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

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

References

1.
Alhindi A, Zhang Q (2013) MOEA/D with guided local search: some preliminary experimental results. In: Computer science and electronic engineering conference (CEEC), 2013 5th, IEEE, pp 109–114Google Scholar
2.
Alsheddy A, Tsang E (2011) Empowerment scheduling for a field workforce. J Sched 14(6):639–654Google Scholar
3.
Basharu M, Arana I, Ahriz H (2005) Distributed guided local search for solving binary DisCSPs. In: FLAIRS conference, pp 660–665Google Scholar
4.
Bentley JJ (1992) Fast algorithms for geometric traveling salesman problems. ORSA J Comput 4(4):387–411MathSciNetCrossRefzbMATHGoogle Scholar
5.
Boese KD (1995) Cost versus distance in the traveling salesman problem. UCLA Computer Science Department, Los AngelesGoogle Scholar
6.
Dorigo M, Birattari M, Stutzle T (2006) Ant colony optimization. IEEE Comput Intell Mag 1(4):28–39CrossRefGoogle Scholar
7.
Eiben ÁE, Van Der Hauw JK, van Hemert JI (1998) Graph coloring with adaptive evolutionary algorithms. J Heuristics 4(1):25–46CrossRefzbMATHGoogle Scholar
8.
Hains DR, Whitley LD, Howe AE (2011) Revisiting the big valley search space structure in the TSP. J Oper Res Soc 62(2):305–312CrossRefGoogle Scholar
9.
Hasan SK, Sarker R, Essam D, Cornforth D (2009) Memetic algorithms for solving job-shop scheduling problems. Memet Comput 1(1):69–83CrossRefGoogle Scholar
10.
Helsgaun K (2000) An effective implementation of the Lin–Kernighan traveling salesman heuristic. Eur J Oper Res 126(1):106–130MathSciNetCrossRefzbMATHGoogle Scholar
11.
Jadon SS, Bansal JC, Tiwari R, Sharma H (2015) Accelerating artificial bee colony algorithm with adaptive local search. Memet Comput 7(3):215–230CrossRefGoogle Scholar
12.
Jones T (1995) Evolutionary algorithms, fitness landscapes and search. Ph.D. Thesis, CiteseerGoogle Scholar
13.
Jones T, Forrest S (1995) Fitness distance correlation as a measure of problem difficulty for genetic algorithms. In: Eshelman LJ (ed) Proceedings of the 6th international conference on genetic algorithms. Morgan Kaufmann, San Francisco, CA, pp 184–192Google Scholar
14.
Kauffman SA (1993) The origins of order: self-organization and selection in evolution. Oxford University Press, OxfordGoogle Scholar
15.
Lau T, Tsang E (2001) Guided genetic algorithm and its application to radio link frequency assignment problems. Constraints 6(4):373–398CrossRefzbMATHGoogle Scholar
16.
Lourenço HR, Martin OC, Stützle T (2010) Iterated local search: framework and applications. In: Handbook of metaheuristics, Springer, pp 363–397Google Scholar
17.
Marinaki M, Marinakis Y (2015) A hybridization of clonal selection algorithm with iterated local search and variable neighborhood search for the feature selection problem. Memet Comput 7(3):181–201CrossRefGoogle Scholar
18.
Merz P, Freisleben B (2000) Fitness landscape analysis and memetic algorithms for the quadratic assignment problem. IEEE Trans Evol Comput 4(4):337–352CrossRefGoogle Scholar
19.
Merz P, Freisleben B (2001) Memetic algorithms for the traveling salesman problem. Complex Syst 13(4):297–346MathSciNetzbMATHGoogle Scholar
20.
Merz P, Katayama K (2004) Memetic algorithms for the unconstrained binary quadratic programming problem. BioSyst 78(1):99–118CrossRefGoogle Scholar
21.
Mester D, Bräysy O (2005) Active guided evolution strategies for large-scale vehicle routing problems with time windows. Comput Oper Res 32(6):1593–1614CrossRefzbMATHGoogle Scholar
22.
Mills P, Tsang E, Ford J (2003) Applying an extended guided local search to the quadratic assignment problem. Ann Oper Res 118(1):121–135MathSciNetCrossRefzbMATHGoogle Scholar
23.
Ochoa G, Veerapen N (2016) Deconstructing the big valley search space hypothesis. In: Evolutionary computation in combinatorial optimization, Springer, pp 58–73Google Scholar
24.
Oliveira SM, Hussin MS, Stützle T, Roli A, Dorigo M (2011) A detailed analysis of the population-based ant colony optimization algorithm for the TSP and the QAP. In: Proceedings of the 13th annual conference companion on genetic and evolutionary computation, ACM, pp 13–14Google Scholar
25.
Reeves CR (1999) Landscapes, operators and heuristic search. Ann Oper Res 86:473–490MathSciNetCrossRefzbMATHGoogle Scholar
26.
Reinelt G (1991) TSPLIB—a traveling salesman problem library. ORSA J Comput 3(4):376–384CrossRefzbMATHGoogle Scholar
27.
Tao X, Haubrich HJ (2005) A hybrid metaheuristic method for the planning of medium-voltage power distribution systemsGoogle Scholar
28.
Tarantilis CD, Zachariadis EE, Kiranoudis CT (2008) A hybrid guided local search for the vehicle-routing problem with intermediate replenishment facilities. INFORMS J Comput 20(1):154–168MathSciNetCrossRefzbMATHGoogle Scholar
29.
Vansteenwegen P, Souffriau W, Berghe GV, Van Oudheusden D (2009) A guided local search metaheuristic for the team orienteering problem. Eur J Oper Res 196(1):118–127CrossRefzbMATHGoogle Scholar
30.
Voudouris C, Tsang E (1999) Guided local search and its application to the traveling salesman problem. Eur J Oper Res 113(2):469–499CrossRefzbMATHGoogle Scholar
31.
Voudouris C, Tsang E, Alsheddy A (2010) Guided local search. In: Handbook of metaheuristics, Springer, pp 321–361Google Scholar
32.
Zachariadis EE, Tarantilis CD, Kiranoudis CT (2009) A guided tabu search for the vehicle routing problem with two-dimensional loading constraints. Eur J Oper Res 195(3):729–743CrossRefzbMATHGoogle Scholar
33.
Zhang W (2004) Configuration landscape analysis and backbone guided local search: part i: satisfiability and maximum satisfiability. Artif Intell 158(1):1–26MathSciNetCrossRefzbMATHGoogle Scholar
34.
Zhong Y, Cole MH (2005) A vehicle routing problem with backhauls and time windows: a guided local search solution. Transp Res Part E: Logist Transp Rev 41(2):131–144CrossRefGoogle Scholar

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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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