.800%	56m
LEHD greedy	0.859%	3s	1.560%	0.3m	3.168%	1.6m
LEHD RRC100	0.0761%	1.2m	0.343%	8m	1.218%	43m
Table 7: Comparison with H-TSP [38]. The results of the H-TSP method are directly obtained from the original paper.
TSP1000 (128 instances)
Concorde	0.000%	7.8h
H-TSP with LKH-3	4.06%	7.5m
LEHD greedy	3.168%	1.6m
LEHD RRC100	1.218%	43m
Recently, Cheng et al. [7] and Pan et al. [38] propose two learning-based methods which can tackle large-scale TSP instances. However, these two methods are search-based/improvement-based approaches that require powerful solvers like LKH3 or specifically designed heuristics, and can only tackle TSP but not other VRP variants. In this work, we propose a purely learning-based construction method to tackle the large-scale TSP/VRP instances, and hence mainly compare with other construction-based methods in the original paper.
We conduct a comparison with these methods for a more comprehensive experimental study. The results are reported in Table 6 and Table 7. Our proposed method clearly outperforms those methods in terms of both performance and running time. It highlights the effectiveness of our proposed LEHD+RRC as a purely learning-based method to tackle large-scale problems.
Conclusion In this work, we propose a novel LEHD model for generalizable constructive neural combinatorial optimization. By leveraging the Light Encoder and Heavy Decoder (LEHD) model structure, the LEHD model achieves a strong and robust generalization ability as a purely learning-based method. We also develop a learn to construct partial solution strategy for efficient model training, and a flexible random solution reconstruction mechanism for online solution improvement with customized computational budgets. Extensive experimental comparisons with other representative methods on both synthetic and real-world instances fully demonstrate the promising generalization ability of our proposed LEHD model. Since the generalization ability is one crucial issue for the current constructive NCO solver, we believe LEHD can provide valuable insights and inspire follow-up works to explore the heavy decoder structure for powerful NCO model design.
Limitation and Future Work Although with a strong generalization performance, the current LEHD model can only be properly trained by supervised learning. It could be interesting to develop an efficient reinforcement learning-based method for LEHD model training. Another promising future work is to design a powerful learning-based method for more efficient partial solution reconstruction.
This work was supported in part by the National Natural Science Foundation of China under Grant 62106096, in part by the Shenzhen Technology Plan under Grant JCYJ20220530113013031, in part by the Characteristic Innovation Project of Colleges and Universities in Guangdong Province under Grant 2022KTSCX110, in part by the Research Grants Council of the Hong Kong Special Administrative Region under Grant CityU 11215622, and in part by the 2023 Graduate Innovation Practice Fund Project of Southern University of Science and Technology.
[43] Gerhard Reinelt. TSPLIB-a traveling salesman problem library. ORSA Journal on Computing, 3(4): 376-384, 1991.
[44] Jialin Song, Yisong Yue, Bistra Dilkina, et al. A general large neighborhood search framework for solving integer linear programs. Advances in Neural Information Processing Systems, 33:20012-20023, 2020.
[45] Eduardo Uchoa, Diego Pecin, Artur Pessoa, Marcus Poggi, Thibaut Vidal, and Anand Subramanian. New benchmark instances for the capacitated vehicle routing problem. European Journal of Operational Research, 257(3):845-858, 2017.
[46] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. Advances in neural information processing systems, 30, 2017.
[47] Mel Vecerik, Todd Hester, Jonathan Scholz, Fumin Wang, Olivier Pietquin, Bilal Piot, Nicolas Heess, Thomas Rothörl, Thomas Lampe, and Martin Riedmiller. Leveraging demonstrations for deep reinforcement learning on robotics problems with sparse rewards. arXiv preprint arXiv:1707.08817, 2017.
[48] Matthew Veres and Medhat Moussa. Deep learning for intelligent transportation systems: A survey of emerging trends. IEEE Transactions on Intelligent Transportation Systems, 21(8):3152-3168, 2019.
[49] Natalia Vesselinova, Rebecca Steinert, Daniel F Perez-Ramirez, and Magnus Boman. Learning combinatorial optimization on graphs: A survey with applications to networking. IEEE Access, 8:120388-120416, 2020.
[50] Thibaut Vidal. Hybrid genetic search for the cvrp: Open-source implementation and swap* neighborhood. Computers & Operations Research, 140:105643, 2022.
[51] Oriol Vinyals, Meire Fortunato, and Navdeep Jaitly. Pointer networks. Advances in neural information processing systems, 28, 2015.
[52] Yaoxin Wu, Wen Song, Zhiguang Cao, Jie Zhang, and Andrew Lim. Learning improvement heuristics for solving routing problems. IEEE transactions on neural networks and learning systems, 33(9):5057-5069, 2021.
[53] Liang Xin, Wen Song, Zhiguang Cao, and Jie Zhang. Step-wise deep learning models for solving routing problems. IEEE Transactions on Industrial Informatics, 17(7):4861-4871, 2020.
[54] Liang Xin, Wen Song, Zhiguang Cao, and Jie Zhang. Multi-decoder attention model with embedding glimpse for solving vehicle routing problems. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pages 12042-12049, 2021.
[55] Liang Xin, Wen Song, Zhiguang Cao, and Jie Zhang. Neurolkh: Combining deep learning model with linkernighan-helsgaun heuristic for solving the traveling salesman problem. Advances in Neural Information Processing Systems, 34:7472-7483, 2021.
[56] Shunyu Yao, Xi Lin, Zhenkun Wang, and Qingfu Zhang. Data-efficient supervised learning is powerful for neural combinatorial optimization. 2023. URL https://openreview.net/forum?id=a_yFkJ4-uEK.
A Ablation Study of Normalization
A.1 LEHD Model
In Table 8, we explore the effects of eliminating normalization from the attention layer in our LEHD model. We train three LEHD models with the same training scheme and training budget, differing solely in the attention layer: one with batch normalization (BN), one with instance normalization (IN), and one without normalization (w/o). Our experimental results demonstrate that the LEHD model without normalization in the attention layer can significantly outperform the other two models with normalization.
Table 8: Effect of normalization for LEHD model.
TSP100	TSP200	TSP500	TSP1000
BN	0.775%	1.312%	3.808%	12.209%
IN	0.640%	1.197%	34.391%	222.730%
w/o	0.577%	0.859%	1.560%	3.168%
A.2 POMO Model
We also compare the performance of POMO [29] with different types of normalization: one with batch normalization (BN), one with instance normalization (IN), and one without normalization (w/o) in Table 9. We train all three POMO models with the same reinforcement learning method with POMO strategy and training budget (1,000 epochs). The results show that different types of normalization have little effect on POMO.
Table 9: Effect of normalization for POMO model.
TSP100	TSP200	TSP500	TSP1000
BN	1.325%	5.502%	27.616%	41.631%
IN	1.449%	5.602%	27.454%	41.748%
w/o	1.321%	4.990%	28.598%	45.747%
The results in Table 9 show that removing normalization from the attention layer has little impact on the model with a heavy encoder and a light decoder. However, the results in Table 8 show that removing normalization from the attention layer has a positive impact on the performance of the LEHD model, but it is not the critical factor for the LEHD model's strong generalization ability since the LEHD model with batch normalization still performs significantly better than POMO and SGBS in the case of TSP1000. Instead, we can conclude that the underlying reason for the model's strong generalization ability lies in the heavy decoder structure.
B Efficient Self-Improved Training
LEHD uses SL for training, which raises concerns that the performance of LEHD may be upper bounded by the provided (nearly) optimal solutions and the requirement of optimal solutions for LEHD training may make it hard to solve more complex VRPs. We tackle these two concerns as follows.
B.1 RRC can break the performance upper limit of the labeled solution
Firstly, the performance of our proposed LEHD model with RRC is not limited by the quality of labeled solutions. To demonstrate this point, we train a new LEHD model with labeled solutions generated by OR-Tools of which the quality is far from the optimal solutions (e.g., with a  $6.762\%$  optimal gap). The results are shown in Table 10.
In this experiment, we use OR-Tools to generate suboptimal solutions for 100,000 CVRP100 instances, and use them to train the LEHD model with only 10 epochs (about 1.25 hours). Then we compare their performance on the same test set with 10k instances. According to the results,
Table 10: The LEHD model's performance trained with suboptimal label outputted by OR-Tools.
CVRP100
HGS	greedy	0.000%	4.5h
OR-Tools	6.762%	2h
LEHD	10.428%	0.5m
RRC 20	6.249%	3m
RRC 50	4.811%	7m
RRC 100	3.915%	17m
RRC 300	2.862%	52m
RRC 500	2.525%	1.4h
although LEHD with greedy inference is outperformed by OR-Tools, LEHD+RRC can significantly outperform OR-Tools by a large margin with a reasonable runtime. If more training budget is allowed, the performance of LEHD(+RRC) can be further improved.
This result also demonstrates the importance and effectiveness of RRC in our proposed method.
B.2 LEHD can be trained by RL without any labeled solution
Table 11: The LEHD model's performance trained by RL method on TSP20.
(10k instances)
TSP20	TSP100
Concorde LEHD-RL	0.000%	3m	0%	34m
greedy	0.274%	2s	5.463%	0.4m
RRC100	0.014%	0.6m	1.271%	13m
RRC200	0.006%	1.2m	0.983%	26m
RRC300	0.005%	1.7m	0.865%	40m
RRC500	0.003%	3m	0.740%	1.1h
Secondly, it is possible to directly train the LEHD model using reinforcement learning that does not require any labeled data. Experimental results can be found in Table 11 where we train the LEHD model by RL on the small-scale TSP20 instances, and then test its generalization performance on TSP100. The training budget is 40 epochs each with 100,000 instances on TSP20 (roughly 9 hours). According to the results, trained by RL, LEHD can achieve a nearly optimal solution for TSP20, and RRC makes LEHD have a reasonably good generalization performance on TSP100.
One concern for the purely RL training for LEHD is the high computational cost and slow convergence speed, as analyzed in Table 4 in the main paper. We design an efficient training method to tackle this concern as explained in the next point.
B.3 An efficient self-improved training method for LEHD without any labeled solution
Finally, combining the properties from the above two points, we can design an efficient self-improved training method without any labeled solutions for LEHD that can generalize well to large-scale problems. The three key steps are:
- Train LEHD by reinforcement learning with a reasonable computational budget;
- Use LEHD+RRC to generate good solutions for a set of problem instances;
Further train LEHD by supervised learning with the solutions generated in step 2.
The results of this new training method (LEHD-SI) can be found in Table 12. For the TSP, we first train LEHD by RL and use LEHD+RRC to generate solutions for 200,000 TSP100 instances as labels for supervised learning. The training budget is 40 RL epochs and 185 SL epochs, which costs 2.7 days in total. The entire training process on TSP does not require external solvers to generate labeled solutions.
Table 12: Experimental results on TSP. 'LEHD-SI' means that the LEHD model is trained by the self-improved training method.
(10k instances) TSP100	(1k instances) TSP200	TSP500	TSP1000
Concorde Att-GCN+MCTS*	0.000% 0.037%	34m 2h	0.000% 0.884%	23m 16m	0.000% 2.536%	4h 47m	0.000% 3.223%	61h 1.7h
MDAM bs50 POMO augx8 SGBS EAS BQ greedy BQ bs16	0.388% 0.134% 0.060% 0.057% 0.579% 0.046%	21m 1m 40m 6h 0.6m 11m	1.944% 1.459% 0.516% 0.619% 0.892% 0.221%	23m 0.6m 30m 3.6h 0.4m 6m	9.853% 21.948% 11.398% 20.322% 1.862% 0.945%	1.4h 8m 7.3h 61h 3m 46m	20.306% 40.551% 25.997% -	5.8h 1.1h 58h -
LEHD-SI greedy LEHD-SI RRC	100 200	1.073% 0.053% 0.028%	0.4m 13.7m 27m	1.445% 0.183% 0.129%	0.2m 6.5m 12m	2.583% 0.756% 0.642%	2.1m 58m 2h	4.523% 2.022% 1.721%	13m 5.3h 10.5h
LEHD greedy LEHD RRC	50 100 1000	0.577% 0.0284% 0.0114% 0.0016%	0.4m 7.4m 13.7m 2.2h	0.849% 0.110% 0.053% 0.015%	0.2m 3m 6.5m 1h	1.585% 0.475% 0.357% 0.179%	2.1m 31m 58m 9.5h	3.089% 1.400% 1.195% 0.735%	13m 3.1h 5.3h 57h