EB-GLS: an improved guided local search based on the big valley structure

Jialong Shi; Qingfu Zhang; Edward Tsang

doi:10.1007/s12293-017-0242-5

Memetic Computing

September 2018, Volume 10, Issue 3, pp 333–350 | Cite as

EB-GLS: an improved guided local search based on the big valley structure

Authors
Authors and affiliations

Jialong Shi
Qingfu Zhang
Edward Tsang

Regular Research Paper

First Online: 26 July 2017

Received: 31 October 2016

Accepted: 19 July 2017

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Abstract

Local search is a basic building block in memetic algorithms. Guided local search (GLS) can improve the efficiency of local search. By changing the guide function, GLS guides a local search to escape from locally optimal solutions and find better solutions. The key component of GLS is its penalizing mechanism which determines which feature is selected to penalize when the search is trapped in a locally optimal solution. The original GLS penalizing mechanism only makes use of the cost and the current penalty value of each feature. It is well known that many combinatorial optimization problems have a big valley structure, i.e., the better a solution is, the more the chance it is closer to a globally optimal solution. This paper proposes to use big valley structure assumption to improve the GLS penalizing mechanism. An improved GLS algorithm called elite biased GLS (EB-GLS) is proposed. EB-GLS records and maintains an elite solution as an estimate of the globally optimal solutions, and reduces the chance of penalizing the features in this solution. We have systematically tested the proposed algorithm on the symmetric traveling salesman problem. Experimental results show that EB-GLS is significantly better than GLS.

Keywords

Combinatorial optimization Metaheuristics Traveling salesman problem Guided local search Elitism

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Notes

Acknowledgements

The work described in this paper was supported by a grant from ANR/RCC Joint Research Scheme sponsored by the Research Grants Council of the Hong Kong Special Administrative Region, China and France National Research Agency (Project No. A-CityU101/16).

Appendix

In Sect. 4, we state that att532 has the big valley structure, while u2319 does not have the big valley structure. Our statements are based on the landscape sampling experiment conducted on these two instances, in which 1000 runs of GLS and 1000 runs of EB-GLS are executed until finding the global optimum. During each run, the best solutions found so far are recorded and the final global optimum is also recorded. In fact, we conduct the same landscape sampling experiment on another eight instances. By analyzing the results, we conclude that all these eight instances satisfy the requirements of the big valley structure we defined in Sect. 4. Table 5 shows the landscape sampling results on these eight instances. Figure 13 shows the scatter plots of the recorded best solutions found so far.

Table 5

The landscape sampling results on eight selected TSPLIB instances

Instances	City num.	\(N_{opt}\)	\(D_{o,min}\)	\(D_{o,avr}\)	\(D_{o,max}\)	FDC
rd400	400	8	3	21	36	0.83
gr431	431	2	13	13	13	0.76
pcb442	442	2000	6	40	69	0.79
pa561	561	2000	6	44	82	0.83
u574	574	4	2	4	6	0.84
rat575	575	2	3	3	3	0.85
rat783	783	811	2	15	31	0.89
u1432	1432	2000	191	269	342	0.82

\(N_{opt}\) is the number of the unique global optima found by these 2000 runs. \(D_{o,min}\) (\(D_{o,avr}\),\(D_{o,max}\)) is the minimum (average, maximum) distance between the global optima. The last column is the FDC value of the recorded best solutions found so far during the 2000 runs

Fig. 13
The scatter plots of the recorded best solutions found so far during 1000 runs of GLS and 1000 runs of EB-GLS on eight selected instances. The cost difference to the globally optimal cost (*vertical axis*) is plotted against the distance to the nearest globally optimal solution (*horizontal axis*). a rd400. b gr431. c pcb442. d pa561. e u574. (f) rat575. (g) rat783. h u1432

In Table 1 we present the comparison results between EB-GLS and GLS on the 33 TSPLIB instances with more than 1000 cities. Table 6 shows the comparison results on the other 76 TSPLIB instances with less than 1000 cities.

Table 6

Comparison results between EB-GLS and GLS on the TSPLIB instances with less than 1000 cities, the better metric values are marked by bold texts

Instance	Max runtime (s)	Success of 100		Average excess (%)		Excess P value	Average runtime (s)		Runtime P value
Instance	Max runtime (s)	GLS	EB-GLS	GLS	EB-GLS	Excess P value	GLS	EB-GLS	Runtime P value
burma14	2	100	100	0.0000	0.0000 \(\approx \)	-	0.0000	0.0000 \(\approx \)	-
ulysses16	2	100	100	0.0000	0.0000 \(\approx \)	-	0.0000	0.0000 \(\approx \)	-
gr17	2	100	100	0.0000	0.0000 \(\approx \)	-	0.0000	0.0000 \(\approx \)	-
gr21	3	100	100	0.0000	0.0000 \(\approx \)	-	0.0000	0.0000 \(\approx \)	-
ulysses22	3	100	100	0.0000	0.0000 \(\approx \)	-	0.0000	0.0000 \(\approx \)	-
gr24	3	100	100	0.0000	0.0000 \(\approx \)	-	0.0000	0.0000 \(\approx \)	-
fri26	3	100	100	0.0000	0.0000 \(\approx \)	-	0.0002	0.0006 \(\approx \)	1.51e−01
bayg29	3	100	100	0.0000	0.0000 \(\approx \)	-	0.0006	0.0022 −	1.15e−03
bays29	3	100	100	0.0000	0.0000 \(\approx \)	-	0.0009	0.0007 \(\approx \)	6.05e−01
dantzig42	5	100	100	0.0000	0.0000 \(\approx \)	-	0.0006	0.0010 \(\approx \)	3.00e−01
swiss42	5	100	100	0.0000	0.0000 \(\approx \)	-	0.0016	0.0006 \(+\)	2.43e−02
att48	5	100	100	0.0000	0.0000 \(\approx \)	-	0.0048	0.0037 \(\approx \)	2.63e−01
gr48	5	100	100	0.0000	0.0000 \(\approx \)	-	0.0067	0.0042 \(\approx \)	1.86e−01
hk48	5	100	100	0.0000	0.0000 \(\approx \)	-	0.0017	0.0030 −	4.17e−02
eil51	6	100	100	0.0000	0.0000 \(\approx \)	-	0.0103	0.0106 \(\approx \)	9.98e−01
berlin52	6	100	100	0.0000	0.0000 \(\approx \)	-	0.0016	0.0010 \(\approx \)	2.09e−01
brazil58	6	100	100	0.0000	0.0000 \(\approx \)	-	0.0032	0.0031 \(\approx \)	7.99e−01
st70	7	100	100	0.0000	0.0000 \(\approx \)	-	0.0185	0.0119 \(+\)	5.81e−04
eil76	8	100	100	0.0000	0.0000 \(\approx \)	-	0.0099	0.0082 \(\approx \)	9.03e−02
pr76	8	100	100	0.0000	0.0000 \(\approx \)	-	0.0407	0.0148 \(+\)	8.73e−05
gr96	10	100	100	0.0000	0.0000 \(\approx \)	-	0.0262	0.0267 \(\approx \)	5.56e−01
rat99	10	100	100	0.0000	0.0000 \(\approx \)	-	0.0325	0.0191 \(+\)	9.54e−07
kroA100	10	100	100	0.0000	0.0000 \(\approx \)	-	0.0158	0.0103 \(+\)	4.46e−05
kroB100	10	100	100	0.0000	0.0000 \(\approx \)	-	0.0458	0.0280 \(+\)	7.80e−03
kroC100	10	100	100	0.0000	0.0000 \(\approx \)	-	0.0247	0.0069 \(+\)	2.55e−19
kroD100	10	100	100	0.0000	0.0000 \(\approx \)	-	0.0287	0.0133 \(+\)	4.78e−12
kroE100	10	100	100	0.0000	0.0000 \(\approx \)	-	0.0534	0.0314 \(+\)	2.02e−05
rd100	10	100	100	0.0000	0.0000 \(\approx \)	-	0.0354	0.0206 \(+\)	4.10e−07
eil101	11	100	100	0.0000	0.0000 \(\approx \)	-	0.0195	0.0204 \(\approx \)	9.70e−01
lin105	11	100	100	0.0000	0.0000 \(\approx \)	-	0.0202	0.0125 \(+\)	3.36e−06
pr107	11	100	100	0.0000	0.0000 \(\approx \)	-	0.7517	0.1438 \(+\)	2.35e−23
gr120	12	100	100	0.0000	0.0000 \(\approx \)	-	0.1438	0.0493 \(+\)	2.36e−15
pr124	13	100	100	0.0000	0.0000 \(\approx \)	-	0.0246	0.0337 −	3.15e−04
bier127	13	100	100	0.0000	0.0000 \(\approx \)	-	0.3955	0.1055 \(+\)	7.47e−21
ch130	13	100	100	0.0000	0.0000 \(\approx \)	-	0.0895	0.0642 \(+\)	1.67e−05
pr136	14	99	100	0.0001	0.0000 \(\approx \)	3.22e−01	1.7463	0.1631 \(+\)	3.44e−18
gr137	14	100	100	0.0000	0.0000 \(\approx \)	-	0.0907	0.0429 \(+\)	2.01e−17
pr144	15	100	100	0.0000	0.0000 \(\approx \)	-	0.0811	0.1134 −	3.58e−03
ch150	15	100	100	0.0000	0.0000 \(\approx \)	-	0.2003	0.1230 \(+\)	4.34e−07
kroA150	15	100	100	0.0000	0.0000 \(\approx \)	-	0.3707	0.0848 \(+\)	9.34e−14
kroB150	15	100	100	0.0000	0.0000 \(\approx \)	-	0.3672	0.2334 \(+\)	1.12e−06
pr152	16	100	100	0.0000	0.0000 \(\approx \)	-	1.8919	0.6639 \(+\)	1.68e−09
u159	16	100	100	0.0000	0.0000 \(\approx \)	-	0.0658	0.0628 \(\approx \)	9.21e−01
si175	18	100	100	0.0000	0.0000 \(\approx \)	-	5.4633	1.8940 \(+\)	1.65e−20
brg180	18	100	100	0.0000	0.0000 \(\approx \)	-	0.0044	0.0049 \(\approx \)	8.02e−01
rat195	20	100	100	0.0000	0.0000 \(\approx \)	-	0.4984	0.2696 \(+\)	2.83e−05
d198	20	75	96	0.0016	0.0004 \(+\)	3.28e−05	14.7688	4.7853 \(+\)	2.51e−22
kroA200	20	100	100	0.0000	0.0000 \(\approx \)	-	1.7318	0.1695 \(+\)	1.26e−32
kroB200	20	100	100	0.0000	0.0000 \(\approx \)	-	1.2219	0.0886 \(+\)	5.37e−32
gr202	21	100	100	0.0000	0.0000 \(\approx \)	-	1.8843	0.6493 \(+\)	6.34e−19
ts225	23	100	100	0.0000	0.0000 \(\approx \)	-	0.7552	0.4194 \(+\)	4.46e−05
tsp225	23	99	100	0.0008	0.0000 \(\approx \)	3.22e−01	2.9281	0.6930 \(+\)	2.74e−15
pr226	23	93	98	0.0008	0.0006 \(\approx \)	9.29e−02	6.5625	3.0864 \(+\)	1.89e−08
gr229	23	89	100	0.0010	0.0000 \(+\)	6.76e−04	8.0847	1.2958 \(+\)	1.25e−22
gil262	27	100	100	0.0000	0.0000 \(\approx \)	-	2.3682	1.0298 \(+\)	3.09e−10
pr264	27	100	100	0.0000	0.0000 \(\approx \)	-	0.9685	0.4527 \(+\)	1.68e−15
a280	28	100	100	0.0000	0.0000 \(\approx \)	-	0.1767	0.1731 \(\approx \)	3.19e−01
pr299	30	94	100	0.0001	0.0000 \(+\)	1.33e−02	9.7268	0.4571 \(+\)	1.12e−31
lin318	32	100	100	0.0000	0.0000 \(\approx \)	-	2.3201	1.2127 \(+\)	8.33e−09
rd400	40	87	93	0.0009	0.0005 \(\approx \)	1.59e−01	20.4316	6.8556 \(+\)	2.67e−19
fl417	42	1	83	0.0333	0.0123 \(+\)	1.05e−18	41.7864	18.7175 \(+\)	2.85e−29
gr431	44	4	71	0.0072	0.0005 \(+\)	3.25e−28	43.1186	22.0183 \(+\)	9.38e−22
pr439	44	1	58	0.0598	0.0140 \(+\)	3.97e−26	43.8654	26.0844 \(+\)	3.09e−18
pcb442	45	28	94	0.0087	0.0014 \(+\)	1.22e−18	37.2381	8.9825 \(+\)	2.66e−24
d493	50	1	15	0.0135	0.0065 \(+\)	2.10e−20	49.9264	47.3198 \(+\)	2.66e−04
att532	54	6	98	0.0229	0.0012 \(+\)	1.33e−32	52.0478	10.4495 \(+\)	1.17e−35
ali535	54	0	96	0.0337	0.0010 \(+\)	2.81e−36	54.0000	16.4560 \(+\)	1.12e−36
si535	54	0	2	0.2140	0.0387 \(+\)	1.77e−33	54.0000	53.4750 \(\approx \)	1.58e−01
pa561	57	69	99	0.0119	0.0004 \(+\)	7.99e−09	33.7124	10.0504 \(+\)	7.84e−20
u574	58	31	100	0.0076	0.0000 \(+\)	2.44e−23	50.9637	7.4151 \(+\)	3.01e−34
rat575	58	6	53	0.0337	0.0080 \(+\)	4.89e−24	57.1423	38.1065 \(+\)	8.13e−14
p654	66	0	5	0.1657	0.0461 \(+\)	1.30e−31	66.0000	65.1685 \(+\)	2.42e−02
d657	66	0	0	0.0258	0.0038 \(+\)	1.68e−31	66.0000	66.0000 \(\approx \)	-
gr666	67	0	25	0.0417	0.0158 \(+\)	1.81e−16	67.0000	59.0446 \(+\)	1.07e−07
u724	73	2	58	0.0428	0.0069 \(+\)	3.68e−26	72.5028	50.4334 \(+\)	1.59e−17
rat783	79	24	100	0.0151	0.0000 \(+\)	1.07e−26	70.4217	11.3429 \(+\)	6.42e−35

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Authors and Affiliations

Jialong Shi
- 1
Email author
Qingfu Zhang
- 1
Edward Tsang
- 2

1.Department of Computer ScienceCity University of Hong KongHong KongHong Kong
2.Centre for Computational Finance and Economic Agents, School of Computer Science and Electronic EngineeringUniversity of EssexColchesterUK

EB-GLS: an improved guided local search based on the big valley structure

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