SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Table 6: Solution quality for methods trained on random 100-node problems and evaluated on TSPLIB instances with 50-200 nodes. The comparison includes baselines in [68]'s Table 3. Results of DIFUSCO and T2T are based on  $4\times$  sampling decoding with 2OPT post-processing. Results of other baselines are from [68].
INSTANCES	AM	GCN	Learn2OPT	GNNGLS	DIFUSCO	T2T
eil51	16.767%	40.025%	1.725%	1.529%	0.314%	0.314%
berlin52	4.169%	33.225%	0.449%	0.142%	0.000%	0.000%
st70	1.737%	24.785%	0.040%	0.764%	0.172%	0.000%
eil76	1.992%	27.411%	0.096%	0.163%	0.217%	0.163%
pr76	0.816%	27.793%	1.228%	0.039%	0.043%	0.039%
rat99	2.645%	17.633%	0.123%	0.550%	0.016%	0.000%
kroA100	4.017%	28.828%	18.313%	0.728%	0.050%	0.000%
kroB100	5.142%	34.686%	1.119%	0.147%	0.000%	0.000%
kroC100	0.972%	35.506%	0.349%	1.571%	0.000%	0.000%
kroD100	2.717%	38.018%	0.866%	0.572%	0.000%	0.000%
kroE100	1.470%	26.589%	1.832%	1.216%	0.000%	0.000%
rd100	3.407%	50.432%	1.725%	0.003%	0.000%	0.000%
eil101	2.994%	26.701%	0.387%	1.529%	0.124%	0.000%
lin105	1.739%	34.902%	1.867%	0.606%	0.441%	0.393%
pr107	3.933%	80.564%	0.898%	0.439%	0.714%	0.155%
pr124	3.677%	70.146%	10.322%	0.755%	0.997%	0.584%
bier127	5.908%	45.561%	3.044%	1.948%	1.064%	0.718%
ch130	3.182%	39.090%	0.709%	3.519%	0.077%	0.077%
pr136	5.064%	58.673%	0.000%	3.387%	0.182%	0.000%
pr144	7.641%	55.837%	1.526%	3.581%	1.816%	0.000%
ch150	4.584%	49.743%	0.312%	2.113%	0.473%	0.324%
kroA150	3.784%	45.411%	0.724%	2.984%	0.193%	0.193%
kroB150	2.437%	56.745%	0.886%	3.258%	0.366%	0.021%
pr152	7.494%	33.925%	0.029%	3.119%	0.687%	0.687%
u159	7.551%	38.338%	0.054%	1.020%	0.000%	0.000%
rat195	6.893%	24.968%	0.743%	1.666%	0.887%	0.018%
d198	373.020%	62.351%	0.522%	4.772%	0.000%	0.000%
kroA200	7.106%	40.885%	1.441%	2.029%	0.259%	0.000%
kroB200	8.541%	43.643%	2.064%	2.589%	0.171%	0.171%
Mean	16.767%	40.025%	1.725%	1.529%	0.319%	0.133%
Generalization Results on Distribution Shift. The generalization results of the diffusion-based methods (e.g., DIFUSCO and T2T) on the distribution (e.g., cluster, gaussian, etc., instead of uniform) are presented in Table. 8 and Table. 9. The TSP data of different distributions are from [78].
Effect of Number of Iterations. Table. 10 shows the results of T2T with varying #iterations. Each iteration contributes to a single process of solution improvement, and employing more iterations can lead to improved solutions. However, too many iterations could lead to additional computational overhead. As a result, we aim to strike a balance in this trade-off and set #iterations to 3.
Effect of Graph Sparsification Hyperparameter  $k$ . Table. 10 shows the results of T2T with varying  $k$ . A larger value of  $k$  can yield improved results, yet it can also lead to increased computational overhead. Thus, we strike a balance in this trade-off and set  $k$  to 50.
Table 7: Solution quality for methods trained on random 500-node problems and evaluated on TSPLIB instances with 200-1000 nodes. Results of DIFUSCO and T2T are based on  $4 \times$  sampling decoding with 2OPT post-processing.
INSTANCES	DIFUSCO	T2T	INSTANCES	DIFUSCO	T2T
pr439	1.28%	0.26%	d493	0.00%	0.00%
pcb442	0.00%	0.00%	p654	0.00%	0.00%
fl417	0.00%	0.00%	d657	0.00%	0.00%
tsp225	0.99%	0.59%	rat783	1.67%	1.22%
u724	0.00%	0.00%	ts225	0.32%	0.32%
pr299	0.57%	0.14%	lin318	0.30%	0.30%
rd400	0.12%	0.00%	u574	0.00%	0.00%
rat575	0.83%	0.26%	pr1002	2.21%	1.42%
a280	1.11%	0.00%	pr226	0.34%	0.22%
pr264	1.12%	0.48%	Mean	0.57%	0.27%
Table 8: Generalization results from the model trained with Uniform-100 data.
Methods	Decoding	Uniform-100	Gaussian-100
DIFUSCO	Greedy	7.78, 0.20%	5.72, 0.72%
T2T	Greedy	7.76, 0.06%	5.69, 0.28%
DIFUSCO	Sampling	7.76, 0.00%	5.68, 0.09%
T2T	Sampling	7.76, 0.00%	5.68, 0.02%
B Applicability of T2T
We provide a summary concerning the types of problems that T2T can effectively manage:
- Mandatory Requirements:
- The decision variables are limited to a countable and finite set of values. Currently, we focus on binary decision variables and the model predicts a heatmap for the solution. But it can be easily extended to prediction through the multinomial diffusion. Note that even binary variables can cover a wide range of CO problems especially on graphs, since most problems can be attributed to edge-selecting problems like TSP, CVRP, and node-selecting problems like MIS, MAX Clique.
- A problem-specific post-processing procedure is required to transform the prediction into feasible solutions. For problems featuring binary decision variables, this prerequisite can be easily met with the greedy decoding strategy, which sequentially inserts variables (edges or nodes) with the highest confidence if there are no conflicts.
- Nice-to-have Requirements:
- The constraints can be transformed to some sort of penalty with meaningful gradient. While it is preferable to include a constraint penalty term in Eq. 7, the utilization of the optimization objective alone is also effective (as evidenced by the TSP results). This is due to the diffusion's capacity to implicitly learn constraints.
Proof. Since  $p_{\theta}(\mathbf{x}_{t - 1}|\mathbf{x}_t,G,y^*)$  estimates  $q(\mathbf{x}_{t - 1}|\mathbf{x}_t,G,y^*)$ , we aim to show  $q(\mathbf{x}_{t - 1}|\mathbf{x}_t,G,y^*) \propto q(\mathbf{x}_{t - 1}|\mathbf{x}_t,G)p(y^*|\mathbf{x}_{t - 1},G)$ , such that as  $q(\mathbf{x}_{t - 1}|\mathbf{x}_t,G)$  can be readily estimated by  $p_{\theta}(\mathbf{x}_{t - 1}|\mathbf{x}_t,G)$ ,  $q(\mathbf{x}_{t - 1}|\mathbf{x}_t,G,y^*)$ 's neural estimation  $p_{\theta}(\mathbf{x}_{t - 1}|\mathbf{x}_t,G,y^*)$  can be achieved by  $Zp_{\theta}(\mathbf{x}_{t - 1}|\mathbf{x}_t,G)p(y^*|\mathbf{x}_{t - 1},G)$ , where  $Z$  is a normalizing constant.
Table 9: Generalization results from the model trained with Uniform-500 data.
Methods	Decoding	Uniform-500	Rotation-500	Explosion-500	\(Cluster_{3}^{10}\)-300	\(Cluster_{7}^{50}\)-300
DIFUSCO	Greedy	16.82, 1.63%	12.69, 2.44%	12.06, 2.80%	9.72, 2.64%	5.80, 3.09%
T2T	Greedy	16.68, 0.78%	12.51, 1.00%	11.94, 1.76%	9.61, 1.49%	5.75, 2.17%
DIFUSCO	Sampling	16.68, 0.81%	12.60, 1.65%	11.94, 1.78%	9.62, 1.57%	5.75, 2.18%
T2T	Sampling	16,62, 0.42%	12.45, 0.48%	11.87, 1.18%	9.55, 0.85%	5.72, 1.67%
Table 10: TSP-500 Greedy results of the hyperparameter #iterations.
#iterations	0	1	3	5	10
Tour Length↓	16.813	16.705	16.683	16.678	16.669
Drop↓	1.59%	0.94%	0.80%	0.77%	0.72%
We first show that  $p(y^{*}|\mathbf{x}_{t - 1},G)$  does not depend on  $\mathbf{x}_t$ :
$$
\begin{array}{l} p \left(y ^ {*} \mid \mathbf {x} _ {t - 1}, \mathbf {x} _ {t}, G\right) = q \left(\mathbf {x} _ {t} \mid \mathbf {x} _ {t - 1}, y ^ {*}, G\right) \frac {p \left(y ^ {*} \mid \mathbf {x} _ {t - 1} , G\right)}{q \left(\mathbf {x} _ {t} \mid \mathbf {x} _ {t - 1} , G\right)} (14) \\ = q \left(\mathbf {x} _ {t} \mid \mathbf {x} _ {t - 1}\right) \frac {p \left(y ^ {*} \mid \mathbf {x} _ {t - 1} , G\right)}{q \left(\mathbf {x} _ {t} \mid \mathbf {x} _ {t - 1}\right)} (15) \\ = p \left(y ^ {*} \mid \mathbf {x} _ {t - 1}, G\right) (16) \\ \end{array}
$$
Then the probability distribution  $q(\mathbf{x}_{t - 1}|\mathbf{x}_t,G,y^*)$  can be derived by:
$$
\begin{array}{l} q \left(\mathbf {x} _ {t - 1} \mid \mathbf {x} _ {t}, G, y ^ {*}\right) = \frac {q \left(\mathbf {x} _ {t - 1} , \mathbf {x} _ {t} , G , y ^ {*}\right)}{q \left(\mathbf {x} _ {t} , G , y ^ {*}\right)} (17) \\ = \frac {q \left(\mathbf {x} _ {t - 1} , \mathbf {x} _ {t} , G , y ^ {*}\right)}{q \left(y ^ {*} \mid \mathbf {x} _ {t} , G\right) q \left(\mathbf {x} _ {t} , G\right)} (18) \\ = \frac {q \left(y ^ {*} \mid \mathbf {x} _ {t - 1} , \mathbf {x} _ {t} , G\right) q \left(\mathbf {x} _ {t - 1} \mid \mathbf {x} _ {t} , G\right) q \left(\mathbf {x} _ {t} , G\right)}{q \left(y ^ {*} \mid \mathbf {x} _ {t} , G\right) q \left(\mathbf {x} _ {t} , G\right)} (19) \\ = \frac {q \left(y ^ {*} \mid \mathbf {x} _ {t - 1} , G\right) q \left(\mathbf {x} _ {t - 1} \mid \mathbf {x} _ {t} , G\right)}{q \left(y ^ {*} \mid \mathbf {x} _ {t} , G\right)} (20) \\ \end{array}
$$
$q(y^{*}|\mathbf{x}_{t},G)$  is not depend on  $\mathbf{x}_{t - 1}$ , thus it can be regarded as a constant. Now we can obtain  $q(\mathbf{x}_{t - 1}|\mathbf{x}_t,G,y^*)\propto q(\mathbf{x}_{t - 1}|\mathbf{x}_t,G)p(y^* |\mathbf{x}_{t - 1},G)$ , completing the proof.
D Discussion from Another Aspect: Constructive and Improvement Solvers
Existing CO solvers including both traditional and emerging ML-based methods, generally can be categorized into two groups as summarized in detail in Table 12. Constructive solvers solve the problem from scratch in one shot or multiple steps, and the algorithm ends once a complete solution is obtained. However, effectively enhancing the final solution quality (e.g. the objective score) while ensuring the constraints can be challenging for constructive solvers due to the absence of explicit reference to complete solutions (as the constraints are often obeyed globally across the dimensions of the solution). While in contrast, improvement solvers yield and maintain a series of complete solutions during the improvement procedure to optimize (namely minimize) the objective in a score decreasing trend (with fluctuation). Compared with constructive solvers, improvement (search-based) solvers [36; 37; 5; 37; 7; 38] seek an enhanced solution by iteratively performing local search within its neighborhood. The algorithm pipelines generally involve selecting specific regions and locally optimizing sub-solutions. This paradigm often offers better tractability and a more direct correlation to objective score thanks to the explicit involvement of the objective and accessibility of complete solutions. While improvement-based solvers dominate in traditional heuristics (e.g., LKH [60; 63]), for learning class methods, the constructive solvers remain relatively major in number, given their more direct compatibility with network prediction tasks.
For constructive ML-solvers, the training procedures can be classified into reinforcement learning (RL) and supervised learning (SL). RL methods [12; 6; 8] try to directly optimize the objective
Table 11: TSP-500 Greedy results of the graph sparsification hyperparameter  $k$  .
k	10	50	100	200
Tour Length↓	16.687	16.681	16.678	16.674
Drop↓	0.83%	0.79%	0.77%	0.74%
Table 12: Constructive and improvement solvers. Sol. Dist.: solution distribution learning capability; LS: using local search scheme. A constructive solver can be combined with a search-based solver, where the latter is responsible for the search component being not part of the constructive approach.
Neural solver	Paradigm	Training	Sol. Dist.?	LS?
AM [12]	Constructive	Reinforce	No	No
POMO [6]	Constructive	Reinforce	No	No
Sym-NCO [8]	Constructive	Reinforce	No	No
GCN [33]	Constructive	Supervised	No	No
DIMES [35]	Constructive	Reinforce	No	No
NeuRewriter [5]	Improvement	Reinforce	No	Yes
Learning 2OPT [36]	Improvement	Reinforce	No	Yes
Learn2delegate [7]	Improvement	Reinforce	No	Yes
TAM [38]	Improvement	Reinforce	No	Yes
CVAE-Opt [23]	Constructive	Supervised	Yes	No
NeuRouter [28]	Constructive	Supervised	Yes	No
DIFUSCO [19]	Constructive	Supervised	Yes	No
T2T (ours)	Improvement	Supervised	Yes	Yes
function, but since a complete solution is not obtained until the end, the intermediate process lacks a valid reference. SL methods [33; 35; 19] typically fit the network prediction to the reference solutions. The lack of objective awareness and local search procedures limits their ability to surpass the supervised solvers. The training procedures of current improvement ML-solvers are based on RL.
Generative modeling for CO [23; 19] currently falls into a subclass of constructive solvers. Generative modeling for CO currently falls into a subclass of constructive solvers, which leverages the powerful expressive and modeling capabilities of generative models, e.g., VAE [23] and the diffusion model [19]. However, as mentioned above the model lacks awareness of the optimization objective and does not introduce a search procedure to fully utilize the estimated solution distribution. In contrast to the constructive way for CO problem solving, our T2T serves as an improvement solver the strong compatibility between the diffusion model and the diffuse-and-denoise local search paradigm with objective feedback in the loop. In our approach, generative diffusion model and local search are both used (which is the first time to our best knowledge), in a complementary and synergistic way to achieve state-of-the-art performance in both objective score and solving time. Compared to existing improvement solvers, such a design facilitates parallel optimization across multiple sub-regions of the entire solution, greatly improving efficiency while also eliminating the need for the iterative selection of optimization regions.
E Network Architecture Details
E.1 Input Embedding Layer
Given node vector  $x \in \mathbb{R}^{N \times 2}$ , weighted edge vector  $e \in \mathbb{R}^E$ , denoising timestep  $t \in \{\tau_1, \dots, \tau_M\}$ , where  $N$  denotes the number of nodes in the graph, and  $E$  denotes the number of edges, we compute the sinusoidal features of each input element respectively:
$$
\tilde {x} _ {i} = \operatorname {c o n c a t} \left(\tilde {x} _ {i, 0}, \tilde {x} _ {i, 1}\right) \tag {21}
$$
$$
\tilde {x} _ {i, j} = \operatorname {c o n c a t} \left(\sin \frac {x _ {i , j}}{T ^ {\frac {0}{d}}}, \cos \frac {x _ {i , j}}{T ^ {\frac {0}{d}}}, \sin \frac {x _ {i , j}}{T ^ {\frac {2}{d}}}, \cos \frac {x _ {i , j}}{T ^ {\frac {2}{d}}}, \dots , \sin \frac {x _ {i , j}}{T ^ {\frac {d}{d}}}, \cos \frac {x _ {i , j}}{T ^ {\frac {d}{d}}} \right) \tag {22}
$$
$$
\tilde {e} _ {i} = \operatorname {c o n c a t} \left(\sin \frac {e _ {i}}{T _ {d} ^ {0}}, \cos \frac {e _ {i}}{T _ {d} ^ {0}}, \sin \frac {e _ {i}}{T _ {d} ^ {2}}, \cos \frac {e _ {i}}{T _ {d} ^ {2}}, \dots , \sin \frac {e _ {i}}{T _ {d} ^ {\underline {{d}}}}, \cos \frac {e _ {i}}{T _ {d} ^ {\underline {{d}}}}\right) \tag {23}
$$
$$
\tilde {t} = \operatorname {c o n c a t} \left(\sin \frac {t}{T _ {d} ^ {0}}, \cos \frac {t}{T _ {d} ^ {0}}, \sin \frac {t}{T _ {d} ^ {2}}, \cos \frac {t}{T _ {d} ^ {2}}, \dots , \sin \frac {t}{T _ {d} ^ {\frac {d}{d}}}, \cos \frac {t}{T _ {d} ^ {\frac {d}{d}}}\right) \tag {24}
$$
where  $d$  is the embedding dimension,  $T$  is a large number (usually selected as 10000),  $\operatorname{concat}(\cdot)$  denotes concatenation.
Next, we compute the input features of the graph convolution layer:
$$
x _ {i} ^ {0} = W _ {1} ^ {0} \tilde {x} _ {i} \tag {25}
$$
$$
e _ {i} ^ {0} = W _ {2} ^ {0} \tilde {e} _ {i} \tag {26}
$$
$$
t ^ {0} = W _ {4} ^ {0} \left(\operatorname {R e L U} \left(W _ {3} ^ {0} \tilde {t}\right)\right) \tag {27}
$$
where  $t^0\in \mathbb{R}^{d_t}$ ,  $d_{t}$  is the time feature embedding dimension. Specifically, for TSP, the embedding input edge vector  $e$  is a weighted adjacency matrix, which represents the distance between different nodes, and  $e^0$  is computed as above. For MIS, we initialize  $e^0$  to a zero matrix  $0^{E\times d}$ .
E.2 Graph Convolution Layer
Following [33], the cross-layer convolution operation is formulated as:
$$
x _ {i} ^ {l + 1} = x _ {i} ^ {l} + \operatorname {R e L U} \left(\mathrm {B N} \left(W _ {1} ^ {l} x _ {i} ^ {l} + \sum_ {j \sim i} \eta_ {i j} ^ {l} \odot W _ {2} ^ {l} x _ {j} ^ {l}\right)\right) \tag {28}
$$
$$
e _ {i j} ^ {l + 1} = e _ {i} ^ {l} + \operatorname {R e L U} \left(\mathrm {B N} \left(W _ {3} ^ {l} e _ {i j} ^ {l} + W _ {4} ^ {l} x _ {i} ^ {l} + W _ {5} ^ {l} x _ {j} ^ {l}\right)\right) \tag {29}
$$
$$
\eta_ {i j} ^ {l} = \frac {\sigma \left(e _ {i j} ^ {l}\right)}{\sum_ {j ^ {\prime} \sim i} \sigma \left(e _ {i j ^ {\prime}} ^ {l}\right) + \epsilon} \tag {30}
$$
where  $x_{i}^{l}$  and  $e_{ij}^{l}$  denote the node feature vector and edge feature vector at layer  $l$ ,  $W_{1},\dots ,W_{5}\in \mathbb{R}^{h\times h}$  denote the model weights,  $\eta_{ij}^{l}$  denotes the dense attention map. The convolution operation integrates the edge feature to accommodate the significance of edges in routing problems.
For TSP, we aggregate the timestep feature with the edge convolutional feature and reformulate the update for edge features as follows:
$$
e _ {i j} ^ {l + 1} = e _ {i j} ^ {l} + \operatorname {R e L U} \left(\mathrm {B N} \left(W _ {3} ^ {l} e _ {i j} ^ {l} + W _ {4} ^ {l} x _ {i} ^ {l} + W _ {5} ^ {l} x _ {j} ^ {l}\right)\right) + W _ {6} ^ {l} \left(\operatorname {R e L U} \left(t ^ {0}\right)\right) \tag {31}
$$
For MIS, we aggregate the timestep feature with the node convolutional feature and reformulate the update for node features as follows:
$$
x _ {i} ^ {l + 1} = x _ {i} ^ {l} + \operatorname {R e L U} \left(\operatorname {B N} \left(W _ {1} ^ {l} x _ {i} ^ {l} + \sum_ {j \sim i} \eta_ {i j} ^ {l} \odot W _ {2} ^ {l} x _ {j} ^ {l}\right)\right) + W _ {6} ^ {l} \left(\operatorname {R e L U} \left(t ^ {0}\right)\right) \tag {32}
$$
E.3 Output Layer
The prediction of the edge heatmap in TSP and node heatmap in MIS is as follows:
$$
e _ {i, j} = \operatorname {S o f t m a x} (\operatorname {n o r m} \left(\operatorname {R e L U} \left(W _ {e} e _ {i, j} ^ {L}\right)\right)) \tag {33}
$$
$$
x _ {i} = \operatorname {S o f t m a x} (\operatorname {n o r m} \left(\operatorname {R e L U} \left(W _ {n} x _ {i} ^ {L}\right)\right)) \tag {34}
$$
where  $L$  is the number of GCN layers and norm is layer normalization.
E.4 Hyper-parameters
For both TSP and MIS tasks, we construct a 12-layer GCN derived above. We set the node, edge, and timestep embedding dimension  $d = 256$ , 128 for TSP and MIS tasks, respectively.
F Fast Inference for Discrete Diffusion Models
The technique of denoising diffusion implicit models (DDIMs) [49] is introduced for accelerating inference and solution reconstruction. Let  $x_{1:T}$  denotes the latent variables and  $\{\tau_1,\dots ,\tau_M\}$  is one increasing subset of time sequences  $\{1,\ldots ,T\}$ .  $\tau_{M} = T$  for the inference processes and  $\alpha T$  for the reconstruction processes. The diffusion processes can be generated to a class non-Markovian processes with the same training objective, where the inference processes traverse only on a subset of latent variables. The inference processes are reformulated as:
$$
\begin{array}{l} q \left(\mathbf {x} _ {\tau - 1} \mid \mathbf {x} _ {\tau}, \mathbf {x} _ {0}\right) = \frac {q \left(\mathbf {x} _ {\tau} \mid \mathbf {x} _ {\tau - 1} , \mathbf {x} _ {0}\right) q \left(\mathbf {x} _ {\tau - 1} \mid \mathbf {x} _ {0}\right)}{q \left(\mathbf {x} _ {\tau} \mid \mathbf {x} _ {0}\right)} \\ = \operatorname {C a t} \left(\mathbf {x} _ {\tau - 1}; \mathbf {p} = \frac {\mathbf {x} _ {\tau} \mathbf {Q} _ {\tau} ^ {\top} \odot \mathbf {x} _ {0} \overline {{\mathbf {Q}}} _ {\tau - 1}}{\mathbf {x} _ {0} \overline {{\mathbf {Q}}} _ {\tau} \mathbf {x} _ {\tau} ^ {\top}}\right) \tag {35} \\ \end{array}
$$
For both TSP and MIS tasks, we adopt the cosine denosing scheduler such that  $\tau_{i} = \left\lfloor \cos \left(\frac{1 - \pi \cdot c i}{2}\right) \cdot T\right\rfloor$  for some constant  $c$ .
G.1 Training Resource Requirement
Table 13: Details about the training resource requirement of T2T framework. The results are calculated on A100 GPUs. We provide the details about the offline training resource requirement of T2T framework in Table 13.
The results are calculated on A100 GPUs. For comparison reference: AM [12] requires 128M instances generated on the fly for training TSP-100, and it requires 45.8h on 2 1080Ti GPUs; POMO [6] requires 200M instances generated on the fly for training TSP-100, and it requires 1 week on a single Titan RTX; Sym-NCO [8] based on POMO requires 2 weeks on a single A100,
Sym-NCO [8] based on AM requires 3 days on 4 A100 GPUs.
Problem Scale	Dataset Size	1 GPU	2 GPUs	4 GPUs	GPU Mem
TSP-50	1,502 k	56h 24m	30h 49m	19h 8m	13.2 GB
TSP-100	1,502 k	206h 20m	109h 45m	59h 55m	24.0 GB
TSP-500	128 k	64h 26m	34h 37m	22h 46m	19.1 GB
TSP-1000	64 k	123h 31m	65h 46m	38h 35m	35.8 GB
G.2 Hyperparamters
We conduct experiments on TSP and MIS benchmarks with our methods and compare the performance with SOTA learning-based solvers, heuristics, and exact solvers. We select hyperparameters to balance the inference time and the solution quality. Specifically, the inference time of our method is approximate with other learning-based frameworks. For each benchmark, we adopt 20 inference steps and 10 objective-guided denoising steps for reconstruction, which involves 3 iterations and selects the optimal solutions. The disruption ratio  $\alpha$  associated with each benchmark is listed in Table. 14.
Table 14: Disruption Ratio for each benchmark.
Benchmark	TSP-50	TSP-100	TSP-500	TSP-1000	SATLIB	ER-[700-800]
α	0.25	0.25	0.4	0.4	0.1	0.1
G.3 Baseline Settings
G.3.1 TSP Benchmarks
TSP-50/100. For TSP-50 and TSP-100, we compare our proposed T2T with 10 baselines, including one exact solver Concorde [62], two heuristic solvers 2OPT [64] and Farthest Insertion, and 7 learning-based solvers, i.e., AM [12], GCN [33], Transformer [65], POMO [6], Sym-NCO [8], Image Diffusion [59] and DIFUSCO [19]. The post-processing includes greedy sampling and 2OPT refinement. To ensure a fair comparison with similar runtimes, we set the number of inference steps for DIFUSCO to 120.
TSP-500/1000. For TSP-500 and TSP-1000, we compare our method with 2 exact solvers Concorde [62] and Gurobi [66], 2 heuristics solvers LKH-3 [63] and Farthest Insertion, and 6 learning-based methods including EAN [67], AM [12], GCN [33], POMO+EAS [17], DIMES [35] and DIFUSCO [19]. The learning-based methods can further be categorized into supervised learning (SL) and reinforcement learning (RL). The post-processing includes greedy sampling (Grdy, G), multiple sampling (S), 2OPT refinement (2OPT), beam search (BS), active search (AS), and their combinations. To ensure a fair comparison with similar runtimes, we set the number of inference steps for DIFUSCO to 120.
G.3.2 MIS Benchmarks
We evaluate our method on two benchmarks, namely SATLIB and ER-[700-800]. For both benchmarks, we compare T2T with one exact solver Gurobi [66], one heuristic solver KaMIS [69] and 5 learning-based frameworks Intel [70], DGL [70], LwD [72], DIMES [35] and DIFUSCO [19]. The post-processing includes greedy sampling (Grdy) and tree search (TS). Specifically, we set the inference steps 50 for DIFUSCO on both benchmarks.
G.4 Graph Sparsification
In TSP, the number of edges in the graph increases quadratically with the number of nodes, which poses scalability challenges. To address this, we adopt graph sparsification by limiting each node to connect only with its  $k$  nearest neighbors (KNN) based on Euclidean distance. Specifically, we set  $k = 50$  for TSP-500 and TSP-1000.
Fig. 6 visualizes the specific process of T2T with 5 rounds of gradient search on TSP-50.
Figure 6: The visualization of T2T solving TSP-50, as corresponding to the orange part in Fig. 3.