SlowGuess/NeurIPS_2023_OCR huggingface dataset:
The proof is given in the Appendix. As  $p_{\theta}(\mathbf{x}_{t - 1}|\mathbf{x}_t,G)$  can be readily obtained from the trained network, the main challenge of estimating  $p_{\theta}(\mathbf{x}_{t - 1}|\mathbf{x}_t,G,y^*)$  lies in estimating  $p(y^{*}|\mathbf{x}_{t - 1},G)$ . Since  $\mathbf{x}_{t - 1}$  is not accessible at step  $t$ , we utilize Taylor expansion to approximate  $\log p(y^{*}|\mathbf{x}_{t - 1},G)$  around  $\mathbf{x}_{t - 1} = \mathbf{x}_t$ , given that  $\mathbf{x}_{t - 1}\sim \mathbf{x}_t$ :
$$
\begin{array}{l} \log p \left(y ^ {*} | \mathbf {x} _ {t - 1}, G\right) \approx \log p \left(y ^ {*} | \mathbf {x} _ {t}, G\right) + \left[ \nabla_ {\mathbf {x} _ {t}} \log p \left(y ^ {*} | \mathbf {x} _ {t}, G\right) \right] ^ {\top} \left(\mathbf {x} _ {t - 1} - \mathbf {x} _ {t}\right) \\ = \left[ \nabla_ {\mathbf {x} _ {t}} \log p \left(y ^ {*} \mid \mathbf {x} _ {t}, G\right) \right] ^ {\top} \mathbf {x} _ {t - 1} + \underbrace {\log p \left(y ^ {*} \mid \mathbf {x} _ {t} , G\right) - \left[ \nabla_ {\mathbf {x} _ {t}} \log p \left(y ^ {*} \mid \mathbf {x} _ {t} , G\right) \right] ^ {\top} \mathbf {x} _ {t}} _ {C (\mathbf {x} _ {t})} \tag {8} \\ \end{array}
$$
where  $C(\mathbf{x}_t)$  is irrelevant to  $\mathbf{x}_{t - 1}$ . Applying exponentiation, we obtain:
$$
p \left(y ^ {*} | \mathbf {x} _ {t - 1}, G\right) \propto \exp \left(\left[ \nabla_ {\mathbf {x} _ {t}} \log p \left(y ^ {*} | \mathbf {x} _ {t}, G\right) \right] ^ {\top} \mathbf {x} _ {t - 1}\right) \tag {9}
$$
To determine  $p(y^{*}|\mathbf{x}_{t},G)$ , we utilize energy-based modeling [58] to model the distribution  $p(y|\mathbf{x}_t,G)$ . In this context, we define the energy function as:
$$
E (y, \mathbf {x} _ {t}, G) = | y - l \left(\mathbf {x} _ {0} \left(\mathbf {x} _ {t}\right); G\right) | \tag {10}
$$
where  $\mathbf{x}_0(\mathbf{x}_t)$  denotes the clean target sample  $\mathbf{x}_0$  corresponding to  $\mathbf{x}_t$ . The energy function quantifies the compatibility between  $y$  and  $(\mathbf{x}_t, G)$ , and it reaches zero when  $y$  is exactly the final objective score originating from  $\mathbf{x}_t$  with respect to  $G$ . Such a design enables the best  $y$  matching the inputs to maintain the highest probability density, and the probability density is positively correlated with the matching degree. Then we employ the Gibbs distribution to characterize the probability distribution over a collection of arbitrary energies:
$$
p (y | \mathbf {x} _ {t}, G) = \frac {\exp (- E (y , \mathbf {x} _ {t} , G))}{\int_ {y ^ {\prime}} \exp (- E (y ^ {\prime} , \mathbf {x} _ {t} , G))} = \frac {\exp (- | y - l (\mathbf {x} _ {0} (\mathbf {x} _ {t}) ; G) |)}{\int_ {y ^ {\prime}} \exp (- | y ^ {\prime} - l (\mathbf {x} _ {0} (\mathbf {x} _ {t}) ; G) |)} \tag {11}
$$
Let  $Z = \int_{y'} \exp(-|y' - l(\mathbf{x}_0(\mathbf{x}_t); G)|)$  and substitute in  $y^*$ , we have:
$$
\log p \left(y ^ {*} \mid \mathbf {x} _ {t}, G\right) = y ^ {*} - l \left(\mathbf {x} _ {0} \left(\mathbf {x} _ {t}\right); G\right) - \log Z \quad \text {a n d} \quad \nabla_ {\mathbf {x} _ {t}} \log p \left(y ^ {*} \mid \mathbf {x} _ {t}, G\right) = - \nabla_ {\mathbf {x} _ {t}} l \left(\mathbf {x} _ {0} \left(\mathbf {x} _ {t}\right); G\right) \tag {12}
$$
Note the gradient operator has removed to effect of  $y^{*}$  as a constant, thereby eliminating the need to access  $y^{*}$  for achieving the target. And  $Z$  as the integral over the entire distribution is not affected by  $\mathbf{x}_t$  and thus can also be removed by the gradient operator. Recall that the network  $\theta$  is trained to predict the logits of  $\tilde{p}_{\theta}(\tilde{\mathbf{x}}_0|\mathbf{x}_t)\in [0,1]^{N\times 2}$  which includes the logits of  $N$  Bernoulli distributions, satisfying  $[\tilde{p}_{\theta}(\tilde{\mathbf{x}}_0|\mathbf{x}_t)]_i = (1 - [\mathbb{E}\tilde{\mathbf{x}}_0]_i,[\mathbb{E}\tilde{\mathbf{x}}_0]_i)$ . Thus  $l(\mathbf{x}_0(\mathbf{x}_t);G)$  can be estimated using  $\mathbb{E}\tilde{\mathbf{x}}_0$  as:
$$
l \left(\mathbf {x} _ {0} \left(\mathbf {x} _ {t}\right); G\right) \approx l \left(\mathbb {E} _ {\tilde {\mathbf {x}} _ {0} \sim \tilde {p} _ {\theta} \left(\tilde {\mathbf {x}} _ {0} \mid \mathbf {x} _ {t}\right)} \tilde {\mathbf {x}} _ {0}; G\right) \tag {13}
$$
By Eq. 9, 12 and 13, we obtain  $p(y^{*}|\mathbf{x}_{t - 1},G)\propto \exp \left(\left[-\nabla_{\mathbf{x}_t}l(\mathbb{E}_{\tilde{\mathbf{x}}_0\sim \tilde{p}_{\theta}(\tilde{\mathbf{x}}_0|\mathbf{x}_t)}\tilde{\mathbf{x}}_0;G)\right]^\top \mathbf{x}_{t - 1}\right)$  and the guided denoising probability estimation can now be achieved by  $p_{\theta}(\mathbf{x}_{t - 1}|\mathbf{x}_t,G,y^*) = Zp_{\theta}(\mathbf{x}_{t - 1}|\mathbf{x}_t,G)p(y^* |\mathbf{x}_{t - 1},G)$  where  $Z$  is the normalizing constant, to realize the guided denoising.
3.3.2 The T2T Framework for Efficient Differentiable Neural Search
With the objective gradient incorporated denoising process in Sec. 3.3.1, we now delve into T2T.
Overview. As shown in Fig. 3, the algorithm starts with an initial solution  $\mathbf{s}_0$  and conducts several iterations to enhance the given solution. Each iteration involves adding a certain degree of noise to disrupt the structure, denoising with objective gradient guidance to obtain a lower-cost soft-constrained solution, and subsequently post-processing to decode a feasible solution  $\mathbf{s} \in \Omega$ . The algorithm eventually reports the solution with the lowest objective score ever achieved.
Structure Disruption. Recall that  $\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_T$  are situated in spaces characterized by progressively escalating levels of disruption. To introduce a controlled degree of disruption to the given solution, we employ  $\mathbf{s}_0\overline{\mathbf{Q}}_{\alpha T}$  to derive the distribution of the disrupted solution  $q(\mathbf{s}_{\alpha T}|\mathbf{s}_0)$  as  $N$  Bernoulli distributions, where  $\alpha$  serves as a hyperparameter to control the degree of noise. Subsequently, the disrupted solution  $\mathbf{s}_{\alpha T}$  can be sampled from  $q(\mathbf{s}_{\alpha T}|\mathbf{s}_0)$ .
Objective Guided Solution Reconstruction. From  $\mathbf{s}_{\alpha T}$ , we employ  $p_{\theta}(\mathbf{s}_{t-1}|\mathbf{s}_t, G, y^*)$  proposed in Sec. 3.3.1 to perform denoising, which aimed at recovering a potentially lower-cost  $\mathbf{s}_0'$ . Additionally,
Figure 3: T2T iteratively adds noise to the solution and performs denoising guided by the problem objective (e.g. TSP), to improve the solution as shown in the orange part. The bottom blue part shows the capability of diffusion to establish a smooth transition from random noises to feasible solutions.  $\mathbf{s}^*$ : optimal solution;  $y^*$ : optimal objective score;  $\theta$ : model parameters;  $G$ : given instance.
we utilize the fast sampling algorithm introduced in DDIM [49] to accelerate sampling. The algorithm determines a  $M$ -element subset from latent variables  $\mathbf{s}_{1:\alpha T}$  to obtain  $\{\mathbf{s}_{\tau_1},\mathbf{s}_{\tau_2},\dots ,\mathbf{s}_{\tau_M}\} (\tau_1 = 1$  and  $\tau_{M} = \alpha T)$ , and directly models  $p_{\theta}(\mathbf{s}_{\tau_{i - 1}}|\mathbf{s}_{\tau_i},G,y^*)$ , which effectively reduces the number of denoising steps. In our implementation, we uniformly perform a 10-step reconstruction, i.e.,  $M = 10$ .
Post-inference Decoding. Note the mechanism guarantees that the recovered solution  $\mathbf{s} \in \{0,1\}^{N \times 2}$  (consisting of  $N$  one-hot vectors), but it does not necessarily satisfy  $\mathbf{s} \in \Omega$ . Thus, in the last step's inference, we directly output the logits of its distribution  $p_{\theta}(\mathbf{s}_0')$  to produce  $\mathbb{E}\mathbf{s}_0'$  as the heatmap  $H \in [0,1]^N$  where each element denotes each edge/node's confidence to be selected, and then we adopt greedy decoding to obtain a feasible solution. We follow previous works [59; 35; 19] to perform greedy decoding by sequentially inserting edges (for TSP) or nodes (for MIS) with the highest confidence if there are no conflicts. For TSP, the 2OPT heuristic [60] is optionally applied.
Tradeoff Between Guidance Accuracy and Local Rewriting Degree. Note that the objective-guided denoising process relies heavily on the accuracy of the estimation  $\tilde{p}_{\theta}(\tilde{\mathbf{x}}_0|\mathbf{x}_t)$ . Intuitively, it is easier to restore the original solution from the data that has lower noise content. And with excessive noise interference, the estimation will deviate significantly from the ground truth, resulting in faulty guidance. This is also one of the reasons why we use a local rewriting mechanism. However, while the accuracy of the estimation increases as it moves closer to the original solution space, it also limits the degree of reconstruction in terms of the solution structure. Therefore, it is crucial to strike a balance between the estimation accuracy and the extent of local rewriting. To accomplish this, we employ the aforementioned hyperparameter  $\alpha$  to control an appropriate range of perturbation.
All the experiments are performed on GPUs of NVIDIA Telsla A100. All the test evaluations are performed on a single GPU, while the trainings are conducted in parallel with four A100 GPUs. Source code is publicly available at https://github.com/Thinklab-SJTU/T2TC0.
We test on two CO problems including TSP and MIS. The comparison includes SOTA learning-based solvers, heuristics, and exact solvers for each problem. For comparison fairness of learning solvers, the initial solutions of our approach are obtained by denoising random noises by Eq. 5 (same solution as the constructive diffusion solver DIFUSCO [19]). To balance the inference time, for experiments with runtime measurement, T2T utilizes a reduced number of sampling steps for constructing the initial solutions, specifically 20 steps. While DIFUSCO, as an important baseline, adopts 120 inference steps for TSP and 50 steps for MIS. The models for both TSP and MIS are trained with 1000 denoising steps, i.e.,  $T = 1000$ . For gradient search, T2T generally involves 3 iterations with 10 guided denoising steps for each. See more details in Appendix.
Table 1: Greedy Decoding on TSP-50 and TSP-100. RL: Reinforcement Learning, SL: Supervised Learning, Grdy: Greedy Decoding. * denotes the baseline for the performance drop.
ALGORITHM	TYPE	TSP-50	TSP-100
LENGTH↓	DROP↓	LENGTH↓	DROP↓
Concorde [62]	Exact	5.69*	0.00%	7.76*	0.00%
2OPT [64]	Heuristics	5.86	2.95%	8.03	3.54%
Farthest Insertion	Heuristics	6.12	7.50%	8.72	12.36%
AM [12]	RL+Grdy	5.80	1.76%	8.12	4.53%
GCN [33]	SL+Grdy	5.87	3.10%	8.41	8.38%
Transformer [65]	RL+Grdy	5.71	0.31%	7.88	1.42%
POMO [6]	RL+Grdy	5.73	0.64%	7.84	1.07%
Sym-NCO [8]	RL+Grdy	-	-	7.84	0.94%
Image Diffusion [59]	SL+Grdy	5.76	1.23%	7.92	2.11%
DIFUSCO [19]	SL+Grdy	5.72	0.48%	7.84	1.01%
T2T (Ours)	SL+Grdy	5.69	0.04%	7.77	0.18%
AM [12]	RL+Grdy+2OPT	5.77	1.41%	8.02	3.32%
GCN [33]	SL+Grdy+2OPT	5.70	0.12%	7.81	0.62%
Transformer [65]	RL+Grdy+2OPT	5.70	0.16%	7.85	1.19%
POMO [6]	RL+Grdy+2OPT	5.73	0.63%	7.82	0.82%
Sym-NCO [8]	RL+Grdy+2OPT	-	-	7.82	0.76%
DIFUSCO [19]	SL+Grdy+2OPT	5.69	0.09%	7.78	0.22%
T2T (Ours)	SL+Grdy+2OPT	5.69	0.02%	7.76	0.06%
4.2 Experiments for TSP
Datasets. A TSP instance includes  $N$  2-D coordinates and a reference solution obtained by heuristics. Training and testing instances are generated via uniformly sampling  $N$  nodes from the unit square  $[0,1]^2$ , which is a standard procedure as adopted in [12; 23; 33; 61; 35; 19]. We experiment on various problem scales including TSP-50, TSP-100, TSP-500, and TSP-1000. The reference solutions for TSP-50/100 are labeled by the Concorde exact solver [62] and the solutions for TSP-500/1000 are labeled by the LKH-3 heuristic solver [63]. The test set for TSP-50/100 is taken from [12; 33] with 1280 instances and the test set for TSP-500/1000 is from [34] with 128 instances for the fair comparison. This section also incorporates results on real-world TSPLIB $^4$  dataset.
Metrics. Following [12; 33; 35; 19], we adopt three evaluation metrics: 1) Length: the average total distance or cost of the solved tours w.r.t. the corresponding instances, as directly corresponds to the objective. 2) Drop: the relative performance drop w.r.t. length compared to the global optimality or the reference solution; 3) Time: the average computational time to solve the problems.
Results for TSP-50/100. Table 1 presents the results. Since DIFUSCO [19] has been shown able to attain global optimality for instances with these scales, at the cost of a sampling decoding scheme, we degrade to the greedy decoding setting for a more meaningful evaluation. The baselines include state-of-the-art learning-based methods with greedy decoding, as well as traditional exact and heuristic solvers. Hyperparameter  $\alpha$  is set as 0.25. Both DIFUSCO and T2T adopt 50 inference steps. As evident from the results, in the setting of pure greedy decoding without 2OPT refinement, the proposed search-based paradigm achieves a substantial reduction in performance drops from  $0.48\%$  to  $0.04\%$  for TSP-50 and from  $1.01\%$  to  $0.18\%$  for TSP-100.
Results for TSP-500/1000. Table 2 presents the results. The baselines include recent learning methods with greedy decoding and sampling decoding  $(\times 4)$ , i.e. producing multiple solutions in parallel and selecting the best one, as well as traditional exact and heuristic solvers. DIFUSCO and T2T are compared in the same conditions. DIFUSCO and T2T adopt 120 and 20 inference steps, respectively. The results of other baselines are quoted from [34] and [35], with the runtimes provided for reference. In the sampling decoding setting, compared to previous neural solvers, T2T improves the performance drop from  $0.83\%$  to  $0.37\%$  for TSP-500 and  $1.30\%$  to  $0.78\%$  for TSP-1000.
Effect of Disruption Ratio  $\alpha$ . As discussed in Sec. 3.3.2, the hyperparameter  $\alpha$  is utilized to regulate the extent of perturbation, striking a balance between guidance accuracy and local rewriting degree. Fig. 4 (a) illustrates the impact of  $\alpha$  on the solving performance. The experiments are performed on TSP-500 with both greedy and sampling decoding settings. The results demonstrate the tradeoff existence and indicate that  $0.3 \leq \alpha \leq 0.4$  can attain relatively better performance.
<sup>4</sup>http://comopt.ifi.uni-heidelberg.de/software/TsPLIB95/
Table 2: Results on TSP-500 and TSP-1000. AS: Active Search, S: Sampling Decoding, BS: Beam Search. * denotes the baseline for computing the performance drop. DIFUSCO and T2T are compared in the same running settings, while other numbers are quoted from [34; 35].
ALGORITHM	TYPE	TSP-500	TSP-1000
LENGTH↓	DROP↓	TIME	LENGTH↓	DROP↓	TIME
Concorde [62]	Exact	16.55*	-	37.66m	23.12*	-	6.65h
Gurobi [66]	Exact	16.55	0.00%	45.63h	-	-	-
LKH-3 (default) [63]	Heuristics	16.55	0.00%	46.28m	23.12	0.00%	2.57h
Farthest Insertion	Heuristics	18.30	10.57%	0s	25.72	11.25%	0s
AM [12]	RL+Grdy	20.02	20.99%	1.51m	31.15	34.75%	3.18m
GCN [33]	SL+Grdy	29.72	79.61%	6.67m	48.62	110.29%	28.52m
POMO+EAS-Emb [17]	RL+AS+Grdy	19.24	16.25%	12.80h	-	-	-
POMO+EAS-Tab [17]	RL+AS+Grdy	24.54	48.22%	11.61h	49.56	114.36%	63.45h
DIMES [35]	RL+Grdy	18.93	14.38%	0.97m	26.58	14.97%	2.08m
DIMES [35]	RL+AS+Grdy	17.81	7.61%	2.10h	24.91	7.74%	4.49h
DIFUSCO [19]	SL+Grdy	18.11	9.41%	5.70m	25.72	11.24%	17.33m
T2T (Ours)	SL+Grdy	17.39	5.09%	4.90m	25.17	8.87%	15.66m
DIMES [35]	RL+Grdy+2OPT	17.65	6.62%	1.01m	24.83	7.38%	2.29m
DIMES [35]	RL+AS+Grdy+2OPT	17.31	4.57%	2.10h	24.33	5.22%	4.49h
DIFUSCO [19]	SL+G+2OPT	16.81	1.55%	5.75m	23.55	1.86%	17.52m
T2T (Ours)	SL+G+2OPT	16.68	0.78%	4.98m	23.41	1.25%	15.90m
EAN [67]	RL+S+2OPT	23.75	43.57%	57.76m	47.73	106.46%	5.39h
AM [12]	RL+BS	19.53	18.03%	21.99m	29.90	29.23%	1.64h
GCN [33]	SL+BS	30.37	83.55%	38.02m	51.26	121.73%	51.67m
DIMES [35]	RL+S	18.84	13.84%	1.06m	26.36	14.01%	2.38m
DIMES [35]	RL+AS+S	17.80	7.55%	2.11h	24.89	7.70%	4.53h
DIFUSCO [19]	SL+S	17.48	5.65%	19.02m	25.11	8.61%	59.18m
T2T (Ours)	SL+S	17.02	2.84%	15.98m	24.72	6.92%	53.92m
DIMES [35]	RL+S+2OPT	17.64	6.56%	1.10m	24.81	7.29%	2.86m
DIMES [35]	RL+AS+S+2OPT	17.29	4.48%	2.11h	24.32	5.17%	4.53h
DIFUSCO [19]	SL+S+2OPT	16.69	0.83%	19.05m	23.42	1.30%	59.53m
T2T (Ours)	SL+S+2OPT	16.61	0.37%	16.03m	23.30	0.78%	54.67m
(a)  $\alpha$ -Drop Curve (Greedy and Sampling)
Figure 4: Effect of  $\alpha$  and initialization steps to the performance drop. For each subgraph, the left represents the greedy decoding, while the right represents the sampling decoding.
(b) InitStep-Drop Curve (Greedy and Sampling)
Effect of Initial Solution Quality. Fig. 4 (b) illustrates the performance variation when altering the initial solution quality. The quality is controlled by the number of denoising steps for producing the initial solution. The experiments are performed on TSP-500 with both greedy and sampling decoding settings. As shown, T2T exhibits a relatively stable performance across varying initial solution qualities, based on which we generally utilize 20 steps for constructing the initial solutions.
Generalization Results on Synthetic Data and TSPLIB. Based on the problem set  $\{\mathrm{TSP - 50},\mathrm{TSP - 100},\mathrm{TSP - 500},\mathrm{TSP - 1000}\}$ , we train the model on a specific problem scale and then evaluate it on all problem scales. Table 4 presents the generalization results of T2T and DIFUSCO with greedy decoding. The results show the satisfying cross-domain generalization ability of T2T, e.g., the model trained on TSP-1000 achieves less than a  $2\%$  optimality gap on all other problem scales. Moreover, we evaluate our model trained with random 100-node problems on real-world TSPLIB instances with 50-200 nodes. The compared baselines include DIFUSCO and baselines listed in [68]'s Table 3. Table 3 shows T2T's superiority achieving merely
0.133% optimality gap, 58.3% better than the best previous method. Detailed results of Table 3 and the results on large-scale TSPLIB instances with 200-1000 nodes in Appendix A. For each instance, we normalize the coordinates to [0,1] and solve with the solving algorithms.
Table 3: Evaluation on TSPLIB dataset.
Method	Drop
AM	16.767%
GCN	40.035%
Learn2OPT	1.725%
GNNGLS	1.529%
DIFUSCO	0.319%
T2T	0.133%
Validation of Gradient Search with Worse Distribution Learning in Training. We investigate the performance over  $\frac{1}{15}$  amount of training data and the model trained with lower-quality feasible solutions produced by farthest insertion heuristic (with  $7.50\%$  gap to optimality). Results in Fig. 5 validate the effectiveness of gradient search even if the distribution is poorly learned, suggesting that T2T
Table 4: Generalization results. Tour length and drop with Greedy Decoding are reported.
Testing\Training	TSP-50	TSP-100	TSP-500	TSP-1000
TSP-50	DIFUSCO T2T	5.69, 0.09%	5.70, 0.25%	5.83, 2.55%	5.84, 2.71%
5.69, 0.02%	5.70, 0.11%	5.78, 1.60%	5.75, 1.10%
TSP-100	DIFUSCO T2T	7.87, 1.44%	7.78, 0.23%	8.03, 3.44%	8.02, 3.31%
7.80, 0.55%	7.77, 0.08%	7.95, 2.47%	7.91, 1.96%
TSP-500	DIFUSCO T2T	17.31, 4.61%	17.05, 3.04%	16.78, 1.40%	16.86, 1.85%
17.18, 3.79%	16.92, 2.25%	16.68, 0.81%	16.72, 1.00%
TSP-1000	DIFUSCO T2T	24.17, 4.54%	24.04, 3.98%	23.65, 2.30%	23.63, 2.21%
24.20, 4.66%	23.85, 3.16%	23.47, 1.51%	23.41, 1.23%
Figure 5: Results on TSP-50 for fewer-shot training with lower-quality supervision.
Table 5: Results on MIS. TS: Tree Search, UL: Unsupervised Learning. DIFUSCO and T2T are compared in the same running settings, while other numbers are quoted from [35; 43].
ALGORITHM	TYPE	SATLIB	ER-[700-800]
SIZE↑	DROP↓	TIME	SIZE↑	DROP↓	TIME
KaMIS [69]	Heuristics	425.96*	-	37.58m	44.87*	-	52.13m
Gurobi [66]	Exact	425.95	0.00%	26.00m	41.28	7.78%	50.00m
Intel [70]	SL+Grdy	420.66	1.48%	23.05m	34.86	22.31%	6.06m
DIMES [35]	RL+Grdy	421.24	1.11%	24.17m	38.24	14.78%	6.12m
DIFUSCO [19]	SL+Grdy	424.56	0.33%	8.25m	36.55	18.53%	8.82m
T2T (Ours)	SL+Grdy	425.02	0.22%	8.12m	39.56	11.83%	8.53m
Intel [70]	SL+TS	-	-	-	38.80	13.43%	20.00m
DGL [71]	SL+TS	-	-	-	37.26	16.96%	22.71m
LwD [72]	RL+S	422.22	0.88%	18.83m	41.17	8.25%	6.33m
GFlowNets [43]	UL+S	423.54	0.57%	23.22m	41.14	8.53%	2.92m
DIFUSCO [19]	SL+S	425.13	0.19%	26.32m	40.35	10.07%	32.98m
T2T (Ours)	SL+S	425.22	0.17%	23.80m	41.37	7.81%	29.73m
stands less sensitive to the quality of labels in training than other peer methods. It further demonstrates the generality of T2T especially when high-quality solution labels are not easily available.
4.3 Experiments for MIS
Datasets. Two datasets are tested for the MIS problem following [70; 72; 71; 35; 19], include SATLIB [73] and Erdős–Rényi (ER) graphs [74]. SATLIB contains SAT instances encoded in DIMACS CNF format, which are then reduced to MIS instances. ER graphs are randomly generated with each edge maintaining a fixed probability of being present or absent, independently of the other edges. We adopt ER graphs of 700 to 800 nodes with the pairwise connection probability set as 0.15.
Metrics. Following previous works [12; 33; 35; 19], we adopt three evaluation metrics to measure model performance: 1) Size: the average size of the solutions w.r.t. the corresponding instances, i.e. the objective. 2) Drop: the relative performance drop w.r.t. length compared to the optimal solution or the reference solution; 3) Time: the average computational time required to solve the problems.
Main Results. Table 5 presents results. The baselines include state-of-the-art neural methods with greedy decoding and sampling decoding  $(\times 4)$ , as well as exact solver Gurobi [66] and heuristic solver KaMIS [69]. The solving time of Gurobi is set as comparable to neural solvers, thus it does not reach optimality. DIFUSCO and T2T are compared in the same conditions. The results of other baselines are quoted from [35]. DIFUSCO and T2T adopt 50 and 20 inference steps, respectively. In the greedy decoding setting, compared to previous neural solvers, T2T improves the performance drop from  $0.33\%$  to  $0.22\%$  for SATLIB dataset and from  $18.53\%$  to  $11.83\%$  for ER-[700-800] dataset.
We have presented a principled learning paradigm for solving hard-constrained optimization problems specifically combinatorial optimization, which bridges the offline training with reference solution as supervision to the instance-wise solving in testing which calls for more direct gradient feedback from the objective score. Experiments on TSP and MIS show the strong performance of our approach. With the potential generality, further validation across a wider range of CO problems is the immediate further work, including those with general representing capabilities such as Mixed Integer Programming (MIP) and ML-dominant special problems like Quadratic Assignment Problem (QAP). Another promising future direction is to explore the potential of recent (scalable) graph Transformers [75; 76; 77] as expressive encoder backbones for solving complex CO problems.
A Supplementary Experiments
Results on TSPLIB. We evaluate our model trained with random 100-node problems on real-world TSPLIB instances with 50-200 nodes. The compared baselines include DIFUSCO and baselines listed in [68]'s Table 3. Table 6 presents the results and shows T2T's superiority achieving merely  $0.133\%$  optimality gap,  $58.3\%$  better than the best previous method. We also supplement the results (optimality drop) of DIFUSCO and T2T on large-scale TSPLIB benchmark instances with 200-1000 nodes, as shown in Table 7. For each instance, we normalize the coordinates to  $[0,1]$  and solve with the solving algorithms.