Research Article
An Integrated Method Based on PSO and EDA for the Max-Cut Problem
Table 6
Comparison of the results obtained by the GRASP-TS/PM, PR2, BLS, TSHEA, and PSO-EDA on instances from the first set.
| | Instances | | Best | GRASP-TS/PM [6] | PR2 [11] | BLS [12] | TSHEA [13] | PSO-EDA | | | | | | | | | | | Gap (%) |
| | G1 | 800 | 11624 | 11624 | 11624.0 | 11624 | 11624.0 | 11624 | 11612.4 | 11624 | 11624.0 | 11624 | 11624.0 | 0 | | G2 | 800 | 11620 | 11620 | 11620.0 | 11620 | 11620.0 | 11620 | 11615.0 | 11620 | 11620.0 | 11620 | 11614.0 | 0 | | G3 | 800 | 11622 | 11620 | 11620.0 | 11620 | 11620.0 | 11622 | 11621.1 | 11622 | 11622.0 | 11622 | 11622.0 | 0 | | G4 | 800 | 11646 | 11646 | 11646.0 | 11646 | 11646.0 | 11646 | 11642.8 | 11646 | 11646.0 | 11646 | 11642.3 | 0 | | G5 | 800 | 11631 | 11631 | 11631.0 | 11631 | 11631.0 | 11631 | 11631.0 | 11631 | 11631.0 | 11631 | 11631.0 | 0 | | G6 | 800 | 2178 | 2178 | 2177.9 | 2178 | 2178.0 | 2178 | 2178.0 | 2178 | 2178.0 | 2178 | 2178.0 | 0 | | G7 | 800 | 2006 | 2006 | 2006.0 | 2006 | 2006.0 | 2006 | 2001.05 | 2006 | 2006.0 | 2006 | 2006.0 | 0 | | G8 | 800 | 2005 | 2005 | 2004.9 | 2005 | 2005.0 | 2005 | 2004.4 | 2005 | 2005.0 | 2005 | 2002.7 | 0 | | G9 | 800 | 2054 | 2054 | 2053.6 | 2054 | 2054.0 | 2054 | 2049.95 | 2054 | 2054.0 | 2054 | 2047.1 | 0 | | G10 | 800 | 2000 | 2000 | 1999.3 | 2000 | 1999.8 | 2000 | 1996.05 | 2000 | 2000.0 | 2000 | 1997.5 | 0 | | G11 | 800 | 564 | 564 | 564.0 | 564 | 564.0 | 564 | 564.0 | 564 | 564.0 | 564 | 564.0 | 0 | | G12 | 800 | 556 | 556 | 556.0 | 556 | 556.0 | 556 | 556.0 | 556 | 556.0 | 556 | 556.0 | 0 | | G13 | 800 | 582 | 582 | 581.8 | 582 | 582.0 | 582 | 582.0 | 582 | 582.0 | 582 | 582.0 | 0 | | G14 | 800 | 3064 | 3063 | 3062.1 | 3064 | 3062.6 | 3064 | 3062.85 | 3064 | 3064.0 | 3062 | 3060.4 | 0.03 | | G15 | 800 | 3050 | 3050 | 3049.1 | 3050 | 3049.3 | 3050 | 3050.0 | 3050 | 3050.0 | 3050 | 3047.6 | 0 | | G16 | 800 | 3052 | 3052 | 3050.9 | 3052 | 3051.4 | 3052 | 3051.1 | 3052 | 3052.0 | 3052 | 3051.4 | 0 | | G17 | 800 | 3047 | 3047 | 3045.8 | 3047 | 3046.4 | 3047 | 3046.7 | 3047 | 3047.0 | 3047 | 3045.1 | 0 | | G18 | 800 | 992 | 992 | 992.0 | 992 | 992.0 | 992 | 991.7 | 992 | 992.0 | 992 | 990.5 | 0 | | G19 | 800 | 906 | 906 | 906.0 | 906 | 906.0 | 906 | 904.55 | 906 | 906.0 | 906 | 904.4 | 0 | | G20 | 800 | 941 | 941 | 941.0 | 941 | 941.0 | 941 | 941.0 | 941 | 941.0 | 941 | 941.0 | 0 | | G21 | 800 | 931 | 931 | 930.6 | 931 | 931.0 | 931 | 930.2 | 931 | 931.0 | 931 | 930.3 | 0 | | G22 | 2000 | 13359 | 13349 | 13342.4 | 13359 | 13354.5 | 13359 | 13344.45 | 13359 | 13359.0 | 13359 | 13353.1 | 0 | | G23 | 2000 | 13344 | 13332 | 13322.4 | 13342 | 13331.6 | 13344 | 13340.6 | 13344 | 13344.0 | 13344 | 13331.8 | 0 | | G24 | 2000 | 13337 | 13324 | 13317.3 | 13333 | 13325.3 | 13337 | 13329.8 | 13337 | 13337.0 | 13337 | 13325.1 | 0 | | G25 | 2000 | 13340 | 13326 | 13318.1 | 13339 | 13328.2 | 13340 | 13333.4 | 13340 | 13340.0 | 13338 | 13324.2 | 0.01 | | G26 | 2000 | 13328 | 13313 | 13303.3 | 13326 | 13312.3 | 13328 | 13320.0 | 13328 | 13328.0 | 13326 | 13319.1 | 0 | | G27 | 2000 | 3341 | 3325 | 3318.1 | 3336 | 3326.9 | 3341 | 3332.25 | 3341 | 3341.0 | 3341 | 3323.7 | 0 | | G28 | 2000 | 3298 | 3287 | 3277.4 | 3296 | 3288.9 | 3298 | 3293.85 | 3298 | 3298.0 | 3298 | 3290.3 | 0 | | G29 | 2000 | 3405 | 3394 | 3384.5 | 3405 | 3391.9 | 3405 | 3388.2 | 3405 | 3405.0 | 3405 | 3388.5 | 0 | | G30 | 2000 | 3413 | 3402 | 3393.4 | 3411 | 3404.8 | 3412 | 3404.85 | 3413 | 3413.0 | 3412 | 3405.4 | 0.03 | | G31 | 2000 | 3310 | 3299 | 3287.7 | 3306 | 3299.5 | 3309 | 3305.3 | 3310 | 3310.0 | 3308 | 3301.3 | 0.06 | | G32 | 2000 | 1410 | 1406 | 1397.3 | 1410 | 1404.6 | 1410 | 1409.3 | 1410 | 1410.0 | 1410 | 1408.6 | 0 | | G33 | 2000 | 1382 | 1374 | 1369.1 | 1382 | 1376.1 | 1382 | 1380.1 | 1382 | 1382.0 | 1382 | 1380.4 | 0 | | G34 | 2000 | 1384 | 1376 | 1372.5 | 1384 | 1378.2 | 1384 | 1384.0 | 1384 | 1384.0 | 1384 | 1383.2 | 0 | | G35 | 2000 | 7687 | 7661 | 7657.4 | 7679 | 7670.8 | 7684 | 7680.85 | 7687 | 7685.6 | 7685 | 7673.5 | 0.03 | | G36 | 2000 | 7680 | 7660 | 7652.1 | 7671 | 7658.7 | 7678 | 7673.6 | 7680 | 7677.5 | 7671 | 7660.2 | 0.12 | | G37 | 2000 | 7691 | 7670 | 7662.0 | 7682 | 7667.9 | 7689 | 7685.85 | 7691 | 7688.05 | 7678 | 7668.2 | 0.17 | | G38 | 2000 | 7688 | 7670 | 7659.8 | 7682 | 7670.4 | 7687 | 7684.95 | 7688 | 7688.0 | 7688 | 7670.8 | 0 | | G39 | 2000 | 2408 | 2397 | 2387.1 | 2407 | 2391.1 | 2408 | 2405.35 | 2408 | 2408.0 | 2408 | 2396.7 | 0 | | G40 | 2000 | 2400 | 2397 | 2384.3 | 2399 | 2383.3 | 2400 | 2394.6 | 2400 | 2399.6 | 2395 | 2385.3 | 0.21 | | G41 | 2000 | 2405 | 2398 | 2383.7 | 2404 | 2388.9 | 2405 | 2403.0 | 2405 | 2405.0 | 2405 | 2387.8 | 0 | | G42 | 2000 | 2481 | 2474 | 2461.7 | 2478 | 2466.2 | 2481 | 2475.4 | 2481 | 2478.45 | 2478 | 2470.6 | 0.12 | | G43 | 1000 | 6660 | 6660 | 6659.4 | 6660 | 6659.9 | 6660 | 6658.15 | 6660 | 6659.0 | 6660 | 6658.7 | 0 | | G44 | 1000 | 6650 | 6649 | 6647.7 | 6650 | 6649.9 | 6650 | 6647.7 | 6650 | 6650.0 | 6650 | 6649.4 | 0 | | G45 | 1000 | 6654 | 6654 | 6652.6 | 6654 | 6653.9 | 6654 | 6652.15 | 6654 | 6654.0 | 6654 | 6650.1 | 0 | | G46 | 1000 | 6649 | 6649 | 6646.0 | 6649 | 6648.8 | 6649 | 6647.75 | 6649 | 6649.0 | 6649 | 6646.2 | 0 | | G47 | 1000 | 6657 | 6656 | 6655.4 | 6657 | 6656.8 | 6657 | 6654.35 | 6657 | 6657.0 | 6657 | 6650.8 | 0 | | G48 | 3000 | 6000 | 6000 | 6000.0 | 6000 | 6000.0 | 6000 | 6000.0 | 6000 | 6000.0 | 6000 | 6000.0 | 0 | | G49 | 3000 | 6000 | 6000 | 6000.0 | 6000 | 6000.0 | 6000 | 6000.0 | 6000 | 6000.0 | 6000 | 6000.0 | 0 | | G50 | 3000 | 5880 | 5880 | 5880.0 | 5880 | 5880.0 | 5880 | 5879.9 | 5880 | 5880.0 | 5880 | 5880.0 | 0 | | G51 | 1000 | 3848 | 3847 | 3843.8 | 3848 | 3846.4 | 3848 | 3847.85 | 3848 | 3848.0 | 3848 | 3844.6 | 0 | | G52 | 1000 | 3851 | 3850 | 3846.8 | 3851 | 3848.4 | 3851 | 3850.85 | 3851 | 3851.0 | 3851 | 3845.5 | 0 | | G53 | 1000 | 3850 | 3848 | 3845.8 | 3850 | 3847.7 | 3850 | 3849.5 | 3850 | 3849.55 | 3850 | 3845.1 | 0 | | G54 | 1000 | 3852 | 3850 | 3847.8 | 3851 | 3847.8 | 3852 | 3850.6 | 3852 | 3851.25 | 3850 | 3846.1 | 0.05 | | G55 | 5000 | 10299 | — | — | 10265 | 10234.0 | 10294 | 10282.4 | 10299 | 10291.75 | 10293 | 10267.3 | 0.06 | | G56 | 5000 | 4017 | — | — | 3981 | 3959.2 | 4012 | 3998.65 | 4017 | 4008.6 | 4004 | 3990.0 | 0.32 | | G57 | 5000 | 3494 | — | — | 3472 | 3462.0 | 3492 | 3488.6 | 3494 | 3488.7 | 3492 | 3486.6 | 0.06 | | G58 | 5000 | 19276 | — | — | 19205 | 19182.0 | 19263 | 19255.6 | 19276 | 19266.0 | 19251 | 19213.8 | 0.13 | | G59 | 5000 | 6085 | — | — | 6027 | 6006.2 | 6078 | 6067.9 | 6085 | 6070.45 | 6060 | 6028.2 | 0.41 | | G60 | 7000 | 14186 | — | — | 14112 | 14091.8 | 14176 | 14166.8 | 14186 | 14173.5 | 14161 | 14142.2 | 0.18 | | G61 | 7000 | 5796 | — | — | 5730 | 5695.7 | 5789 | 5773.35 | 5796 | 5776.0 | 5769 | 5756.7 | 0.47 | | G62 | 7000 | 4868 | — | — | 4836 | 4830.2 | 4868 | 4863.8 | 4866 | 4860.2 | 4860 | 4856.2 | 0.16 | | G63 | 7000 | 27018 | — | — | 26916 | 26879.3 | 26997 | 26980.7 | 27018 | 26993.6 | 26958 | 26908.6 | 0.22 | | G64 | 7000 | 8735 | — | — | 8641 | 8594.1 | 8735 | 8735.0 | 8735 | 8717.95 | 8710 | 8672.2 | 0.29 | | G65 | 8000 | 5560 | — | — | 5526 | 5515.9 | 5558 | 5551.2 | 5560 | 5555.4 | 5546 | 5543.4 | 0.25 | | G66 | 9000 | 6364 | — | — | 6314 | 6302.4 | 6360 | 6350.2 | 6364 | 6353.7 | 6350 | 6338.4 | 0.22 | | G67 | 10000 | 6944 | — | — | 6902 | 6884.6 | 6940 | 6935.3 | 6944 | 6937.3 | 6932 | 6927.6 | 0.17 | | G70 | 10000 | 9548 | — | — | 9463 | 9434.0 | 9541 | 9527.1 | 9548 | 9539.6 | 9530 | 9518.4 | 0.19 | | G72 | 10000 | 6998 | — | — | 6946 | 6933.8 | 6998 | 6935.3 | 6990 | 6979.7 | 6984 | 6977.2 | 0.20 | | G77 | 14000 | 9926 | — | — | — | — | 9926 | 9916.1 | 9902 | 9890.8 | 9904 | 9896.0 | 0.22 | | G81 | 20000 | 14030 | — | — | — | — | 14030 | 14021.7 | 14010 | 13993.2 | 13980 | 13976.5 | 0.36 |
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