SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Benchmarks We evaluate DeepACO on eight representative COPs, including the Traveling Salesman Problem (TSP), Capacitated Vehicle Routing Problem (CVRP), Orienteering Problem (OP), Prize Collecting Traveling Salesman Problem (PCTSP), Sequential Ordering Problem (SOP), Single Machine Total Weighted Tardiness Problem (SMTWTP), Resource-Constrained Project Scheduling Problem (RCPSP), and Multiple Knapsack Problem (MKP). They cover routing, assignment, scheduling, and subset COP types. Their definitions and setups are given in Appendix D.
Hardware Unless otherwise stated, we conduct experiments on 48-core Intel(R) Xeon(R) Platinum 8350C CPU and an NVIDIA GeForce RTX 3090 Graphics Card.
5.2 DeepACO as an enhanced ACO algorithm
In this part, we do not apply NLS and set  $W$  in Eq. (4) to 0 for training. It only entails minutes of training to provide substantial neural enhancement as shown in Fig. 2 (also see Appendix C.4). On the other hand, the extra inference time for DeepACO is negligible; for example, it takes less than 0.001 seconds for a TSP100.
DeepACO for fundamental ACO algorithms We implement DeepACO based on three fundamental ACO meta-heuristics: Ant System [25], Elitist Ant System [25], and MAX-MIN Ant System [76]. They are widely recognized as the basis for many SOTA ACO variations [44, 2, 5, 3, 93]. We compare the heuristic measures learned by DeepACO to the expert-designed ones (detailed separately for eight COPs in Appendix D) on 100 held-out test instances for each benchmark COP. The results in Fig. 2 show that DeepACO can consistently outperform its ACO counterparts, verifying the universal neural enhancement DeepACO can bring. We defer the results on more COP scales to Appendix C.1.
DeepACO for advanced ACO algorithms We also apply DeepACO to Adaptive Elitist Ant System (AEAS) [2], a recent ACO algorithm with problem-specific adaptations. In Fig. 3, we plot the evolution curves of AEAS based on the original and learned heuristic measures, respectively, and DeepACO shows clearly better performance. In light of this, we believe that it is promising to exploit DeepACO for designing new ACO SOTAs.
Figure 2: Evolution curves of fundamental DeepACO and ACO algorithms on eight different COPs. For each COP, we plot the best-so-far objective value (averaged over 100 held-out test instances and 3 runs) w.r.t. the number of used evaluations along the ACO iterations.
Figure 3: Evolution curves of DeepACO and ACO based on Adaptive elitist-ant system (AEAS) [2].
DeepACO for different pheromone models The pheromone model used in ACO can vary, even for the same problem [63]. Following the naming convention in [63], we use  $PH_{suc}$  to denote the pheromone model that captures the successively chosen items and  $PH_{items}$  the one that directly captures how good each item is. We extend DeepACO from  $PH_{suc}$  to  $PH_{items}$  using a Transformer [79] encoder equipped with an MLP decoder. We showcase such flexibility of DeepACO on MKP in Fig. 7, where DeepACO using  $PH_{items}$  can still outperform its ACO counterparts. It validates that DeepACO can be readily extended to variations of pheromone models using properly aligned neural models. We defer further discussions to Appendix A and C.5.
DeepACO for better robustness to hyperparameter choice Increasing the robustness of ACO to hyperparameter choices has long been an important research topic [84, 92]. In Fig. 4, we adjust two important hyperparameters, i.e., Alpha that controls the transition probability of solution construction and Decay that controls the pheromone updates, evaluate DeepACO (AS) and ACO (AS) on the TSP100 test dataset, and plot their best objective values within the evolution of 4K evaluations. The results indicate that DeepACO is more robust to the hyperparameter choice given its much lower color variance. We thereby argue that data-driven training can also dispense with high-demanding expert knowledge for hyperparameter tuning.
DeepACO for real-world instances We draw all 49 real-world symmetric TSP instances featuring  $EUC\_2D$  and containing less than 1K nodes from TSPLIB. We infer instances with  $n < 50$  nodes using the model trained on TSP20, those with  $50 \leq n < 200$  nodes using the model trained on TSP100, and the rest using the model trained on TSP500. The evolution curves of DeepACO and ACO are shown in Fig. 5, suggesting that DeepACO can consistently outperform its ACO counterparts even when generalizing across scales and distributions.
Figure 4: Sensitivity of DeepACO and ACO to the hyperparameter choice.
Figure 5: Evolution curves of DeepACO and ACO averaged over TSPLIB instances.
5.3 DeepACO as an NCO method
Comparison on routing problems As shown in Table 1, DeepACO demonstrates competitive performance against the SOTA NCO methods for TSP. Here, we apply the generic 2-opt for NLS  $(T_{NLS} = 10, T_p = 20)$  and set  $W$  in Eq. (4) to 9. Note that most NCO baselines are specialized for TSP or routing problems, while DeepACO is a general-purpose meta-heuristic and validated across eight different COPs. In addition, the results of ablation studies in Table 2 further validate our designs; that is, both neural-guided perturbation and training with LS effectively strengthen DeepACO. More results on TSP100 and CVRP are given in Appendix C.2.
Table 1: Comparison results of DeepACO and NCO baselines. We report per-instance execution time.
Method	TSP500	TSP1000
Obj.	Gap(%)	Time(s)	Obj.	Gap(%)	Time(s)
LKH-3 [39]	16.55	0.00	2.6	23.12	0.00	11
ACO	17.57	6.16	7.3	24.94	7.87	54
AM* [54]	21.46	29.7	0.6	33.55	45.1	1.5
POMO* [56]	20.57	24.4	0.6	32.90	42.3	4.1
GCN+MCTS* [32]	16.97	2.54	50	23.86	3.22	100
DIMES+MCTS [66]	17.01	2.78	11	24.45	5.75	32
DIFUSCO+MCTS* [78]	16.63	0.46	4.7	23.39	1.17	11
SO* [18]	16.94	2.40	15	23.77	2.80	26
Pointerformer [46]	17.14	3.56	1.4	24.80	7.30	4.0
H-TSP* [65]	-	-	-	24.65	6.62	0.3
DeepACO {T=2}	16.94	2.36	2.0	23.99	3.76	6.4
DeepACO {T=10}	16.86	1.84	10	23.85	3.16	32
Results of methods with \* are drawn from [18, 65, 78].
Table 2: Ablation studies of DeepACO. We additionally evaluate DeepACO 1) with vanilla LS and more iterations, 2) implementing random perturbation for LS, and 3) trained with  $W = 0$  for Eq. (4).
DeepACO	TSP500	TSP1000
Obj.	Time(s)	Obj.	Time(s)
w/ vanilla LS	16.90	17	23.89	65
w/ random perturbation	16.92	11	23.90	39
w/ {W = 0} for training	17.05	10	24.23	32
Original	16.86	10	23.85	32
For heatmaps with better exploration We implement the three extended designs introduced in Section 4.4 without LS and compare their evolution curves with vanilla DeepACO in Fig. 6. We apply 4 decoder heads for Multihead DeepACO and set  $K$  to 5 for the top-K entropy loss. The coefficients for the additional loss terms (i.e.,  $\mathcal{L}_{KL},\mathcal{L}_H,$  and  $\mathcal{L}_I$ ) are set to 0.05, 0.05, and 0.02, respectively. The results indicate that these designs lead to better performance on TSP20, 50, and 100 since small-scale COPs typically desire more exploration.
For enhancing heatmap decoding with pheromone update Though we can conduct pure solution samplings similar to [47, 66, 78] given the learned heuristic measures, we demonstrate the superiority of ACO evolution (without NLS) with self-learning pheromone update in Fig. 8. Furthermore, coupling heatmaps with ACO can benefit from extensive ACO research, such as advanced ACO variations, algorithmic hybridizations, and seamless incorporation of local search operators.
Figure 6: Evolution curves of vanilla DeepACO and its three extended implementations on TSP20, 50, and 100. The results are averaged over 3 runs on held-out test datasets each with 100 instances.
Figure 7: Evolution curves of DeepACO and ACO on MKP300 and MKP500 using the pheromone model  $PH_{items}$ .
Figure 8: Solution comparison between using Ant System evolution and pure sampling.
In this work, we propose DeepACO to provide universal neural enhancement for ACO algorithms and automate heuristic designs for future ACO applications. We show that DeepACO performs
consistently better than its ACO counterparts and is on par with the specialized NCO methods. For the Operations Research (OR) community, DeepACO demonstrates a promising avenue toward leveraging deep reinforcement learning for algorithmic enhancement and design automation. For the Machine Learning (ML) community, DeepACO presents a versatile and adaptable NCO framework that can seamlessly integrate with SOTA ACO and EA techniques, including improved rules for solution construction and pheromone update [76, 11], algorithmic hybridizations [67, 91], and incorporation of sophisticated local search operators [3, 93].
However, DeepACO is potentially limited by compressing all learned heuristic information into an  $n \times n$  matrix of heuristic measures. Due to the complexity of COPs and this restricted way of expressing problem patterns, DeepACO may fail to produce close-to-optimal solutions when not incorporating LS components. To address such limitations, we plan to investigate 3D or dynamic heuristic measures in our future work.
The authors appreciate the helpful discussions with Yu Hong, Juntao Li, and anonymous reviewers. Yu Hu, Jiusi Yin, and Tao Yu also contributed to this work. This work was supported by the National Natural Science Foundation of China (NSFC) [grant numbers 61902269, 52074185]; the National Key R&D Program of China [grant number 2018YFA0701700]; the Undergraduate Training Program for Innovation and Entrepreneurship, Soochow University [grant number 202210285001Z, 202310285041Z]; the Priority Academic Program Development of Jiangsu Higher Education Institutions, China; and Provincial Key Laboratory for Computer Information Processing Technology, Soochow University [grant number KJS1938].
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DeepACO: Neural-enhanced Ant Systems for Combinatorial Optimization (Appendix)
A Neural architecture
A.1 GNN
For pheromone model  $PH_{suc}$ , we employ the neural architecture recommended by Joshi et al. [48] and Qiu et al. [66] as the GNN backbone, which relies on an anisotropic message passing scheme and an edge gating mechanism. The GNN we use has 12 layers. For the  $l$ -th layer, we denote the node feature of  $v_i$  as  $\mathbf{h}_i^l$  and the edge feature of  $\text{edge} < i, j >$  as  $\mathbf{e}_{ij}^l$ . The propagated features are defined as
$$
\mathbf {h} _ {i} ^ {l + 1} = \mathbf {h} _ {i} ^ {l} + \alpha (\mathrm {B N} \left(\mathbf {U} ^ {l} \mathbf {h} _ {i} ^ {l} + \mathcal {A} _ {j \in \mathcal {N} _ {i}} \left(\sigma \left(\mathbf {e} _ {i j} ^ {l}\right) \odot \mathbf {V} ^ {l} \mathbf {h} _ {j} ^ {l}\right)\right)), \tag {9}
$$
$$
\mathbf {e} _ {i j} ^ {l + 1} = \mathbf {e} _ {i j} ^ {l} + \alpha (\mathrm {B N} \left(\mathbf {P} ^ {l} \mathbf {e} _ {i j} ^ {l} + \mathbf {Q} ^ {l} \mathbf {h} _ {i} ^ {l} + \mathbf {R} ^ {l} \mathbf {h} _ {j} ^ {l}\right)), \tag {10}
$$
where  $\mathbf{U}^l, \mathbf{V}^l, \mathbf{P}^l, \mathbf{Q}^l, \mathbf{R}^l \in \mathbb{R}^{d \times d}$  are learnable parameters,  $\alpha$  denotes the activation function, BN denotes batch normalization [43],  $\mathcal{A}_{j \in \mathcal{N}_i}$  denotes the aggregation operation over the neighbourhood of  $v_i$ ,  $\sigma$  is the sigmoid function, and  $\odot$  is the Hadamard product. In this paper, we define  $\alpha$  as SiLU [28] and  $\mathcal{A}$  as mean pooling.
We map the extracted edge features into real-valued heuristic measures using a 3-layer MLP with skip connections [38] to the embedded node feature  $\mathbf{h}_i^{12}$ . We apply SiLU [28] as the activation function except for the output layer, where the sigmoid function is utilized to obtain normalized outputs.
A.2 Transformer
For pheromone model  $PH_{items}$ , the architecture mostly follows the transformer encoder (num_hidden_layers = 3, hidden_size = 32, num_attention_heads = 2) but depreciates its positional encoding. On top of it, we add position-wise feedforward layers (num_hidden_layers = 3, hidden_size = 32) that map the hidden representations of each solution component into its real-valued heuristic measure. Overall, it is similar to the neural networks used in [23, 35].
B Details of NLS
In NLS, neural-guided perturbation does not involve a new neural network other than that introduced in Section 4.1. It directly leverages the learned heuristic measures to guide the perturbation process. Specifically, each time a solution reaches local optima, it gets perturbed with local search (LS) which iteratively maximizes the heuristic measures of its solution components. Note that NLS can be implemented using an arbitrary LS operator.
Take TSP as an example. With 2-opt as the LS operator, NLS alternates between (a) optimizing a solution to minimize its tour length and (b) perturbing it to maximize the total heuristic measures of its edges. The two processes are similar, except that we use the inverse of heuristic measures in (b) while the distance matrix in (a) as indicators of good edges (line 6 in Algorithm 1). Our training is customized for NLS. The heuristic learner is trained to minimize the expected TSP length of the NLS-refined sampled solutions.
C Extended results
C.1 DeepACO for more COP scales
We evaluate DeepACO on more COP scales and present the results in Fig. 9. The evolution curves demonstrate that DeepACO can provide consistent neural enhancement across COP scales.
Figure 9: Evolution curves of fundamental DeepACO and ACO algorithms on eight different COPs with more scales. For each COP, we plot the best-so-far objective value (averaged over 100 held-out test instances) w.r.t. the number of used evaluations along the ACO iterations.
Table 3: Comparison results on TSP100.
Method	TSP100
Obj.	Time (s)
AM [54]	7.945	0.36
GCN [47]	7.907	6.13
da Costa et al. [22]	7.821	30.66
Hudson et al. [42]	7.815	10.11
Att-GCRN+MCTS [32]	7.764	0.53
DeepACO (NLS, T=4)	7.767	0.50
DeepACO (NLS, T=10)	7.763	1.23
Table 4: The comparison results on CVRP.
Method	CVRP100	CVRP400	CVRP1000	CVRP2000
Obj.	Time (s)	Obj.	Time (s)	Obj.	Time (s)	Obj.	Time (s)
AM [54]	16.42	0.06	29.33	0.20	61.42	0.59	114.36	1.87
TAM-LKH3 [41]	16.08	0.86	25.93	1.35	46.34	1.82	64.78	5.63
DeepACO (NLS,T=4)	16.07	2.97	25.31	3.65	45.00	10.21	61.89	14.53
DeepACO (NLS,T=10)	15.77	3.87	25.27	5.89	44.82	15.87	61.66	35.94
C.2 TSP and CVRP
Table 3 and 4 gather the comparative experimental results on more diverse routing tasks, i.e., TSP100, CVRP100, 400, 1000, and 2000. They demonstrate DeepACO's consistent and competitive performance. "Time" in the tables refers to the duration needed to solve a single instance, and the results of the baselines are drawn from their respective papers or [41]. NLS for CVRP is based on the local search strategy in the HGS-CVRP algorithm [80, 81].
NLS for CVRP The local search strategy proposed by Vidal [80] (denoted as  $LS_{HGS}$ ) consists of searching among 5 collections of neighborhoods, including SWAP, RELOCATE, 2-opt, 2-opt*, and SWAP* neighborhood. NLS for CVRP follows Algorithm 1 and apply  $LS_{HGS}$  as the local search strategy. In practice, we implement the code provided by Vidal [80] and set  $T_{NLS}$  to 1. To be noted that  $LS_{HGS}$  occasionally produces infeasible solutions, and such solutions are discarded in NLS. We refer the reader to [80] for more details.
C.3 Generalizability of DeepACO
Across COPs The generalizability of DeepACO across COPs is rooted in the generalizability of the ACO meta-heuristic and the feasibility to represent many COPs' solutions using binary variables. In our paper, we evaluated DeepACO across 8 diverse COPs, including routing, assignment, scheduling, and subset COP types. Furthermore, we extend DeepACO to the Bin Packing Problem (BPP), a grouping problem, aiming at optimally splitting items into groups. We follow the experimental setup in [58] and demonstrate the consistent neural enhancement provided by DeepACO in Table 5.
Across scales Table 6 compares the performance of ACO, DeepACO trained on TSP100, and DeepACO trained on the respective test scale, all implementing vanilla LS (instead of NLS to ensure the same execution time for DeepACO and ACO). The results show that DeepACO still outperforms its ACO counterpart even with a significant distributional shift (i.e., from TSP100 to TSP1000).
Table 5: The evolution trend of ACO and DeepACO on the Bin Packing Problem.
T	1	5	10	20	30	40
ACO ↑	0.877	0.896	0.902	0.907	0.909	0.910
DeepACO ↑	0.947	0.952	0.954	0.956	0.957	0.958
Table 6: The generalization performance of DeepACO across TSP scales.
Method	TSP500	TSP1000
ACO (LS)	17.55	24.93
DeepACO (LS, trained on TSP100)	17.18	24.69
DeepACO (LS)	16.98	23.96
C.4 Training duration
As presented in Table 7, DeepACO entails only minutes of training to provide substantial neural enhancement in Fig. 2. It is mainly because our training hybridizes 1) heatmap-based on-policy sampling without costly step-by-step neural decoding [66], and 2) parallel multi-start sampling leveraging the solution symmetries [56, 52].
Table 7: Training settings for DeepACO. "m" denotes minutes.
COP	Problem scale	Total training instances	Total training time
TSP	20 / 100 / 500	640 / 640 / 640	1m / 2m / 8m
CVRP	20 / 100 / 500	640 / 640 / 640	2m / 4m / 18m
OP	100 / 200 / 300	320 / 320 / 320	4m / 6m / 7m
PCTSP	20 / 100 / 500	320 / 320 / 640	1m / 1m / 10m
SOP	20 / 50 / 100	640 / 640 / 640	1m / 2m / 4m
SMTWTP	50 / 100 / 500	640 / 640 / 640	1m / 2m / 11m
RCPSP	30 / 60 / 120	3200 / 3200 / 3200	2m / 4m / 8m
MKP	300 / 500	640 / 640	26m / 30m
C.5 Flexibility for a different pheromone model
The pheromone model used in ACO can vary, even for the same problem [63]. The pheromone model used in this work models the successive relationships of the nodes in a COP graph, which applies to most COPs but is by no means exhaustive. This section aims to show the flexibility of DeepACO for different pheromone models.
Here, we use the Multiple Knapsack Problem (MKP) (formally defined in Appendix D.8) as an example. In Section 5.2, we mostly use the pheromone model  $PH_{suc}$  that models successively chosen items [63], where the pheromone trails are deposited on the edges connecting two items. One alternative, i.e.,  $PH_{items}$ , is to directly model how good each item is, where the pheromone trails are deposited on the items, i.e., the graph nodes. On the one hand,  $PH_{suc}$  is reasonable because the quality of an item must be considered in conjunction with how it is combined with other items. On the other hand,  $PH_{items}$  makes sense because a solution to MKP is order-invariant. Both pheromone models have their own advantages [63] and are used in the recent literature [30, 8].
Isomorphic GNN [48] is applied in this work to  $PH_{suc}$  due to its ability to extract edge features. To apply DeepACO to variations of pheromone models, it is necessary to use a neural model properly aligned with the pheromone model. As a showcase, we use a lightweight Transformer [79] encoder (without positional encoding) with an MLP decoder on top of it to model  $PH_{items}$  for MKP. Under the same experimental setup as in Section 5.1, we compare DeepACO with its ACO counterparts using  $PH_{items}$  for MKP. The results presented in Fig. 7 demonstrate that DeepACO using  $PH_{items}$  can also outperform its ACO counterparts. It validates that DeepACO can be readily extended to variations of pheromone models using properly aligned neural models.
C.6 Visualization of the learned heuristic
We visualize the learned heuristic in Fig. 10. After the graph sparsification (see Appendix D), we scatter the remaining solution components (i.e., the TSP edges in this case) in a unit square by normalizing (in a row-wise fashion) their manually designed and learned heuristic measures. We can observe that the solution components with a large learned heuristic measure do not necessarily
Figure 10: Visualization of the learned heuristic on TSP. We randomly sample a TSP instance  $\pmb{\rho}$  and scatter its learned heuristic measure  $h_{\theta}(\pmb{\rho})$  w.r.t. the manually designed ones, both after normalization. The dashed line represents an identical mapping.
obtain an equivalently large  $\frac{1}{d}$ , and vice versa. In particular, DeepACO learns a set of more aggressive heuristic measures, where most solution components are associated with normalized measures close to zero.
D Benchmark problems
D.1 Traveling Salesman Problem