- Email: [email protected]
Variable neighborhood search for the strong metric dimension problem
Available online at www.sciencedirect.com
Electronic Notes in Discrete Mathematics 39 (2012) 51–57 www.elsevier.com/locate/endm
Variable neighborhood search for the strong metric dimension problem Nenad Mladenovi´c a,1 Jozef Kratica b,2 ˇ Vera Kovaˇcevi´c-Vujˇci´c c,3 Mirjana Cangalovi´ c c,4 a b
c
Department of Mathematics, SISCM, Brunel University, London, UK
Mathematical Institute, Serbian Academy of Sciences and Arts, Kneza Mihaila 36/III, 11000 Belgrade, Serbia
Faculty of Organizational Sciences, University of Belgrade, Jove Ili´ca 154, 11000 Belgrade, Serbia
Abstract We consider a variable neighborhood search approach for solving the strong metric dimension problem. The proposed method is based on the idea of decomposition and it is characterized by suitably chosen neighborhood structures and efficient local search. Computational experiments on ORLIB instances show that the new approach outperformes a genetic algorithm, the only existing heuristic in the literature for solving this problem. Keywords: Strong metric dimension, metaheuristics, combinatorial optimization.
This research was partially supported by Serbian Ministry of Science under grants 174010 and 174033. 1 Email: [email protected] 2 Email: [email protected] 3 Email: [email protected] 4 Email: [email protected] 1571-0653/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2012.10.008
52
1
N. Mladenovi´c et al. / Electronic Notes in Discrete Mathematics 39 (2012) 51–57
Introduction
The strong metric dimension problem (SMDP) was introduced by Sebo and Tannier [9] and further investigated by Oellermann and Peters-Fransen [8]. A vertex w strongly resolves two vertices u and v if u belongs to a shortest v − w path or v belongs to a shortest u − w path. A vertex set S of G is a strong resolving set of G if every two distinct vertices of G are strongly resolved by some vertex of S. A strong metric basis of G is a strong resolving set of the minimum cardinality. Now, the strong metric dimension of G, denoted by sdim(G) is defined as the cardinality of its strong metric basis. The strong metric dimension is NP-hard in a general case, see [8]. The first metaheuristic approach based on the genetic algorithm (GA) is proposed in [2]. Recently, the strong metric dimension of some special classes of graphs has been studied theoretically, such as distance hereditary graphs [5], Hamming graphs [3] and some convex polytopes [4]. Example 1.1 Consider the graph G of Figure 1. The set S = {A, B} is a strong resolving set of G. Indeed, in the case when v ∈ S then w = v strongly resolves u and v, for all u. Similarly, when u ∈ S then w = u strongly resolves u and v, for all v. In the case when u = C and v = D, then w = B strongly resolves u and v. Similar result holds for u = D and v = C. The last case when u = v ∈ {C, D} is trivial, since the shortest path of zero length can be extended to the shortest u − w path for any w ∈ S. It is easy to see that a singleton set cannot be a strong resolving set, since the u − v path with u = A, v = D or u = B, v = C cannot be extended to a shortest u − w path, with w in a singleton set. Therefore sdim(G) = 2.
Fig. 1. Graph G in Example 1.1
N. Mladenovi´c et al. / Electronic Notes in Discrete Mathematics 39 (2012) 51–57
2
53
Mathematical formulation
The integer linear programming (ILP) formulation of the strong metric dimension problem was proposed in [3]. Given a simple connected undirected graph G = (V ,E), where V = {1, 2, . . . , n}, |E| = m, it is easy to determine the length d(u, v) of a shortest u − v path for all u, v ∈ V , using any shortest path algorithm. Then, the coefficient matrix can be defined as:
A(u,v),i
⎧ ⎪ ⎨1, d(u, i) = d(u, v) + d(v, i) = 1, d(v, i) = d(v, u) + d(u, i) ⎪ ⎩ 0, otherwise
where 1 ≤ u < v ≤ n, 1 ≤ i ≤ n. Let the variable yi determine whether vertex i belongs to a strong resolving set S, i.e. yi = 1 for i ∈ S, yi = 0 otherwise. The ILP model of the strong metric dimension problem can now be formulated as follows, min
n
yi
(1)
i=1
subject to: n
A(u,v),i · yi ≥ 1
1≤u
(2)
i=1
yi ∈ {0, 1}
1≤i≤n
(3)
The objective function (1) represents the cardinality of a feasible solution S. Constraints (2) make sure that for each two vertices u and v there exists at least one vertex w from S which strongly resolves u and v, i.e. that S is a strong resolving set. Constraints (3) reflect the binary nature of decision variables yi .
3
VNS for the strong metric dimension problem
Variable neighborhood search (VNS) is an effective metaheuristic introduced in [6]. The basic idea of VNS is to use more than one neighborhood structure and to proceed with their systematic change within a local search. A detailed description of different VNS variants is out of the scope of this paper and can be found in [1].
54
N. Mladenovi´c et al. / Electronic Notes in Discrete Mathematics 39 (2012) 51–57
The VNS approach for the SMDP is based on the idea of decomposition described in [7]. The initial set S is obtained by a simple procedure, which starts from the empty set and adds randomly chosen vertices from V until S becomes a strong resolving set. For a given strong resolving set S the last element is deleted, to obtain set S . The objective function value ObjF (S ) is computed as the number of pairs of vertices from V which are not strongly resolved with respect to S . The following steps are repeated until the stopping criterion is met. For a given k set S in k-th neighborhood N k (S ) is obtained using the shaking function. Starting from S the local search procedure tries to improve S and updates S whenever a new strong resolving set with smaller cardinality is generated. The neighborhood N k (S ) contains all sets obtained from S by deleting k of its elements and replacing them by k elements from V \S . In the local search procedure, starting with S , one element from set S is interchanged with one element of its complement. The best improvement strategy is used, i.e. in every step an interchange is performed, which gives the maximal decrease of the objective function. Whenever the improved set is a strong resolving set, the current set S is updated, the new set S is obtained by deleting the last element, and the procedure continues. The local search procedure stops when there is no improvement. After the local search procedure, there are three possibilities. If |S | < |S | or ObjF (S ) < ObjF (S ) then set S := S and continue the search with the same neighborhood N k . If |S | = |S | and ObjF (S ) > ObjF (S ), then repeat the search with the same S and the next neighborhood. If |S | = |S | and ObjF (S ) = ObjF (S ), then with probability pmove set S := S and continue search with the same neighborhood N k , and with probability 1 − pmove repeat the search with the same S and the next neighborhood.
4
Experimental results
All tests were performed on a single processor from an Intel Quad 2.5 GHz computer with 1GB memory, under Windows XP operating system. The VNS implementations were coded in C programming language. In our experiments, the following values for the VNS parameters were used: kmin = 2, kmax = 20, pmove = 0.2, itermax = 100. In this section we present the computational results on crew scheduling (CSP) and graph coloring (GCOL) [2] ORLIB graph instances. The VNS has been run 20 times for each instance and the results are summarized in Table
N. Mladenovi´c et al. / Electronic Notes in Discrete Mathematics 39 (2012) 51–57
55
Table 1 Comparison of GA and VNS on CSP instances Inst
n
m
GA
V NS
sol
t[sec]
ttot [sec]
sol
t[sec]
ttot [sec]
csp50
50
173
29
4.108
26.769
29
0.032
0.350
csp100
100
715
61
100.787
528.21
61
0.587
4.901
csp150
150
1355
98
540.584
3166.3
98
3.883
25.258
csp200
200
2543
144
4978.7
8047.6
142
11.810
83.261
csp250
250
4152
178
4941.3
17060
172
30.790
197.573
csp300
300
6108
-
-
-
224
135.132
463.696
csp350
350
7882
-
-
-
237
214.257
816.756
csp400
400
10760
-
-
-
288
231.514
1362.773
csp450
450
13510
-
-
-
316
361.747
2096.719
csp500
500
16695
-
-
-
367
859.872
3474.093
1 and Table 2. The tables are organized as follows: • The first three columns contain the test instance name and the number of nodes and edges, respectively; • The fourth column gives the best GA solution value from [2] (named sol) obtained in 20 runs; • The average running time (t, in seconds) used to reach the final GA solution for the first time is given in the fifth column, while the sixth column ttot shows the average total running time for finishing GA; • The next three columns show the VNS results, presented in the same way as for GA. From Table 1 and Table 2, it is clear that the results obtained by the VNS outperform those obtained by the GA approach from [2], both with respect to the solution quality and the running time. The VNS has made improvements of the GA upper bounds in 2 cases and has produced 15 new upper bounds for large scale instances which were out of reach for the GA method. The VNS running times are one to two orders of magnitude smaller than those of GA.
5
Conclusions
In this paper an efficient variable neighborhood search approach for solving the strong metric dimension problem is presented. The objective function gives the number of pairs of vertices not strongly resolved by the vertices of a set with the given cardinality. The corresponding neighborhood structures allow an effective shaking procedure which successfully diversifies the search process. The local search implementation was conducted very efficiently and it resulted in an excellent overall VNS performance for graphs with up to 300 vertices.
56
N. Mladenovi´c et al. / Electronic Notes in Discrete Mathematics 39 (2012) 51–57
Table 2 Comparison of GA and VNS on GCOL instances Inst
n
m
GA
V NS
sol
t[sec]
ttot [sec]
sol
t[sec]
ttot [sec]
gcol1
100
2487
91
18.119
172.732
91
3.359
10.773
gcol2
100
2487
91
28.443
182.663
91
2.950
10.771
gcol3
100
2482
91
49.108
198.800
91
1.740
10.813
gcol4
100
2503
91
33.926
173.360
91
3.101
10.799
gcol5
100
2450
91
22.734
166.339
91
3.078
10.968
gcol6
100
2537
91
46.603
194.784
91
1.994
10.505
gcol7
100
2505
91
15.826
170.525
91
2.746
10.820
gcol8
100
2479
90
16.111
177.131
90
3.994
10.783
gcol9
100
2486
91
15.676
178.285
91
2.762
10.856
gcol10
100
2506
91
17.327
167.539
91
2.958
10.816
gcol11
100
2467
91
18.700
169.150
91
2.496
10.883
gcol12
100
2531
91
20.990
166.088
91
1.370
10.707
gcol13
100
2467
91
25.398
171.383
91
3.337
10.926
gcol14
100
2524
91
27.909
169.664
91
3.156
10.698
gcol15
100
2528
91
44.315
182.577
91
2.148
10.617
gcol16
100
2493
91
33.121
173.897
91
2.892
10.849
gcol17
100
2503
91
25.073
172.929
91
1.356
10.841
gcol18
100
2472
91
34.893
173.495
91
2.895
10.838
gcol19
100
2527
91
14.407
163.039
91
1.534
10.611
gcol20
100
2420
91
31.080
175.908
91
3.065
11.114
gcol21
300
22482
-
-
-
288
245.705
905.704
gcol22
300
22569
-
-
-
288
334.413
901.708
gcol23
300
22393
-
-
-
289
223.233
908.390
gcol24
300
22446
-
-
-
288
198.009
914.817
gcol25
300
22360
-
-
-
288
292.655
906.743
gcol26
300
22601
-
-
-
288
279.909
912.178
gcol27
300
22327
-
-
-
288
307.033
913.269
gcol28
300
22472
-
-
-
288
226.901
902.212
gcol29
300
22520
-
-
-
288
268.638
973.929
gcol30
300
22543
-
-
-
288
319.898
964.425
An experimental comparison with the only existing heuristic approach based on the genetic algorithm indicates the superiority of the VNS approach with respect to the solution quality and running time. Further research could be directed to testing on more powerful and/or parallel computers as well as to using the computational results to generate theoretical hypotheses about the strong metric dimension.
References [1] Hansen, P., N. Mladenovi´c and J.A. Moreno-P´erez, Variable neighbourhood search: algorithms and applications, Annals Oper. Res. 175 (2010), 367–407. ˇ [2] Kratica, J., V. Kovaˇcevi´c-Vujˇci´c, and M. Cangalovi´ c, Computing strong metric
N. Mladenovi´c et al. / Electronic Notes in Discrete Mathematics 39 (2012) 51–57
57
dimension of some special classes of graphs by genetic algorithms, Yugoslav J. Oper. Res. 18 (2008), 43–51. ˇ [3] Kratica, J., V. Kovaˇcevi´c-Vujˇci´c, M. Cangalovi´ c, M. Stojanovi´c, Minimal doubly resolving sets and the strong metric dimension of Hamming graphs, Applicable Analysis and Discrete Mathematics 6 (2012), 63–71. ˇ [4] Kratica, J., V. Kovaˇcevi´c-Vujˇci´c, M. Cangalovi´ c, M. Stojanovi´c, Minimal doubly resolving sets and the strong metric dimension of some convex polytopes, Appl. Math. Comput. 218 (2012), 9790–9801. [5] May, T., and O.R. Oellermann, The strong metric dimension of distance hereditary graphs, J. Comb. Math. Comb. Comput., in press [6] Mladenovi´c, N., and P. Hansen, Variable neighbourhood search, Comp. & Oper. Res. 24 (1997), 1097–1100. ˇ [7] Mladenovi´c, N., J. Kratica, V. Kovaˇcevi´c-Vujˇci´c, and M. Cangalovi´ c, Variable neighborhood search for metric dimension and minimal doubly resolving set problems, Euro. J. Oper. Res. 220 (2012), 328–337. [8] Oellermann, O.R., and J. Peters-Fransen, The strong metric dimension of graphs and digraphs, Discr. Appl. Math. 155 (2007), 356–364. [9] Seb¨o, A., and E. Tannier, On metric generators of graphs, Math. & Oper. Res. 29 (2004), 383–393.
Electronic Notes in Discrete Mathematics 39 (2012) 51–57 www.elsevier.com/locate/endm
Variable neighborhood search for the strong metric dimension problem Nenad Mladenovi´c a,1 Jozef Kratica b,2 ˇ Vera Kovaˇcevi´c-Vujˇci´c c,3 Mirjana Cangalovi´ c c,4 a b
c
Department of Mathematics, SISCM, Brunel University, London, UK
Mathematical Institute, Serbian Academy of Sciences and Arts, Kneza Mihaila 36/III, 11000 Belgrade, Serbia
Faculty of Organizational Sciences, University of Belgrade, Jove Ili´ca 154, 11000 Belgrade, Serbia
Abstract We consider a variable neighborhood search approach for solving the strong metric dimension problem. The proposed method is based on the idea of decomposition and it is characterized by suitably chosen neighborhood structures and efficient local search. Computational experiments on ORLIB instances show that the new approach outperformes a genetic algorithm, the only existing heuristic in the literature for solving this problem. Keywords: Strong metric dimension, metaheuristics, combinatorial optimization.
This research was partially supported by Serbian Ministry of Science under grants 174010 and 174033. 1 Email: [email protected] 2 Email: [email protected] 3 Email: [email protected] 4 Email: [email protected] 1571-0653/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2012.10.008
52
1
N. Mladenovi´c et al. / Electronic Notes in Discrete Mathematics 39 (2012) 51–57
Introduction
The strong metric dimension problem (SMDP) was introduced by Sebo and Tannier [9] and further investigated by Oellermann and Peters-Fransen [8]. A vertex w strongly resolves two vertices u and v if u belongs to a shortest v − w path or v belongs to a shortest u − w path. A vertex set S of G is a strong resolving set of G if every two distinct vertices of G are strongly resolved by some vertex of S. A strong metric basis of G is a strong resolving set of the minimum cardinality. Now, the strong metric dimension of G, denoted by sdim(G) is defined as the cardinality of its strong metric basis. The strong metric dimension is NP-hard in a general case, see [8]. The first metaheuristic approach based on the genetic algorithm (GA) is proposed in [2]. Recently, the strong metric dimension of some special classes of graphs has been studied theoretically, such as distance hereditary graphs [5], Hamming graphs [3] and some convex polytopes [4]. Example 1.1 Consider the graph G of Figure 1. The set S = {A, B} is a strong resolving set of G. Indeed, in the case when v ∈ S then w = v strongly resolves u and v, for all u. Similarly, when u ∈ S then w = u strongly resolves u and v, for all v. In the case when u = C and v = D, then w = B strongly resolves u and v. Similar result holds for u = D and v = C. The last case when u = v ∈ {C, D} is trivial, since the shortest path of zero length can be extended to the shortest u − w path for any w ∈ S. It is easy to see that a singleton set cannot be a strong resolving set, since the u − v path with u = A, v = D or u = B, v = C cannot be extended to a shortest u − w path, with w in a singleton set. Therefore sdim(G) = 2.
Fig. 1. Graph G in Example 1.1
N. Mladenovi´c et al. / Electronic Notes in Discrete Mathematics 39 (2012) 51–57
2
53
Mathematical formulation
The integer linear programming (ILP) formulation of the strong metric dimension problem was proposed in [3]. Given a simple connected undirected graph G = (V ,E), where V = {1, 2, . . . , n}, |E| = m, it is easy to determine the length d(u, v) of a shortest u − v path for all u, v ∈ V , using any shortest path algorithm. Then, the coefficient matrix can be defined as:
A(u,v),i
⎧ ⎪ ⎨1, d(u, i) = d(u, v) + d(v, i) = 1, d(v, i) = d(v, u) + d(u, i) ⎪ ⎩ 0, otherwise
where 1 ≤ u < v ≤ n, 1 ≤ i ≤ n. Let the variable yi determine whether vertex i belongs to a strong resolving set S, i.e. yi = 1 for i ∈ S, yi = 0 otherwise. The ILP model of the strong metric dimension problem can now be formulated as follows, min
n
yi
(1)
i=1
subject to: n
A(u,v),i · yi ≥ 1
1≤u
(2)
i=1
yi ∈ {0, 1}
1≤i≤n
(3)
The objective function (1) represents the cardinality of a feasible solution S. Constraints (2) make sure that for each two vertices u and v there exists at least one vertex w from S which strongly resolves u and v, i.e. that S is a strong resolving set. Constraints (3) reflect the binary nature of decision variables yi .
3
VNS for the strong metric dimension problem
Variable neighborhood search (VNS) is an effective metaheuristic introduced in [6]. The basic idea of VNS is to use more than one neighborhood structure and to proceed with their systematic change within a local search. A detailed description of different VNS variants is out of the scope of this paper and can be found in [1].
54
N. Mladenovi´c et al. / Electronic Notes in Discrete Mathematics 39 (2012) 51–57
The VNS approach for the SMDP is based on the idea of decomposition described in [7]. The initial set S is obtained by a simple procedure, which starts from the empty set and adds randomly chosen vertices from V until S becomes a strong resolving set. For a given strong resolving set S the last element is deleted, to obtain set S . The objective function value ObjF (S ) is computed as the number of pairs of vertices from V which are not strongly resolved with respect to S . The following steps are repeated until the stopping criterion is met. For a given k set S in k-th neighborhood N k (S ) is obtained using the shaking function. Starting from S the local search procedure tries to improve S and updates S whenever a new strong resolving set with smaller cardinality is generated. The neighborhood N k (S ) contains all sets obtained from S by deleting k of its elements and replacing them by k elements from V \S . In the local search procedure, starting with S , one element from set S is interchanged with one element of its complement. The best improvement strategy is used, i.e. in every step an interchange is performed, which gives the maximal decrease of the objective function. Whenever the improved set is a strong resolving set, the current set S is updated, the new set S is obtained by deleting the last element, and the procedure continues. The local search procedure stops when there is no improvement. After the local search procedure, there are three possibilities. If |S | < |S | or ObjF (S ) < ObjF (S ) then set S := S and continue the search with the same neighborhood N k . If |S | = |S | and ObjF (S ) > ObjF (S ), then repeat the search with the same S and the next neighborhood. If |S | = |S | and ObjF (S ) = ObjF (S ), then with probability pmove set S := S and continue search with the same neighborhood N k , and with probability 1 − pmove repeat the search with the same S and the next neighborhood.
4
Experimental results
All tests were performed on a single processor from an Intel Quad 2.5 GHz computer with 1GB memory, under Windows XP operating system. The VNS implementations were coded in C programming language. In our experiments, the following values for the VNS parameters were used: kmin = 2, kmax = 20, pmove = 0.2, itermax = 100. In this section we present the computational results on crew scheduling (CSP) and graph coloring (GCOL) [2] ORLIB graph instances. The VNS has been run 20 times for each instance and the results are summarized in Table
N. Mladenovi´c et al. / Electronic Notes in Discrete Mathematics 39 (2012) 51–57
55
Table 1 Comparison of GA and VNS on CSP instances Inst
n
m
GA
V NS
sol
t[sec]
ttot [sec]
sol
t[sec]
ttot [sec]
csp50
50
173
29
4.108
26.769
29
0.032
0.350
csp100
100
715
61
100.787
528.21
61
0.587
4.901
csp150
150
1355
98
540.584
3166.3
98
3.883
25.258
csp200
200
2543
144
4978.7
8047.6
142
11.810
83.261
csp250
250
4152
178
4941.3
17060
172
30.790
197.573
csp300
300
6108
-
-
-
224
135.132
463.696
csp350
350
7882
-
-
-
237
214.257
816.756
csp400
400
10760
-
-
-
288
231.514
1362.773
csp450
450
13510
-
-
-
316
361.747
2096.719
csp500
500
16695
-
-
-
367
859.872
3474.093
1 and Table 2. The tables are organized as follows: • The first three columns contain the test instance name and the number of nodes and edges, respectively; • The fourth column gives the best GA solution value from [2] (named sol) obtained in 20 runs; • The average running time (t, in seconds) used to reach the final GA solution for the first time is given in the fifth column, while the sixth column ttot shows the average total running time for finishing GA; • The next three columns show the VNS results, presented in the same way as for GA. From Table 1 and Table 2, it is clear that the results obtained by the VNS outperform those obtained by the GA approach from [2], both with respect to the solution quality and the running time. The VNS has made improvements of the GA upper bounds in 2 cases and has produced 15 new upper bounds for large scale instances which were out of reach for the GA method. The VNS running times are one to two orders of magnitude smaller than those of GA.
5
Conclusions
In this paper an efficient variable neighborhood search approach for solving the strong metric dimension problem is presented. The objective function gives the number of pairs of vertices not strongly resolved by the vertices of a set with the given cardinality. The corresponding neighborhood structures allow an effective shaking procedure which successfully diversifies the search process. The local search implementation was conducted very efficiently and it resulted in an excellent overall VNS performance for graphs with up to 300 vertices.
56
N. Mladenovi´c et al. / Electronic Notes in Discrete Mathematics 39 (2012) 51–57
Table 2 Comparison of GA and VNS on GCOL instances Inst
n
m
GA
V NS
sol
t[sec]
ttot [sec]
sol
t[sec]
ttot [sec]
gcol1
100
2487
91
18.119
172.732
91
3.359
10.773
gcol2
100
2487
91
28.443
182.663
91
2.950
10.771
gcol3
100
2482
91
49.108
198.800
91
1.740
10.813
gcol4
100
2503
91
33.926
173.360
91
3.101
10.799
gcol5
100
2450
91
22.734
166.339
91
3.078
10.968
gcol6
100
2537
91
46.603
194.784
91
1.994
10.505
gcol7
100
2505
91
15.826
170.525
91
2.746
10.820
gcol8
100
2479
90
16.111
177.131
90
3.994
10.783
gcol9
100
2486
91
15.676
178.285
91
2.762
10.856
gcol10
100
2506
91
17.327
167.539
91
2.958
10.816
gcol11
100
2467
91
18.700
169.150
91
2.496
10.883
gcol12
100
2531
91
20.990
166.088
91
1.370
10.707
gcol13
100
2467
91
25.398
171.383
91
3.337
10.926
gcol14
100
2524
91
27.909
169.664
91
3.156
10.698
gcol15
100
2528
91
44.315
182.577
91
2.148
10.617
gcol16
100
2493
91
33.121
173.897
91
2.892
10.849
gcol17
100
2503
91
25.073
172.929
91
1.356
10.841
gcol18
100
2472
91
34.893
173.495
91
2.895
10.838
gcol19
100
2527
91
14.407
163.039
91
1.534
10.611
gcol20
100
2420
91
31.080
175.908
91
3.065
11.114
gcol21
300
22482
-
-
-
288
245.705
905.704
gcol22
300
22569
-
-
-
288
334.413
901.708
gcol23
300
22393
-
-
-
289
223.233
908.390
gcol24
300
22446
-
-
-
288
198.009
914.817
gcol25
300
22360
-
-
-
288
292.655
906.743
gcol26
300
22601
-
-
-
288
279.909
912.178
gcol27
300
22327
-
-
-
288
307.033
913.269
gcol28
300
22472
-
-
-
288
226.901
902.212
gcol29
300
22520
-
-
-
288
268.638
973.929
gcol30
300
22543
-
-
-
288
319.898
964.425
An experimental comparison with the only existing heuristic approach based on the genetic algorithm indicates the superiority of the VNS approach with respect to the solution quality and running time. Further research could be directed to testing on more powerful and/or parallel computers as well as to using the computational results to generate theoretical hypotheses about the strong metric dimension.
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57
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