Scientific Reports

Table 4 The test results of the algorithms on large “kings” graphs.

From: An efficient swarm evolution algorithm with probability learning for the black and white coloring problem

Instance	\|V\|	b	Swarm-E-1			Swarm-E-2			VNS1			VNS2
Instance	\|V\|	b	best	avg	time(s)	best	avg	time(s)	best	avg	time(s)	best	avg	time(s)
C₃₁ \({ \boxtimes }\) C₃₀	930	150	*724*	701	1178.85	*724*	721	554.79	281	270	1576.59	281	270	620.68
C₃₁ \({ \boxtimes }\) C₃₀	930	350	422	384	1792.94	*518*	497	220.54	49	42	1654.69	48	43	977.63
C₃₁ \({ \boxtimes }\) C₃₀	930	680	40	17	1796.43	186	184	228.85	3	2	1036.85	3	2	965.88
C₄₁ \({ \boxtimes }\) C₄₀	1640	200	1200	1159	1659.87	*1374*	1357	1048.02	621	605	1592.29	625	611	493.88
C₄₁ \({ \boxtimes }\) C₄₀	1640	450	359	308	1798.44	1092	1057	506.00	167	151	284.14	164	153	1366.87
C₄₁ \({ \boxtimes }\) C₄₀	1640	1200	2	1	1041.38	*352*	346	1651.24	3	1	1122.08	4	2	1207.60
C₅₁ \({ \boxtimes }\) C₅₀	2550	250	1575	1523	1776.12	*2226*	2205	132.24	1146	1124	113.96	1150	1126	521.34
C₅₁ \({ \boxtimes }\) C₅₀	2550	580	404	363	1786.18	*1856*	1783	664.53	353	333	148.68	348	335	522.95
C₅₁ \({ \boxtimes }\) C₅₀	2550	1900	1	1	42.89	*540*	483	1543.37	3	1	1489.64	3	1	1367.92
C₆₁ \({ \boxtimes }\) C₆₀	3660	560	1013	940	1738.63	*2990*	2878	1296.43	986	941	515.27	967	947	1389.92
C₆₁ \({ \boxtimes }\) C₆₀	3660	850	421	372	1783.17	*2658*	2536	1798.66	442	428	1216.56	462	438	1179.09
C₆₁ \({ \boxtimes }\) C₆₀	3660	1100	215	167	1723.06	*2342*	2229	1795.07	220	207	268.0	242	210	1542.21
C₇₁ \({ \boxtimes }\) C₇₀	4970	680	1441	1379	1799.92	*4156*	4028	1730.41	1484	1456	458.10	1506	1464	1538.03
C₇₁ \({ \boxtimes }\) C₇₀	4970	960	815	759	1788.27	*3827*	3616	1771.45	860	829	1333.87	886	844	1493.99
C₇₁ \({ \boxtimes }\) C₇₀	4970	1300	391	341	1516.68	*3338*	3225	1777.74	437	405	528.10	433	410	451.89

Instance	\|V\|	b	SA			Greedy_One			Greedy_Two
Instance	\|V\|	b	best	avg	time(s)	best	avg	time(s)	best	avg	time(s)
C₃₁ \({ \boxtimes }\) C₃₀	930	150	717	704	262.36	720	718	893.11	654	654	22.91
C₃₁ \({ \boxtimes }\) C₃₀	930	350	486	470	1659.76	517	516	1031.21	444	444	65.70
C₃₁ \({ \boxtimes }\) C₃₀	930	680	152	131	1358.54	*187*	186	1044.02	97	97	57.17
C₄₁ \({ \boxtimes }\) C₄₀	1640	200	1336	1169	1034.57	1360	1355	1331.80	1274	1274	104.32
C₄₁ \({ \boxtimes }\) C₄₀	1640	450	1024	900	471.81	*1107*	1104	1322.78	1015	1015	187.13
C₄₁ \({ \boxtimes }\) C₄₀	1640	1200	178	105	360.27	0	0	–	237	236	577.84
C₅₁ \({ \boxtimes }\) C₅₀	2550	250	2136	1825	486.08	2197	2194	1282.09	2094	2094	304.77
C₅₁ \({ \boxtimes }\) C₅₀	2550	580	1625	1083	548.89	0	0	–	1755	1755	576.16
C₅₁ \({ \boxtimes }\) C₅₀	2550	1900	112	39	440.48	0	0	–	395	394	1155.36
C₆₁ \({ \boxtimes }\) C₆₀	3660	560	2673	1720	1050.36	0	0	–	2848	2848	1078.34
C₆₁ \({ \boxtimes }\) C₆₀	3660	850	2268	1221	1023.74	0	0	–	2551	2550	1426.31
C₆₁ \({ \boxtimes }\) C₆₀	3660	1100	1918	966	1682.25	0	0	–	2295	1376	1535.45
C₇₁ \({ \boxtimes }\) C₇₀	4970	680	3677	1999	1066.58	0	0	–	3998	399	1733.22
C₇₁ \({ \boxtimes }\) C₇₀	4970	960	3257	1956	1184.14	0	0	–	0	0	–
C₇₁ \({ \boxtimes }\) C₇₀	4970	1300	2643	811	1793.56	0	0	–	0	0	–

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