Fast Polynomial Time Approximate Solution for 0-1 Knapsack Problem

Table 3

Mean relative error of upper bounds to the best upper bound of (KP) (ppm).

(No, R)	UMT2	UMTM	Ukmax	B	(No, R)	UMT2	UMTM	Ukmax	B
(1, 10^3)	4.6696	4.6696	4.6869	0	(2, 10^3)	0.0138	0.0031	0.0421	0.0174
(1, 10^4)	91.413	91.3869	81.4422	0	(2, 10^4)	0.0345	0.0075	0.0465	0.0189
(1, 10^5)	60.2607	60.2542	43.2765	0	(2, 10^5)	0.0194	0.005	0.0351	0.0164
(1, 10^6)	61.0366	61.0366	0	0	(2, 10^6)	0.0258	0.0057	0.0391	0.0152
(1, 10^7)	61.3372	61.3372	0	0	(2, 10^7)	0.0274	0.0069	0.0466	0.018
(3, 10^3)	0.0116	0.0049	0.0179	0	(4, 10^3)	28.7876	28.7876	0	0
(3, 10^4)	0.0031	0	0.0196	0.0044	(4, 10^4)	34.656	34.6518	0	0
(3, 10^5)	0.0148	0.0018	0.027	0.016	(4, 10^5)	31.5535	31.5456	0	0
(3, 10^6)	0.0114	0.0016	0.0166	0.0071	(4, 10^6)	27.5082	27.5001	0	0
(3, 10^7)	0.0066	0.0022	0.0105	0.0031	(4, 10^7)	25.7493	25.7467	0	0
(5, 10^3)	38.9054	38.9054	38.9054	0	(6, 10^3)	0	0	0	0
(5, 10^4)	0	0	0	0	(6, 10^4)	0	0	0	0
(5, 10^5)	0	0	0	0	(6, 10^5)	0	0	0	0
(5, 10^6)	0	0	0	0	(6, 10^6)	0	0	0	0
(5, 10^7)	0	0	0	0	(6, 10^7)	0	0	0	0
(7, 10^3)	0	0	0	0	(8, 10^3)	0	0	0	0
(7, 10^4)	0	0	0	0	(8, 10^4)	0	0	0	0
(7, 10^5)	0	0	0	0	(8, 10^5)	22.8602	22.8602	0	0
(7, 10^6)	0	0	0	0	(8, 10^6)	0	0	0	0
(7, 10^7)	0	0	0	0	(8, 10^7)	0	0	0	0
(9, 10^3)	0	0	0	0	(10, 10^3)	0	0	0	0
(9, 10^4)	0	0	0	0	(10, 10^4)	0	0	0	0
(9, 10^5)	16.1344	16.1344	0	0	(10, 10^5)	0	0	0	0
(9, 10^6)	30.1447	30.1447	0	0	(10, 10^6)	0	0	0	0
(9, 10^7)	8.9156	8.9156	8.9156	0	(10, 10^7)	0	0	0	0

The best upper bound is min{U_MT2, U_MTM, U_kmax, B} in this paper.