SlowGuess/NeurIPS_2023_OCR huggingface dataset:
RUN-CSP is introduced. The work of [Karalias and Loukas, 2020] proposes Erdős Goes Neural (EGN) in which the goal is to minimize a loss that can be regarded as a certificate of the existence of a CO problem solution whose cost is upper bounded by the loss. By relying on an MFA they can calculate this loss efficiently. Also [Qiu et al., 2022] use an MFA to optimize a distribution over the solution space and optimize the corresponding parameters with gradient estimates that are obtained by REINFORCE. The work of [Min et al., 2022] uses the same MFA-based concept as EGN and focuses on alleviating the oversmoothing problem of GNNs by using modified Graph Convolutional Networks [Kipf and Welling, 2017]. An approach based on an entry-wise concave continuous relaxation of the discrete CO loss and a subsequent rounding procedure for integral solution generation is introduced in [Wang et al., 2022]. Here the concept of [Karalias and Loukas, 2020] is generalized to a wider class of problems and rather simple rounding procedure is introduced to generate solutions efficiently. This approach is further extended by [Wang and Li, 2023] to a meta-learning setting called Meta-EGN in which network parameters are updated for individual CO problem instances. They also argue that RL based CO methods suffer from unstable training. We call this claim into question by finding no stability issues with our RL-based method. Further VAG-CO outperforms EGN and even Meta-EGN despite not updating any parameters on test problem instances. The approach of extending functions on sets to continuous supports is taken in Neural Set Function Extensions (NSFE) [Karalias et al., 2022]. In NSFE the discrete CO loss function is replaced with a convex combination of the discrete loss function values at certain integral solutions. These extensions can be regarded as expectation values that can be calculated in closed form without sampling. Whether the lack of sampling noise in NSFE is beneficial for the optimization procedure is not obvious.
Variational Annealing. The concept of using autoregressive models in the variational problem of approximating Boltzmann distributions of Ising models was introduced by [Wu et al., 2019]. They show that variational annealing (VA), i.e. the combination of the variational approach and temperature annealing, is a highly performant method. The work of [Hibat-Allah et al., 2021] compares VA to other ones like Simulated Annealing (SA) [Kirkpatrick et al., 1983] and confirms its strong performance on problems related to spin glasses. In [Khandoker et al., 2023] the strong performance of VA compared to SA is confirmed on CO problems. A crucial aspect of these works on VA with respect to ours and the ones in the previous paragraph is that they optimize the parameters of their models only for individual problem instances. They do not attempt to learn how to generalize over a family of problems. The work of [Sun et al., 2022] aims at a generalizing application of VA in CO by using an MFA. Our experiments indicate that the simplicity of MFA-based methods leads to performance limitations in particular on hard CO problem instances.
We evaluate VAG-CO on various CO problems that are studied in the recent literature. Additionally, we evaluate VAG-CO on synthetic datasets where solving the corresponding CO problem is known to be particularly hard. Finally, we discuss experiments on the impact of entropy regularization and subgraph tokenization. Our result tables also include inference runtimes. A quantitative runtime comparison is, however, difficult since the runtimes reported from other works and for Gurobi 10.0.0 [Gurobi Optimization, LLC, 2023] were obtained with different setups. See App. A.12 for details on the runtime measurements. For Gurobi we report results for various different runtime limits.
Maximum Independent Set. In the following we will compare VAG-CO on the Maximum Independent Set (MIS) problem to recently published results from [Karalias et al., 2022], where the graph datasets COLLAB, ENZYMES and PROTEINS Morris et al. [2020] are used. In the MIS problem the task is to find the largest subset of independent, i.e. unconnected, nodes in a graph. As an optimization objective for VAG-CO we use Eq. 2 with the Ising energy function for MIS (see Tab. 1). Here, the energy function  $E(\boldsymbol{q})$  consists of two terms  $E_A(\boldsymbol{q})$  and  $E_B(\boldsymbol{q})$ , that depend on the binary representation of the solution  $\boldsymbol{q} = \frac{\sigma + 1}{2}$  with  $\boldsymbol{q} \in \{0,1\}^N$ . When  $q_i = 1$  the corresponding node is defined to be included in the set and it is excluded otherwise. The first term  $E_A(\boldsymbol{q})$  is proportional to the number of nodes that are in the set, whereas  $E_B(\boldsymbol{q})$  is proportional to the number of independence violations, i.e. the number of connected nodes within the set. By selecting  $A, B \in \mathbb{R}_+$  such that  $A < B$  we ensure that all minima satisfy  $E_B = 0$  since in this case excluding a violating node from the set always reduces the energy. In our experiments, we choose  $A = 1.0$  and  $B = 1.1$ .
We follow the procedure of [Karalias et al., 2022] and use a  $0.6 / 0.1 / 0.3$  train/val/test split on the aforementioned datasets and use only the first 1000 graphs on the COLLAB dataset. The results
Problem Type	Ising formulation: minqE(q) where E(q) = EA(q) + EB(q)
MVC	E(q) = A ∑iN qi + B, ∑(i,j)∈ε(1 - qi) · (1 - qj)
MIS	E(q) = -A ∑iN qi + B ∑(i,j)∈ε qi · qj
MaxCl	E(q) = -A ∑iN qi + B ∑(i,j)∉ε qi · qj
MaxCut	E(σ) = -∑(i,j)∈ε 1-σiσj/2 where σi = 2 qi - 1
Table 1: Ising formulations of the MVC, MIS, MaxCl and MaxCut problem ([Lucas, 2014]). The term that includes the constant  $A$  corresponds to  $E_{A}(q)$  and the term that includes the constant  $B$  corresponds to  $E_{B}(q)$ .
ENZYMES	PROTEINS	IMDB-BINARY	COLLAB	MUTAG
Evaluation metric A.1	AR* (s/graph )	AR* (s/graph )	AR* (s/graph )	AR* (s/graph )	AR* (s/graph )
EGN (r)	0.821 ± 0.124 (N/A)	0.903 ± 0.114 (N/A)	0.515 ± 0.310 (N/A)	0.886 ± 0.198 (N/A)	0.939 ± 0.069 (N/A)
NSFE (r)	0.775 ± 0.155 (N/A)	0.729 ± 0.205 (N/A)	0.679 ± 0.287 (N/A)	0.392 ± 0.253 (N/A)	0.854 ± 0.132 (N/A)
REINFORCE (r)	0.751 ± 0.301 (N/A)	0.725 ± 0.285 (N/A)	0.881 ± 0.240 (N/A)	1.000 (N/A)	0.781 ± 0.316 (N/A)
Straight-through (r)	0.725 ± 0.268 (N/A)	0.722 ± 0.26 (N/A)	0.917 ± 0.253 (N/A)	0.856 ± 0.221 (N/A)	0.965 ± 0.162 (N/A)
DB-Greedy	0.9810 ± 0.0009 (0.008)	0.9848 ± 0.0003 (0.008)	1.000 (0.007)	0.9968 ± 0.0001 (0.033)	0.994 ± 0.001 (0.005)
MFA: CE	0.9875 ± 0.001 (0.036)	0.9883 ± 0.0007 (0.037)	1.000 (0.022)	0.9985 ± 0.0004 (0.054)	1.000 (0.019)
MFA-Anneal: CE	0.9880 ± 0.0014 (0.036)	0.9881 ± 0.0007 (0.037)	1.000 (0.022)	0.9991 ± 0.0004 (0.054)	1.000 (0.019)
VAG-CO (ours)	0.9960 ± 0.0007 (0.026)	0.9977 ± 0.0005 (0.032)	1.000 (0.017)	0.9988 ± 0.0002 (0.064)	1.000 (0.015)
Gurobi (tmax=0.01)	1.000 (0.001)	0.9996 ± 0.0001 (0.001)	1.000 (0.0003)	0.997 ± 0.001 (0.002)	1.000 (0.0001)
Gurobi (tmax=0.1)	1.000 (0.001)	1.000 (0.001)	1.000 (0.0003)	1.000 (0.002)	1.000 (0.0001)
Evaluation metric A.1	AR (s/graph )	AR (s/graph )	AR (s/graph )	AR (s/graph )	AR (s/graph )
MFA	0.9635 ± 0.0014 (0.0029)	0.9705 ± 0.0012 (0.0025)	0.990 ± 0.002 (0.0028)	0.960 ± 0.007 (0.0024)	0.970 ± 0.003 (0.0028)
MFA-Anneal	0.968 ± 0.001 (0.0029)	0.973 ± 0.002 (0.0025)	0.994 ± 0.001 (0.0028)	0.917 ± 0.045 (0.0024)	0.975 ± 0.002 (0.0028)
VAG-CO (ours)	0.9863 ± 0.0007 (0.026)	0.9908 ± 0.0005 (0.032)	0.995 ± 0.001 (0.017)	0.993 ± 0.001 (0.064)	0.994 ± 0.002 (0.015)
Table 2: Results on the MIS Problem (see Sec. 1). We show test set results on the best approximation ratio  $AR^*$  and the average approximation ratio  $\widehat{AR}$  across different methods and datasets. Values that are closer to one are better. (r) indicates that these are results as reported in [Karalias et al., 2022]. The time for each algorithm is reported in round brackets as seconds per graph.
are shown in Tab. 2 where the test set average of the best approximation ratio  $\mathrm{AR}^*$  out of  $n_S = 8$  sampled solutions per graph is reported (see App. A.1). This metric was originally proposed in [Karalias and Loukas, 2020]. We also report results of our own implementation of an MFA-based method that is trained with REINFORCE. This method is used with (MFA-Anneal) and without (MFA) annealing (App. A.5). As in Wang and Li [2023] and Karalias and Loukas [2020] we use the conditional expectation procedure (CE, App. A.5.5) to sample solutions and report the corresponding results as (MFA: CE) and (MFA-Anneal: CE). We also add results obtained by the Degree Based Greedy (DB-Greedy) algorithm [Wormald, 1995] that is proposed by [Angelini and Ricci-Tersenghi, 2023] as a baseline for machine learning algorithms on MIS. For VAG-CO we greedily sample different states for a given problem instance by using different BSF orderings of the corresponding graph nodes. Our results show that VAG-CO significantly outperforms all competitors including the MFA on ENZYMES and PROTEINS. On IMDB-BINARY and MUTAG the MFA-based approaches and VAG-CO outperform all other machine learning methods and achieve an optimal  $\mathrm{AR}^*$ . On COLLAB MFA-Anneal and VAG-CO exhibit the best results with insignificantly better results for MFA-Anneal. We also report results based on the test set average of the approximation ratio with  $n_S = 30\widehat{\mathrm{AR}}$  (App. A.1). This metric shows the performance of the learned probability distribution for each model, when no post processing is applied. Our results show that VAG-CO always achieves a significantly better performance in terms of  $\widehat{\mathrm{AR}}$  than MFA-based approaches.
Minimum Vertex Cover. We compare VAG-CO to results from Wang and Li [2023] where the Minimum Vertex Cover (MVC) problem is solved on the TWITTER [Leskovec and Krevl, 2014], COLLAB and IMDB-BINARY Morris et al. [2020] graph datasets. The MVC problem is the task of finding the smallest subset of vertices such that each edge has at least one node in the subset. As for the MIS problem we formulate this CO problem in terms using the Ising energy function that is defined in Tab. 1 and set  $A = 1.0$  and  $B = 1.1$ . We follow the procedure of Wang and Li [2023] and use a  $0.7 / 0.1 / 0.2$  train/val/test split on these datasets. Their method Meta-EGN uses meta
learning in combination with the EGN method. They apply CE to generate solutions and report the best out of  $n_S = 8$  sampled solutions. Additionally, they report fine tuning results denoted by (f-t), where they apply one step of fine tuning individually for each CO problem instance in the test dataset. Our results in Tab. 3 show that VAG-CO improves upon the results reported in Wang and Li [2023] across all datasets. We also show that the MFA with and without annealing performs remarkably well on the TWITTER and COLLAB dataset and even outperforms VAG-CO on the COLLAB dataset. The results in terms of  $\widehat{\mathrm{AR}}$  show that the strong performance of the MFA can be attributed to the post-processing method CE rather than to the learned MFA-based model.
Maximum Clique. The Maximum Clique (MaxCl) Problem is the problem of finding the largest set of nodes, in which each node is connected to all other nodes that are contained in the set. The MaxCl problem is equivalent to solving the MIS problem on the complementary graph. The complementary graph can be obtained by connecting nodes that are not connected in the original graph and by disconnecting nodes that are connected. We evaluate our method on the ENZYMES and IMDB-BINARY datasets with the same train/val/test splits
that are used in the MIS problem (see Sec. 6). Results are shown in Tab. 4, where we compare to results that are reported in [Karalias et al., 2022] and to our own implementation of MFA and MFA-Anneal. Here, we see that VAG-CO is always among the best performing method in therms of  $AR^{*}$  and always the single best method in therms of  $\widehat{AR}$ .
Maximum Cut. The Maximum Cut (MaxCut) problem is the problem of finding two sets of nodes so that the number of edges between these two sets is as high as possible. We evaluate our method on BA graphs [Barabási and Albert, 1999], where we follow [Zhang et al., 2023] and generate 4000/500/500 train/val/test graphs between the size of 200-300 nodes and set the generation parameter  $m$  to 4. Results on this dataset are shown in Tab. 4, where we compare our method to MaxCut Values  $(MCut = \sum_{(i,j) \in \mathcal{E}} \frac{1 - \sigma_i \sigma_j}{2})$  that are reported in [Zhang et al., 2023]. Our method achieves the best  $MCut$  results among all learned methods.
TWITTER	COLLAB	IMDB-BINARY
Evaluation metric A.1	AR*(s/graph)	AR*(s/graph)	AR*(s/graph)
EGN (r)	1.033 ± 0.023 (0.29)	1.013 ± 0.022 (0.15)	1.000 (0.08)
EGN f-t (r)	1.028 ± 0.021 (0.80)	1.008 ± 0.015 (0.38)	1.000 (0.32)
RUN-CSP (r)	1.180 ± 0.435 (0.16)	1.208 ± 0.198 (0.19)	1.188 ± 0.178 (0.08)
Meta-EGN (r)	1.019 ± 0.017 (0.29)	1.003 ± 0.010 (0.15)	1.000 (0.08)
Meta-EGN f-t (r)	1.017 ± 0.017 (0.80)	1.002 ± 0.010 (0.38)	1.000 (0.32)
Greedy (r)	1.014 ± 0.014 (1.95)	1.209 ± 0.198 (1.79)	1.180 ± 0.077 (0.02)
MFA: CE	1.0064 ± 0.0004 (0.08)	1.00021 ± 0.00003 (0.05)	1.000 (0.02)
MFA-Anneal: CE	1.0041 ± 0.0001 (0.08)	1.00023 ± 0.00004 (0.05)	1.000 (0.02)
VAG-CO (ours)	1.0024 ± 0.00006 (0.15)	1.0017 ± 0.0002 (0.16)	1.000 (0.016)
Gurobi (tmax=0.01)	1.054 ± 0.001 (0.008)	1.0002 ± 0.0001 (0.002)	1.000 (0.0003)
Gurobi (tmax=0.1)	1.0015 ± 0.00001 (0.025)	1.000 (0.004)	1.000 (0.0003)
Gurobi (tmax=0.2)	1.0001 ± 0.0001 (0.028)	1.000 (0.004)	1.000 (0.0003)
Evaluation metric A.1	AR(s/graph)	AR(s/graph)	AR(s/graph)
MFA:	1.0106 ± 0.0002 (0.004)	1.00106 ± 0.00001 (0.003)	1.0014 ± 0.0004 (0.003)
MFA-Anneal:	1.0090 ± 0.0004 (0.004)	1.0048 ± 0.0002 (0.003)	1.0011 ± 0.0002 (0.003)
VAG-CO (ours)	1.0079 ± 0.0003 (0.15)	1.0056 ± 0.0002 (0.16)	1.0011 ± 0.0006 (0.016)
Table 3: Results on the MVC problem (see Sec. 1). We show test set results on the best approximation ratio  $AR^*$  and the average approximation ratio  $\widehat{AR}$  across different methods and datasets. Values that are closer to one are better. (r) indicates that these results are reported in [Wang and Li, 2023].
Figure 1: Left: Comparison of  $AR^{*}$  for the MIS problem on the RRG-100 dataset with different degrees  $d$ . Middle: Results for the MVC problem in terms of  $AR^{*}$  on the RB-200 dataset at different generation parameters  $p$ . Left, Middle: Values closer to one are better. The runtime  $t$  is given in s/graph and  $t_{\mathrm{max}}$  is the time limit for Gurobi. Right: Ablation on annealing and subgraph tokenization (ST). The  $\hat{\epsilon}_{\mathrm{rel}}$  on the RB-100 MVC is plotted over the number of gradient update steps. Lower values are better.
ENZYMES	IMDB-BINARY
Evaluation metric A.1	AR* (s/graph)	AR* (s/graph)
EGN (r)	0.883 ± 0.156 (N/A)	0.936 ± 0.175 (N/A)
NSFE (r)	0.933 ± 0.148 (N/A)	0.961 ± 0.143 (N/A)
REINFORCE (r)	0.751 ± 0.301 (N/A)	0.881 ± 0.240 (N/A)
Straight-through (r)	0.725 ± 0.268 (N/A)	0.917 ± 0.253 (N/A)
DB-Greedy	0.9485 ± 0.0034 (0.025)	0.9875 ± 0.0008 (0.01)
MFA: CE	0.9766 ± 0.001 (0.04)	0.999 ± 0.009 (0.025)
MFA-Anneal: CE	0.9921 ± 0.0017 (0.04)	0.999 ± 0.0001 (0.025)
VAG-CO (ours)	0.987 ± 0.004 (0.029)	0.9981 ± 0.0013 (0.017)
Gurobi (tmax=0.01)	0.983 ± 0.002 (0.004)	0.996 ± 0.0002 (0.001)
Gurobi (tmax=0.1)	1.000 (0.005)	1.000 (0.002)
Evaluation metric A.1	AR (time in [s/graph])	AR (time in [s/graph])
MFA	0.764 ± 0.013 (0.001)	0.983 ± 0.009 (0.001)
MFA-Anneal	0.806 ± 0.004 (0.001)	0.974 ± 0.004 (0.001)
VAG-CO (ours)	0.943 ± 0.012 (0.029)	0.993 ± 0.0013 (0.017)
BA 200 - 300
MCut (s/graph)
Gurobi (r)	732.47 (1.57)
SDP (r)	700.36 (4.29)
Greedy (r)	688.31 (0.026)
EGN (r)	693.45 (0.092)
Anneal (r)	696.73 (0.09)
GFlowNet (r)	704.30 (0.35)
VAG-CO (ours)	722.31 (0.17)
Gurobi (tmax=0.1)	731.34 (0.1)
Gurobi (tmax=1.)	731.79 (1.)
Gurobi (tmax=2.)	732.00 (2.)
Gurobi (tmax=10.)	732.01 (10.)
Table 4: Left: Results on the Maximum Clique Problem (see Tab. 1). We show test set results on the best approximation ratio  $AR^*$  and the average approximation ratio  $\widehat{AR}$  across different methods and datasets. Values that are closer to one are better. (r) indicates that these are results as reported in [Karalias et al., 2022]. The time for each algorithm is reported in round brackets as seconds per graph. Right: Results on the Maximum Cut Problem (see Tab. 1) on BA 200 - 300 graphs. We show test set results on the average maximum cut value  $\widehat{MCut}$  across different methods. Values that are larger are better. (r) indicates that these are results as reported in [Zhang et al., 2023].
Evaluation on Synthetic Problems for MIS. On many of the benchmarks above nearly optimal results are obtained. Therefore, we conduct additional experiments on synthetic graph datasets that yield hard CO problems.
For graphs with a degree  $d$  larger than 16 the MIS problem on random regular graphs (RRGs) is known to be particularly hard [Barbier et al., 2013]. Therefore, we generate RRGs with an average size of 100 nodes (RRG-100), where we sample 4100 RRGs between the size of 80 to 120 with different degrees  $d \in [3,7,10,20]$ . For training, validation and testing we use 3000/100/1000 graphs. Results for the MIS problem on the RRG-100 dataset are shown in Fig. 1 (Left), where we plot the test set average of the best relative error  $\epsilon_{\mathrm{rel}}^{*}$  out of  $n_S = 8$  sampled solutions per problem instance. Here the relative error is calculated with respect to exact solutions that are obtained with the commercial solver Gurobi. Error bars indicate the standard error over graphs with the corresponding node degree. As expected we find that for all methods  $\epsilon_{\mathrm{rel}}^{*}$  increases for larger degrees. MFA and MFA-Anneal outperform the DB-Greedy method. Furthermore, VAG-CO is the best performing method for all graph degrees. We also show Gurobi performance for each  $d$  at different time limits  $t_{\mathrm{max}}$ . On this dataset VAG-CO outperforms Gurobi with the given compute budged (see App. A.12) by a high margin on hard instances.
Evaluation on Synthetic Problems for MVC. We also conduct experiments on graphs that are generated by the so-called RB method [Xu et al., 2005]. This method allows the generation of graphs that are known to yield hard MVC problem instances [Wang and Li, 2023]. The RB model has three distinct generation parameters:  $n$ ,  $k'$ , and  $p$ . Adjusting the values of  $n$  and  $k'$  allows us to control the expected size of the generated graphs, while  $p$  serves as a parameter to regulate the hardness of the graph instances. Specifically, when  $p$  is close to one, the generated graphs tend to be easier, whereas reducing  $p$  leads to increased difficulty. To ensure diversity in the RB dataset, we generate graphs with varying levels of hardness by randomly sampling  $p$  from the range of 0.25 to 1.0. For our experiments, we utilize the RB-200 dataset, consisting of RB graphs with an average size of 200 nodes with a train/val/test split of 2000/500/1000 graphs. Following [Wang and Li, 2023] each graph in this dataset is generated by randomly selecting values for  $n$  from the range of 20 to 25, and  $k'$  from the
Figure 2: Study on the advantages of larger ST configuration sizes  $k$  for VAGCO on the RB-100 MVC dataset. On the left y-axis we plot  $AR^{*}$  (lower values are better) for VAG-CO models that are trained for values of  $k$ . The left y-axis shows the inference time per graph.
range of 9 to 10. Figure 1 (Middle) shows the corresponding results in terms of  $\mathrm{AR}^*$  in dependence
of  $p$ . We find that the DB-Greedy algorithm outperforms both MFA approaches. The MFA with annealing tends to perform worse than the MFA without annealing on this dataset. Here, we also add our own implementation of EGN and EGN-Anneal [Sun et al., 2022] and find that EGN performs similarly to MFA and EGN-Anneal achieves a similar performance as DB-Greedy. VAG-CO outperforms all other methods for all values of  $p$  on this dataset. We also show Gurobi performance for each  $p$  at different time limits  $t_{\mathrm{max}}$ . Here, we see that at high  $p$  values Gurobi achieves close to optimal results within a very small time budged and is the best performing method. Whereas for low  $p$  values Gurobi performance drops by a high margin and VAG-CO achieves similar results within a comparable time.
Ablation on Annealing. In the following we investigate whether the annealed entropy regularization in Eq. 3 is indeed beneficial. These experiments are conducted for the MVC problem on RB-100 graphs that are generated with  $n \in [9, 15]$ ,  $k' \in [8, 11]$  and  $p \in [0.25, 1]$ . We compare VAG-CO test set learning curves with three different initial temperatures  $T_0 \in \{5 \cdot 10^{-2}, 5 \cdot 10^{-3}, 0\}$ . The run with  $T_0 = 0$  is equivalent to VAG-CO without annealing (App. A.14). Figure 1 (right) shows the average relative error  $\hat{\epsilon}_{\mathrm{rel}}$  for  $n_S = 30$  sampled solutions per problem instance on the test set over the number of gradient update steps. We find that the run without annealing starts to converge rather quickly but at a comparably bad relative energy. In contrast to that, the runs with annealing  $(T_0 \in \{5 \cdot 10^{-2}, 5 \cdot 10^{-3}\})$  keep improving over more steps and achieve a better final performance.
Ablation on Subgraph Tokenization. Next we study the impact of subgraph tokenization (see Sec. 3) on the performance of VAG-CO. The same problem setting as in the annealing ablation above is used. Figure 1 (right) compares VAG-CO with the default  $k = 5$  subgraph tokenization (w/ ST) to VAG-CO without subgraph tokenization (w/o ST). For a fair comparison the numbers of gradient update steps and the annealing schedules are set to be equal. We find that with subgraph tokenization we achieve a considerably better  $\hat{\epsilon}_{\mathrm{rel}}$ . These results underpin that subgraph tokenization does indeed yield an improved efficiency in terms of gradient update steps.
We additionally provide a more detailed investigation on ST, where we compare VAG-CO with four different values of  $k \in \{1, 2, 5, 10\}$  on the RB-100 MVC dataset. In our experiments we keep the number of gradient update steps constant and iteratively tune for the best learning rate and the best initial temperature in each setting of  $k$ . Results are shown in Fig. 2, where  $AR^*$  and the average time per graph are plotted over  $k$ . As  $k$  increases the performance in terms of  $AR^*$  does improve and the inference time is reduced considerably. This experiment underscores the performance and scalability improvement due to ST.
While MFA-based approaches typically generate a solution to a CO problem in a single forward pass VAG-CO requires a number of forward passes that is proportional the problem size  $N$ . However, methods with solution generation in a single forward pass frequently rely on post-processing procedures like CE that have a time complexity which is also linear in  $N$ . In our approach the nodes in the graph are always processed in an order that is determined by a BFS. Therefore studying the impact of other node orderings or making the algorithm invariant to such orderings might be an interesting future research direction. For the annealing schedule we report comparisons for VAG-CO with three different initial temperatures but a comprehensive study on the optimization of the annealing schedule is left to future investigations.
Our results show that learning to solve hard CO problems requires sufficiently expressive approaches like autoregressive models. With VAG-CO we introduce a method that enables stable and efficient training of such a model through an annealed entropy regularization. We provide a theoretical motivation for this regularization method and demonstrate its practical benefit in experiments. In addition, we demonstrate that by systematically grouping solution variables VAG-CO achieves an improved performance with respect to a naive autoregressive model. Importantly VAG-CO outperforms recent approaches on numerous benchmarks and exhibits superior performance on synthetic CO problems that are designed to be hard.