A TABU SEARCH ALGORITHM FOR THE GENERALIZED MINIMUM SPANNING TREE PROBLEM

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1 !#"$% $$ %!& '($! *)!!#% $)$ +-,/ <CB5D/EF6HG <C,/I-9AB5G 91J <C25D/.K9L;=<5,/2MG,F68.12C0 9:6HG J G684 A TABU SEARCH ALGORITHM FOR THE GENERALIZED MINIMUM SPANNING TREE PROBLEM Fernando de Cristo (UFSM/PPGEGP) fernandodecristo@yahoo.com.br Felipe Muller (UFSM/PPGEP) felipe@inf.ufsm.br Luciano Ferreira (Unicruz) lferreira@unicruz.edu.br Denis Borenstein (UFRGS/PPGA) denisb@ea.ufrgs.br The generalized minimum spanning tree problem (GMST) is present in many situations of the real world, such as in the context of telecommunications, transports and data clustering, where a network of clusters needs to be connected using one node from each cluster. In this work is presented the design and implementation of a tabu search algorithm for the GMST problem. In our computational tests on 150 TSPLIB instances generated with the Center and Grid Clustering grouping procedure, our tabu search algorithm found the optimal solution in 146 instances in a very low computational effort. Keywords: Generalized Minimum Spanning Tree. Graph Optimization. Meta-heuristics

2 STUTUTVTUWYX ZC[(WY\X TU]^WY\_=`](W a-zc[(zcw `Z=b^ced c^f^g hi5j^d klm+cc^n^d c^+c+cj(d c^nekc^fob^pv+cj^ki5d b^c(hrqmkc^kn^+-qm+cc^i 1 Introduction The generalized minimum spanning tree problem (GMST) is a generalization of the minimum spanning tree problem. This network design problem finds several practical applications, especially when one considers the design of a large-capacity backbone network connecting several individual networks (GOSH, 2003a; HAOUARI and CHAOUACHI, 2006), some real life agricultural settings related to construction of irrigation networks in environments with lack of water (DROR et al., 2000),. The problem was first introduced by Myung et al. (1995), who have show that it is NP-hard by reduction from the vertex cover problem, and even finding a constant factor approximation algorithm is NP-hard. This result suggests that a polynomial algorithm is unlikely to exist for the GMST problem. The section 3 analyses some papers that propose some heuristic and meta-heuristic approaches to the GMST that suggests that new approaches can be explored and new better results can be obtained mainly considering medium and large size instances. Therefore, a new tabu search and path relinking algorithm for the GMST problem is presented. We did not find anything similar in the literature so far. This paper is organized as follows: section 2 presents a formal definition of GMST problem; section 3 shows a review of literature, presenting some related works, as well some ideas that gave theoretical support to our assumptions; section 4 details the proposed algorithms; section 5 discuss the computational results; and finally section 6 presents some conclusions. 2 The GMST problem The generalized minimum spanning tree problem (GMST) is defined on an undirected graph G = G(V, E), where V = {1,, n} is a vertex set, E = {(i, j): i, j V, i < j} is an edge set, and V is the union of K clusters V k, (k = 1,,K). A non negative cost matrix C = (c ij ) is associated with E. The GMST problem consists of designing a minimum cost tree consisting at least one vertex from each cluster (FEREMANS et al., 2001). The Figure 1 shows a graphic representation for a solution of a GMST problem with 12 vertexes and 4 clusters. Figure 1 Graphic Representation for a Solution of a GMST problem with 12 vertexes and 4 clusters Font: Ferreira et al. (2006) 2

3 STUTUTVTUWYX ZC[(WY\X TU]^WY\_=`](W a-zc[(zcw `Z=b^ced c^f^g hi5j^d klm+cc^n^d c^+c+cj(d c^nekc^fob^pv+cj^ki5d b^c(hrqmkc^kn^+-qm+cc^i 3 Literature Review This section presents some related works that allow us to go deeper in this problem and propose a new tabu search and path relinking algorithm. We will present the related papers in order of importance for our work. Golden et al. (2005) proposed a local search heuristic (LSH) and a genetic algorithm (GA) for the GMST problem which were tested on the TSPLIB instances generated by Feremans (2001). The genetic algorithm generates new solutions (offspring) using one of the four operators crossover, tree separation, (random) mutation, and local search (mutation). The chromosomes represents the generalized spanning tree through an array of size K, where the ith entry (gene value) indicates the node selected to represent the cluster i. To generate the initial population, it selects one node from each cluster randomly and then construct a minimum spanning tree on these nodes using Prim s algorithm. Haouari and Chaouachi (2006) proposed a GA where they used the same solution encoding (chromosome) that Golden et al. (2005). The initial population is generated by the PROGRES heuristic, described in details at Haouari e Chaouachi (2002). The crossover operator is the two-point crossover, where the two points are randomly selected. Such as Golden et al. (2005) they use a random mutation operator. In order to improve the overall computational performance they restrict the search to the subset of edges generated in a preprocessing step of the PROGRES heuristic. In the GA proposed by Dror et al. (2000) a trial solution is represented by a unique coded binary string X of length n (number of the nodes) called individual (chromosome) where x i = 1 if the tree spans node i, and 0 otherwise. The initial population is generated by an insertion heuristic similar to the Prim s algorithm. It uses the single-point crossover operator and a random mutation. Ghosh (2003b) proposed a probabilistic tabu search algorithm where the basic idea behind it was to use preprocessing operations to arrive at a probability value for each vertex which roughly corresponds to its probability of being included in an optimal solution, and to use such probability values to shrink the size of neighborhood of solutions to manageable proportions. This method almost always provides the best quality solution for small to medium sized instances while the large instances are discussed in Ghosh (2003a). Wang et al. (2006) also presented a tabu search algorithm for the GMST problem. The algorithm was implemented by three phases: basic tabu search phase, intensification phase and diversification phase. The procedure randomly generates a feasible initial solution then a cluster is selected and all nodes in the selected cluster are replaced in order to generate new solutions. They use as tabu attribute the cluster visited by recent iterations and a fixed size tabu list. To avoid solutions being trapped in local optimal, they applied a threshold strategy to increase diversity of the intensive search space. Ferreira et al. (2006) proposed versions of GRASP heuristics with adaptive memory to the GMST problem. We focus our attention in the presented constructive heuristics (named, C1, C2, C3 and C4) and in the use of path relinking for the GMST problem. The path relinking technique, present by Glover et al. (2000), in common with other evolutionary methods, operates with a population of solutions, rather than with a single solution, and employ procedures for combining these solutions to create new and sometimes better ones. The main idea is the incorporation of attributes from elite parents in partially or fully constructed solutions aiming another solutions (called solution targets). It is believed that better solutions could be found in the path between those solutions. 4 Proposed Tabu Search (TS) Tabu Search (GLOVER and LAGUNA, 1997) is a meta-heuristic that has as a distinguishing feature, its exploitation adaptive forms of memory, that gives to it the ability to penetrate complexities that often confound alternative approaches. TS guides a local heuristic search 3

4 STUTUTVTUWYX ZC[(WY\X TU]^WY\_=`](W a-zc[(zcw `Z=b^ced c^f^g hi5j^d klm+cc^n^d c^+c+cj(d c^nekc^fob^pv+cj^ki5d b^c(hrqmkc^kn^+-qm+cc^i procedure to explore the solution space beyond local optimality. The local procedure is a search that uses an operation called move to define the neighborhood of any given solution. One of the main components of TS is its use to adaptive memory, which creates a more flexible search behavior. TS is based on the premise that problem solving, in order to qualify as intelligent, must incorporate adaptive memory and responsive exploration. To show these aspects we will present in the sequence the details of our procedure. 4.1 Tabu List Our TS uses a tabu list to keep track of solutions attributes that have changed during the recent past of the search. The tabu list tries to avoid solutions being trapped in local optima and cycling. We consider tabu active the vertexes that participate were evolved in the moves, this tabu status remains for x iterations, where x is randomly chosen in the uniformly distributed interval between 1 and the number of clusters. 4.2 Construction of an Initial Solution TS needs and initial solution to start the procedure of guiding the local search procedure trying to find better solution. The initial solution was generated by the use of three constructive algorithms, that are: - Constructive C1 and C2, by Ferreira et al. (2006); - Adaptation of Prim s, as showed by Ferreira et al. (2006). To each instance to be solved we run the three constructive algorithms followed by the local search procedure proposed by Golden et al. (2005). Then we choose the best solution among the three as initial one for the TS algorithm. The constructive algorithm that presented the best performance was C1, showing an average distance ratio from the optimal solution of 7.45%. However, the best combination was C2+local search procedures with an average distance ratio from the optimal solution of 1.33% 4.3 Interactive Path Relinking Similarly to the tabu search fundamentals, we propose an intensification phase in order to improve solution quality. The Path Relinking (PR) is applied at the end of each TS iteration, using the current solution as the initial point of the path and a population of elite solutions generated and kept updated during the search procedure. 5 Experiments and Results In this section we present the computational experiments performed as well the comparisons with other methods, analysis of the results and some conclusions obtained. Following, we discuss the set of instances that were used and its features to show how representative they are. A huge amount of computational experiments were done to bias the meta-heuristic parameters used in all tests and results presented here. The results obtained by the combination of tabu search and path relinking (TS+PR) are presented and compared with the best results found on the literature. The instances for the GMST problem, proposed by Fischetti et al. (1995), were used in our computational experiments, mainly because they allow comparison with other methods and have optimal solution available. The instances are generated using as base some classical traveling salesman problem instances from TSPLIB, with the clusters generated by two grouping schemes named Center Clustering and Grid Clustering (in which is necessary a user 4

5 STUTUTVTUWYX ZC[(WY\X TU]^WY\_=`](W a-zc[(zcw `Z=b^ced c^f^g hi5j^d klm+cc^n^d c^+c+cj(d c^nekc^fob^pv+cj^ki5d b^c(hrqmkc^kn^+-qm+cc^i defined parameter, called ). The same set of instances, or a subset of it, was used by Feremans (2001 and 2005), Feremans et al. (1999 and 2001), Golden et al. (2005) and Ferreira et al. (2006). The set studied in this paper is composed by 150 instances with the optimal value known. All instances are connected graphs, partitioned in k clusters, where k varies between 7 and 76, while the number of vertexes varies between 47 and 226. The number of vertexes in each cluster is not fixed and each edge as a weight (cost or distance) associated to it. The name of instances follows the code: namev, where v is the number of vertexes. The number of clusters generated is presented as a colum named k. Our TS+PR algorithm was coded in JAVA and compiled with JDK 1.6. The results were obtained using an Athlon 64 X2 2.0GHz, 2GB RAM with operational system Linux The results showed on the tables for TS+PR are the average of 5 executions for each instance. We stop the algorithm when the optimal solution is found or a time limit of 500 seconds is reached. The tabu list parameters, presented on section 4.1, were obtained empirically, aiming a good relationship between solution quality and computation effort. At the begging only TS algorithm was applied and the results showed that some more work were necessary. The PR was included, former only with the incumbent solution obtained at the end of TS and a population of elite solutions generated and kept updated during the search procedure. These strategy reduces the average deviation ration from the optimal solution from 0.12% (obtained by TS only) to 0.06%. Finally, we perform PR at the current solution obtained at the end of each TS iteration, this strategy reaches the optimal solution in all but four tested instances, in a very low computation effort. Table 1 shows a comparison between our TS+PR algorithm and the results obtained by Feremans (2001), Golden et. al. (2005) e Ferreira et al. (2006), for center clustering generated set of instances. All presented times are in seconds. The work of Ghosh (2003) and Wang et al. (2006), were not included in the tests and comparisons because they use a different set of instances that were still not made available to us by some of the authors. All methods found the optimal solution for the tested instances but we can see a large advantage of TS+PR related to the computation effort. Instance k Optimal Value Time Feremans, 2001 Time GRASP+RC (Ferreira, 2006) Time LS (Golden et al.,2005) Time TS+PR spain europ gr gr gr att gr hk eil brazil st eil pr gr

6 STUTUTVTUWYX ZC[(WY\X TU]^WY\_=`](W a-zc[(zcw `Z=b^ced c^f^g hi5j^d klm+cc^n^d c^+c+cj(d c^nekc^fob^pv+cj^ki5d b^c(hrqmkc^kn^+-qm+cc^i rat kroa krob kroc krod kroe rd eil lin pr gr pr bier pr gr pr kroa krob pr u rat kroa krob Table 1 Results Obtained (center clustering generated instances) Table 2 presents the results of our TS+PR algorithm compared with Feremans (2001) and Golden et. al. (2005) for instances generated by Grid Clustering method with = 3 e = 5. Ferreira et al. (2006) did not present results for this kind of instances. The columns named Feremans, Golden and TS+PR presents the execution time in seconds and the column Optm. presents the optimal solution value. TS+PR did not found the optimal solution value only in 4 instances: pr136 ( ), rat195 (1113.8), kroa200 ( ) e krob200 ( ), but still is much faster than the other methods. The cells marked with * indicates that such method did not find the optimal solution for the instance listed in that line. Instance = 3 k Optm. Feremans Golden TS + PR k Optm. Feremans Golden TS + PR att eil st eil pr gr rat kroa krob kroc krod kroe rd = 5 6

7   U U V U Y C ( Y U ^ Y =š ( - C ( C š =œ^ ež ^Ÿ^ 5 ^ž m C ^ ^ž ^ C C (ž ^ e ^Ÿoœ^ V C ^ 5ž œ^ ( r M ^ ^ - M C ^ ªL«v P 8 ±v² ³ Hµ5 P # v 1 ¹vºH» 1«v¼L²# v» ² ½ v ¹ #² ³ v«v» «8 3 P v ² 3» v«1¾ v 8 PºH ²F«1 P¹1 3²» «v² Àv» ½» eil lin pr pr bier pr * * gr pr kroa krob pr u rat * kroa * * krob * * Table 2 Results Obtained (grid clustering generated instances for Á = 3 e Á = 5) Table 3 presents the same comparison of Table 2 using instances generated by Grid Clustering method with Á = 7 e Á = 10. TS+PR found the optimal solution value for all presented instances with less computational effort. Following the same systematic, cells marked with * indicates that such method did not find the optimal solution for the instance listed in that line. Instance = 7 k Optm. Feremans Golden TS + PR k Optm. Feremans Golden TS + PR att eil st eil pr gr rat kroa krob kroc krod kroe rd eil lin pr pr bier pr gr pr kroa krob pr u = 10 7

8 ÃÄUÄUÄVÄUÅYÆ ÇCÈ(ÅYÉÆ ÄUÊ^ÅYÉË=ÌÊ(Å Í-ÇCÈ(ÇCÅ ÌÇ=Î^ÏeÐ Ï^Ñ^Ò ÓÔ5Õ^Ð Ö møcï^ù^ð Ï^ØCØCÕ(Ð Ï^ÙeÖÏ^ÑoÎ^ÚVØCÕ^ÖÔ5Ð Î^Ï(ÓrÛMÖÏ^ÖÙ^Ø-ÛMØCÏ^Ô ÜLÝvÞPß à8áâ ãvä âåæhç5þpè#évß æ1ê ëvìhâ í æ1ývîlä#êví ä ï ævà ë Þ#ä åævývàí Ý8â3ÞPàvß ä â3í ævý1ð évß æ8ñ ÞPìHâäFÝ1êèPë1è â3ä í Ývä òví ï íâ á rat * kroa krob pr Table 3 Results Obtained (grid clustering generated instances for ó = 7 e ó = 10) We can notice that TS+PR found the optimal solution for all Center Clustering generated instances and for all but four, Grid Clustering generated instances. This shows a similar performance than the other methods related to solution quality but TS+PR is much faster. 6 Conclusions This work presented a new meta-heuristic approach for the GMST problem. Since GMST is proved to be NP-hard and the combination of tabu search and path relinking was not yet presented in the literature reviewed, we believed that the proposed method is very robust. Besides that it opens a new way to explore GMST problems and its practical applications, mainly considering from medium to large instances where TS+PR can show its potentiality. The solution quality obtained was very good obtaining the optimal solution in 146 of 150 instances in a very low computational effort as it is desired when we are working in a meta-heuristic field. Even in the four instances that TS+PR was not succeed in find the optimal solution it was so close that we can consider it very good for real life applications. Acknowledgments The authors would like to thanks Cristiane Maria Santos Ferreira for the instances put available to us. The work of Felipe Martins Müller was partially supported by CNPq. References Dror, M., Haouari, M. and Chaouachi, J. Generalized spanning trees. European Journal of Operational Research, v. 120, p , Feremans, C. Generalized Spanning Trees and Extensions. Doctor Thesis, Université Libre de Bruxelles, Feremans, C., Lodi, A., Toth, P. and Tramotani, A. Improving on Branch-and-Cut Algorithms for Generalized Minimum Spanning Trees. Pacific Journal of Optimization, 1, , Feremans, C., Labbé, M. and Laporte, G. On Generalized Minimum Spanning Trees. European Journal of Operational Research, n. 134, p , Feremans, C., Labbé, M. and Laporte, G. The Generalized Minimum Spanning Tree Problem: Polyhedral Analisyzans Branch-and-Cut Algorithm. Eletronic Notes in Discrete Mathematics, n. 3, p , Ferreira, C.M.S., Ochi, L.S. and Macambira, E.M. GRASP com Memória Adaptativa para o Problema da Árvore de Cobertura Mínima Generalizado. Annals of XXXVIII Brazilian Symposium of Operational Research SBPO, Goiânia, Fischetti, M., Salazar, J. J. and Toth, P. The Symmetric Generalized Traveling Salesman Polytope. Networks, n. 26, p , Ghosh D. Solving Medium to Large Sized Euclidean Generalized Minimum Spanning Tree Problems. IIMA Working Papers , Indian Institute of Management Ahmedabad, Research and Publication Department, 2003a. 8

9 ôõuõuõvõuöy øcù(öyú õuû^öyúü=ýû(ö þ-øcù(øcö ýø=î^ïeð Ï^Ñ^Ò ÓÔ5Õ^Ð Ö møcï^ù^ð Ï^ØCØCÕ(Ð Ï^ÙeÖÏ^ÑoÎ^ÚVØCÕ^ÖÔ5Ð Î^Ï(ÓrÛMÖÏ^ÖÙ^Ø-ÛMØCÏ^Ô ÜLÝvÞPß à8áâ ãvä âåæhç5þpè#évß æ1ê ëvìhâ í æ1ývîlä#êví ä ï ævà ë Þ#ä åævývàí Ý8â3ÞPàvß ä â3í ævý1ð évß æ8ñ ÞPìHâäFÝ1êèPë1è â3ä í Ývä òví ï íâ á Ghosh D. A Probabilistic Tabu Search Algorithm for the Generalized Minimum Spanning Tree Problem. IIMA Working Papers , Indian Institute of Management Ahmedabad, Research and Publication Department, 2003b. Glover, F., Laguna, M. and Martí, R. Fundamentals of Scatter Search and Path Relinking. Control and Cybernetics 29(3), , Glover, F. and Laguna, M. Tabu Search. Kluwer Academic Publishers, Golden, B., Raghavan, S. and Stanojevic D. Heuristic Search for the Generalized Minimum Spanning Tree. INFORMS Journal on Computing 17, , Haouari, M. and Chaouachi, J.S. Upper and lower bounding strategies for the generalized minimum spanning tree problem. European Journal of Operational Research 171, , Haouari, M. and Chaouachi, J.S. A probabilistic greedy search algorithm for combinatorial optimization with application to the set covering problem. Journal of the Operational Research Society 53, , Myung, Y.S., Lee, C.H. and Tcha, D.W. On the Generalized Minimum Spanning Tree Problem. Networks, 26, , Wang, Z.; Che, C.H. and Lim, A., Tabu search for generalized minimum spanning tree problem. Trends in Artificial Intelligence, p ,