Table 1: Performance comparison of NeuOpt with various baselines on TSP and CVRP benchmarks.
Method	Model Type	Post (Per-Ins.) Proc.	\(N=20\) Gap↓ Time↓	\(N=50\) Gap↓ Time↓	\(N=100\) Gap↓ Time↓
Obj.-	Gap↓	6m	Obj.-	Gap↓	Time↓	Obj.-	Gap↓	Time↓
TSP	Concorde [54]	Exact	-	3.827	-	2m	5.696	-	9m	7.765	-	43m
LKH-2 [51]	H	-	3.827	0.00%	6m	5.696	0.00%	1.3h	7.765	0.00%	5.7h
GCN+BS [14]#	L2P/SL	BS+H	3.827	0.00%	15m	5.698	0.04%	23m	7.869	1.35%	46m
Att-GCN+MCTS [6]‡, #	L2P/SL	MCTS	(≈3.830)	(≈0.00%)	≈2m	(≈5.691)	(≈0.01%)	≈8m	(≈7.764)	(≈0.04%)	≈15m
GNN+GLS [40] (relocate+2-opt)‡	L2P/SL	GLS	-	≈0.00%	≈2.8h	-	≈0.00%	≈2.8h	-	≈0.58%	≈2.8h
CVAE-Opt-DE [43]‡	L2P/UL	DE	-	≈0.00%	≈1.2d	-	≈0.02%	≈2.5d	-	≈0.34%	≈1.8d
DPDP [42] (100k)	L2P/SL	DP	-	-	-	-	-	-	7.765	0.00%	1.9h
DIMES [7] (T=10)‡, #	L2P/RL	AS+M+M	-	-	-	-	-	-	(≈7.762)	(≈0.01%)	-
DIFUSCO [15] (T=50, S=16)#	L2P/SL	2-opt	-	-	-	5.696	0.01%	5.8h	7.766	0.02%	21.7h
AM+LCP* [33] ( {1280, 45})	L2C/RL	-	3.828	0.01%	2.1h	5.699	0.05%	4.9h	7.811	0.60%	10.9h
Pointerformer [32] (A=8, T=200)	L2C/RL	-	3.827	0.00%	13m	5.697	0.02%	1.1h	7.773	0.11%	5.6h
Sym-NCO [13] (A=8, T=200)	L2C/RL	-	-	-	-	-	-	-	7.771	0.08%	5.6h
POMO [4] (A=8, T=200)	L2C/RL	-	3.827	0.00%	13m	5.696	0.00%	1.1h	7.770	0.07%	5.6h
POMO+EAS [5] (A=8, T=200)	L2C/RL	AS	3.827	0.00%	24m	5.696	0.00%	2h	7.769	0.05%	10.9h
POMO+EAS+SGBS [34] (short)	L2C/RL	AS+BS	-	-	-	-	-	-	7.767	0.04%	6.5h
POMO+EAS+SGBS [34] (long)	L2C/RL	AS+BS	-	-	-	-	-	-	7.767	0.03%	1.1d
Costa et al. [16] (2-opt, T=2k)	L2S/RL	-	3.827	0.00%	31m	5.703	0.12%	40m	7.824	0.77%	1.1h
Sui et al. [17] (3-opt, T=2k)‡	L2S/RL	-	≈3.84	≈0.00%	≈32m	≈5.70	≈0.08%	≈48m	≈7.82	≈0.74%	≈1.3h
Wu et al. [39] (2-opt, T=5k)	L2S/RL	-	-	-	-	5.709	0.23%	1.3h	7.884	1.54%	2h
DACT [9] (2-opt, A=4, T=10k)	L2S/RL	-	3.827	0.00%	1.5h	5.696	0.00%	4.1h	7.772	0.10%	13.5h
NeuOpt (D2A=1, T=1k)	L2S/RL	-	3.827	0.00%	2m	5.697	0.02%	6m	7.790	0.33%	17m
NeuOpt (D2A=1, T=5k)	L2S/RL	-	3.827	0.00%	12m	5.696	0.00%	32m	7.768	0.05%	1.4h
NeuOpt (D2A=1, T=10k)	L2S/RL	-	3.827	0.00%	23m	5.696	0.00%	1.1h	7.766	0.02%	2.8h
NeuOpt (D2A=5, T=1k)	L2S/RL	-	3.827	0.00%	12m	5.696	0.00%	32m	7.767	0.04%	1.4h
NeuOpt (D2A=5, T=3k)	L2S/RL	-	3.827	0.00%	35m	5.696	0.00%	1.6h	7.765	0.01%	4.2h
NeuOpt (D2A=5, T=5k)	L2S/RL	-	3.827	0.00%	1h	5.696	0.00%	2.7h	7.765	0.00%	7h
CVRP	HGS [21]	H	-	6.130	-	10.7h	10.366	-	1.2d	15.563	-	2.5d
LKH-3 [20]	H	-	6.135	0.08%	17.9h	10.375	0.09%	2.8d	15.647	0.54%	5.7d
CVAE-Opt-DE [43]‡	L2P/UL	DE	≈6.14	-	≈2.4d	≈10.40	-	≈4.7d	≈15.75	-	≈11d
DPDP [42] (1000k)	L2P/SL	DP	-	-	-	-	-	-	15.627	0.41%	1.2d
AM+LCP [33] ( {2560, 1})‡	L2C/RL	-	≈6.15	≈0.33%	≈23m	≈10.52	≈1.48%	≈52m	≈16.00	≈2.81%	≈2.1h
Sym-NCO [13] (A=8, T=200)	L2C/RL	-	-	-	-	-	-	-	15.702	0.89%	7.2h
POMO [4] (A=8, T=200)	L2C/RL	-	6.136	0.09%	11m	10.397	0.30%	1.4h	15.672	0.70%	7.2h
POMO+EAS [5] (A=8, T=200)	L2C/RL	AS	6.132	0.04%	38m	10.379	0.13%	3.1h	15.610	0.30%	16h
POMO+EAS+SGBS [34] (short)	L2C/RL	AS+BS	-	-	-	-	-	-	15.587	0.15%	1d
POMO+EAS+SGBS [34] (long)	L2C/RL	AS+BS	-	-	-	-	-	-	15.579	0.10%	4.1d
NLNS [8] (Ruin-Repair, T=5k)	L2S/RL	-	6.175	0.73%	48m	10.506	1.35%	1.4h	15.915	2.26%	2.4h
NCE [37] (CROSS exchange)‡	L2S/SL	-	≈6.13	≈0.00%	≈11h	≈10.41	≈0.42%	≈2.3d	≈15.81	≈1.59%	≈10.4d
Wu et al. [39] (2-opt, T=5k)	L2S/RL	-	-	-	-	10.544	1.72%	4.2h	16.165	3.87%	5h
DACT [9] (2-opt, A=6, T=10k)	L2S/RL	-	6.130	0.01%	4h	10.383	0.16%	16h	15.736	1.11%	1.7d
NeuOpt-GIRE (D2A=1, T=1k)	L2S/RL	-	6.132	0.03%	4m	10.430	0.61%	12m	15.865	1.94%	28m
NeuOpt-GIRE (D2A=1, T=5k)	L2S/RL	-	6.130	0.00%	20m	10.382	0.16%	59m	15.698	0.87%	2.3h
NeuOpt-GIRE (D2A=1, T=10k)	L2S/RL	-	6.130	0.00%	41m	10.375	0.08%	2h	15.656	0.60%	4.6h
NeuOpt-GIRE (D2A=5, T=6k)	L2S/RL	-	6.130	0.00%	2.1h	10.369	0.03%	5.9h	15.610	0.30%	13.8h
NeuOpt-GIRE (D2A=5, T=20k)	L2S/RL	-	6.130	0.00%	6.8h	10.367	0.01%	19.7h	15.586	0.15%	1.9d
NeuOpt-GIRE (D2A=5, T=40k)	L2S/RL	-	6.130	0.00%	13.7h	10.367	0.01%	1.6d	15.579	0.10%	3.8d
We found issues with their code, causing an underestimation of their reported gaps by roughly  $0.02\%$  (TSP-50) and  $0.04\%$  (TSP-100). Thus our reproduced gaps may vary from those in their papers. For not reproduced ones, we use  $(\cdot)$  to indicate the underestimated values sourced directly from their papers. The results with  $\approx$  are based on their original papers since their code is either i) not publicly available (e.g., AM+CLP for CVRP), ii) raises (incompatible) errors that are difficult to fix on our hardware (e.g., Att-GCN+MCTS), or iii) requires long computation time to reproduce (e.g., CVAE-Opt-DE). Note: The abbreviations refer to: A - Augmentation, D2A - Dynamic Data Augmentation, BS - Beam Search, H - Heuristics, MCTS - Monte Carlo Tree Search, DE - Differential Evolution, DP - Dynamic Programming, GLS - Guided Local Search, AS - Active Search, AS+M+M - AS+MCTS+Meta-Learning.