The degree constrained k-cardinality minimum spanning tree problem: a lexisearch

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1 Decision Science Letters 7 (2018) Contents lists available at GrowingScience Decision Science Letters homeage: The degree constrained k-cardinality minimum sanning tree roblem: a lexisearch algorithm Thenealle Jayanth Kumar a and Singamsetty Purusotham a* a Deartment of Mathematics, School of Advanced Sciences, VIT University, Vellore , Tamil Nadu, India C H R O N I C L E A B S T R A C T Article history: Received January 16, 2017 Received in revised format: May 22, 2017 Acceted July 23, 2017 Available online July 25, 2017 Keywords: k-cardinality minimum sanning tree Lexi-search algorithm Degree constraint This aer deals with the degree constrained k-cardinality minimum sanning tree (k-mstpd) roblem defined on a connected, edge weighted and undirected grah. The aim of this roblem is to determine the least weighted sanning tree with exactly k vertices such that excet the root vertex, no other vertex in the resultant sanning tree exceeds the secified degree limit. The k-mstpd roblem has many ractical alications for the design of electric, communication, and transortation networks. The roblem is then formulated as a zero-one rogramming. In this aer, an exact algorithm known as Lexi-search algorithm (LSA) is develoed to tackle the k-mstpd roblem. Furthermore, the develoed LSA is rogrammed in Matlab and tested on some benchmark instances as well as on random instances and the resective results are reorted. The obtained exerimental results showed that the develoed LSA takes significantly less time to find the otimal solutions Growing Science Ltd. All rights reserved. 1. Introduction The minimum sanning tree (MST) roblem is a significant network combinatorial otimization roblem. The MST roblem is an acyclic subgrah, which connects all the vertices/nodes with the minimum overall associated weight to its edges. The k-cardinality minimum sanning tree roblem (k- MST) is a variant of the classic MST roblem and formally defined as follows: Let G ( N, E) be a weighted, connected and undirected grah having N vertices and E edges, the k-mst roblem determines the least weight sanning tree with exactly k ( k N ) vertices and k 1 edges. The k-mst was first coined by Hamacher et al. (1991) and it was roven to be NP-hard by Ravi et al. (1996). It has several otential alications like oil-field leasing, telecommunications, oen it mining and image rocessing, (see Ravi et al., 1996; Garg & Hochbaum, 1997; Philott & Wormald, 1997; Ma et al., 2000) thus it has received a great attention from the researchers in recent years. * Corresonding author. address: drurusotham.or@gmail.com (S. Purusotham) 2018 Growing Science Ltd. All rights reserved. doi: /j.dsl

2 302 Since the k-mst is NP-hard, develoing exact solution methods is again a challenging roblem. Due to this reason, most of the existing works on k-mst roblem concerns aroximate solution rocedures. However, these aroximate solution methods may not assure the exact solutions. The constant factor aroximation aroach for solving the k-mst roblem was first given by Blum et al. (1995). Subsequently, Garg and Hochbaum (1997) studied the k-mst roblem and suggested an O(log k) aroximation algorithm. A 2.5-factor heuristic algorithm for k-mst has been develoed by Arya and Ramesh (1998). Three meta-heuristics algorithms namely Tabu Search (TS), Ant Colony Otimization (ACO) and Evolutionary Comutation (EC) have been roosed by Blum and Blesa (2005) for the k-mst roblem and showed that the ACO and TS aroach works better for lower and higher cardinalities resectively. Recently, Katagiri et al. (2010) roosed a new solution rocedure using Tabu search for k-mst and shown that this method erforms better than the existing methods. An efficient hybrid aroximation algorithm using Tabu search and Ant colony otimization for solving the k-mst roblem was roosed by Katagiri et al. (2012). Purusotham and Sundara Murthy (2012) suggested the iterative techniques to solve some of the variants of network roblems. A hybrid heuristic aroach based on the Memetic algorithm and Tabu search algorithm was develoed by Katagiri and Guo (2013) for the k-mst roblem. Degree constrained minimum sanning tree (DMST) roblem is one of the variants of classic MST roblem and has been roved to be NP-hard by Kar (1972). The DMST roblem has many ractical alications such as design of comuter and communication networks, electric circuits, design of integrated circuits, road networks and energy networks (Martins & De souza, 2009). Several solution methods have been develoed in order to tackle DMST roblem, for few see, Genetic algorithm by Hanr and Wang (2006), an Ant based algorithm by Doan (2007), Learning based automata aroach by Torkestani (2013), and Branch and cut algorithm by Martinez and Cunha (2014). However, the above cited literature revealed that a limited consideration has been given to the variants of the k-mst roblem due to NP-hard nature. Infact, many ractical scenarios can be modeled as variants of k-mst roblem. In this aer, a variant of the k-mst roblem called the degree constrained k-cardinality minimum sanning tree roblem (k-mstpd) is considered and an exact solution rocedure known as Lexi-search algorithm (LSA) is develoed, which rovides the otimal solutions effectively. Without loss of generality, relaxing the degree constraint and assuming k n, the k-mstpd corresonds to classic MST roblem. Our aim is to construct the model and develo an exact Lexisearch algorithm for the k-mstpd. The rest of the aer is structured as follows: The roblem is described and then it is formulated as a zero-one integer rogramming in Section 2. Section 3 includes the concets and solution rocedure of the roosed LSA algorithm. A numerical examle is given in Section 4 to demonstrate the roosed LSA. Section 5 resents the comutational results and subsequently, the conclusions are discussed in Section Mathematical Model Let G ( N, E) be a connected, edge weighted and undirected grah, where N and E be the vertex/node set and edge set resectively. Each edge in E is associated with a non-negative weight known as cost. Given the secified ositive integers k and D, the k-mstpd aims to find the least cost sanning tree

3 T. J. Kumar and S. Purusotham / Decision Science Letters 7 (2018) 303 with exactly k ( k N ) vertices such that excet the root vertex, no other vertex has more than the redefined degree D. The notations and arameters which are useful for formulating the model k- MSTPD are given as follows: N n: Number of vertices/nodes. c : Associated cost fromi th node to j th node, where c c. k : Cardinality i.e. secified number of vertices, where k n D : Secified degree limit on vertices. : Root vertex. x : Binary variable. ji The mathematical model for k-mstpd using zero one rogramming is given as follows: n n min Z c x (1) i 1 j 1 subject to n n x k 1 (2) i 1 j 1 n x D; i j, i, i N (3) j 1 Subtour elimination constraints (4) x 0,1 ; i, j N (5) In this formulation, the constraint (1) reresents the objective function that minimizes the overall cost of the sanning tree. The constraint (2) tells the resultant sanning tree ossess k-1 edges. The constraint (3) ensures that excet the root vertex, no other vertex in the sanning tree is more than a redefined degree limit. The constraint (4) eliminates the subtours. Finally, the integrality constraints are given in (5). 3. Lexi-search Algorithm The Lexi-search algorithm (LSA) is a well organized Branch and Bound (B&B) algorithm suggested by Pandit (1962). In rincile, the LSA has similar features of the Branch and Bound (B&B) algorithm such as checking the feasibility, comuting the bounds, generating a feasible solution. The LSA has been roved to be roductive than the B&B aroach and is based on the following grounds Pandit (1962): It is ossible to list all the solutions in a structural hierarchy which also reflects a hierarchical ordering of the corresonding values of these configurations. Effective bounds can be set to the values of the objective function, when structural combinatorial restraints are laced on the allowable configurations Pattern A two dimensional array [ ] X x = that corresonds to the k- cardinality sanning tree is referred to as a attern. The value of the attern is evaluated by using the Eq. (6).

4 304 n n V( X) c x (6) i 1 j Alhabet Table For any given two dimensional cost array C = [ c ], there exists M = n n ordered airs, which are organized in non-decreasing order and are indexed from 1 to M, stored in an array SN (Sundara Murthy, 1979). Let L = ( b, b,..., b ), b ÎSN be an arranged sequence of indices from SN. The attern 1 2 i indicated through the ordered airs whose indices are reresented by L, which is indeendent of the order of b in the sequence. The indices in the sequence L i can be arranged in increasing order such that b <b, i = 1,2,..., - 1. The ordered sequence L i i+ 1 is said to be a word of length. The set SN is called an Alhabet table. A word L is called sensible word, if b <b, i = 1, 2,..., - 1; in any other case, the i i+ 1 word L becomes non-sensible word Word and Partial Word If = k- 1, then L is said to be a full length word or simly word. If < k- 1, then L indicates artial word. Every full length word L gives a feasible solution Uer and Lower Word The uer bound (UB) of a artial word is assumed to be a very higher value. The lower bound of a artial word ( LB( L )) for the values of the blocks of words indicated by L = ( b, b,..., b ) can be 1 2 determined as follows: LB( L ) = V ( L ) + CC( b + k ) - CC( L ); where V( L ) rovides the value of the artial word and is measured by using V( L ) = V( L ) + C( b ) withv( L ) = Proosed Lexi-search Algorithm The roosed Lexi-search algorithm (LSA) is mainly based on the features such as alhabet table, setting effective bounds, checking the feasibility and backtracking. The algorithmic stes of the roosed LSA for k-mstpd roblem are given as follows: Ste 1: Initialization Inut: C = [ c ] Cost matrix; n number of vertices; k secified number; D secified degree limit; UB an uer bound (99999, say) Ste 2: Alhabet Table Create the alhabet table using the given cost matrix C as discussed in the Section 3.2 and goto Ste 3. Ste 3: Setting effective bounds Usually, the search starts with a artial word L = ( b ) = 1, where = 1. a. Comute the lower bound of the artial word ( LB( L )) as discussed in the Section 3.4 b. If LB( L) < UB, then go to Ste 4.

5 T. J. Kumar and S. Purusotham / Decision Science Letters 7 (2018) 305 c. If LB( L ) ³ UB, then eliminate the artial word L and subsequently remove the blocks of words with L as the leader. Go to Ste 6. Ste 4: Feasibility checking a. If the artial word L satisfies the degree constraint, acyclic and balancing the size of the cardinality k, then it is feasible and infeasible, otherwise. b. If the artial word L is feasible, then accet it and roceed for the subsequent artial word of order + 1 and go to Ste 5. If L is infeasible, ick u the consecutive artial word of same order by considering the letter that succeeds b in its th osition and go to Ste 3. Ste 5: Concatenation a. If L is a artial word, then this will be concatenated by using L = L * b, where * refers the concatenation oeration and go to Ste 3. b. If L is a full length word relace by LB( L ) and go to Ste 7. Ste 6: If all the words of order are exhausted and length of the word L is 1 then move to Ste 9 and go to Ste 7, otherwise. Ste 7: Backtracking Backtracking hels to imrove the solution. The current value ofub is assumed as the uer bound and roceed the search by taking the next letter of the artial word of order -1and go to Ste 3. Ste 8: Reeat the Ste 3 to 7 and discard the feasible or infeasible solutions which are not necessary to obtain the otimal solutions. Continue this rocess untilub has no further imrovement and go to Ste 9. Ste 9: Outut Record currentub and its corresonding full length word L and go to Ste 10. Ste 10: Sto Once the search gets over, the current UB gives an otimal solution and the word L rovides the comlete schedule for k-mstpd roblem. 4. Numerical Illustration In order to illustrate the roosed LSA, a numerical examle is considered for which N n 9, k 7 and D 3. Since the roblem defined on undirected grah, an uer triangular matrix is considered and the cost matrix C = [ c ] is reresented in Table 1. Note that the comonents of the cost matrix C assume the non-negative integral values. For convenience, the artial alhabet table for the cost matrix in Table 1 is shown in Table 2. In Table 2, the arrays SN, C, CC, R and CL reresents the serial number, elements of the cost matrix organized in ascending order, cumulative cost for the elements of the cost matrix, row and column indices resectively. The search table reorted in Table 3 reresents the details about how the roosed LSA comutes the feasible solution and converges to the otimal solution. In Table 3, the column indexed by SN reresents the serial number, the subsequent columns indexed by 1, 2, 3, 4, 5 and 6 indicates the first, second, third, fourth, fifth and sixth osition of the letters in a artial word resectively. The next columns labeled by v and LB tells that value of the artial word and lower bound of a artial word resectively. The successive columns R and CL reresent the row and column indices resectively. Finally, the last column Remark gives whether the artial word is acceted or rejected.

6 306 Since the roblem considered in Table 1 aims to find the least cost sanning tree with exactly k 7 (out of n 9 vertices) vertices and k 6 edges. Thus, the length of the feasible word could be 6 i.e. 6 edges. In Table 3, the LSA algorithm starts with a artial word L = 1, and it satisfies the feasibility conditions, 1 thus the artial word is acceted and denoted by A. It is seen that some of the artial words failed in satisfying the feasibility conditions and hence rejected artial words are reresented by R. Proceeding in this manner it lead to a full length feasible word L = (1, 2, 3, 4, 5,11) which ossess the objective value 6 UB=52 (feasible solution) and it is observed at 11 th row of the Table 3. The grahical reresentation of the feasible solution can be shown in Fig.1. Once the feasible solution is obtained, the current UB is assumed to be the uer bound and roceed for the backtracking in order to enhance the search sace. It is observed that the imroved solution UB=49 is observed at 14 th row of the Table 3 and its corresonding full length word could be L 6 = (1,2,3,4,6,8). Since no other solution is found to be better than the current solution, the current found solution i.e. UB=49 could be the otimal solution and further, the algorithm terminates at 19 th row of the Table 3. The grahical reresentation of otimal solution is given in Fig.2. Table 1 Cost matrix of the k-mstpd roblem C Table 2 Alhabet Table SN C CC R CL SN C CC R CL

7 T. J. Kumar and S. Purusotham / Decision Science Letters 7 (2018) 307 Table 3 Search Table SN V LB R CL Remark A A A A A R R R R R A, UB= A R A, UB = > UB, R > UB, R > UB, R > UB, R > UB, R 5. Comutational Results This section reorts the comutational details of the roosed LSA for the k-mstpd. The LSA was coded in Matlab 2016a and the exerimental analysis was carried out on a 64-bit Microsoft Windows 2010 oerating system with 4 GB of RAM PC and rocessor Intel Core i5 running at a 2.10 GHz. We evaluated the erformance of the roosed LSA by varying distinct arameters in terms of a number of vertices, the cardinality of the tree, the structure of the cost matrix and degree limit on the vertices. To assess the efficiency of the roosed LSA for the k-mstpd, seven benchmark instances taken from TSPLIB which include gr17, gr21, gr24, fri26, bayg29, dantzig42, and gr48. The extensive comutational details are reorted in Table 4. From the results in Table 4, it is seen that for all the 23 cases of seven instances, the exact solutions were found by the LSA within considerable CPU runtime ranging from seconds to seconds. From these results, it is seen that the LSA is caable in finding the otimal solutions for all the cases within ractically allowable runtimes. Note that for all the cases the root vertex is assumed as 1. Furthermore, to measure the effectiveness of the roosed LSA, the comutational exeriments were carried out on a class of 10 randomly generated small and medium test instances ranging from 15 to 100. Each class included 10 instances and overall, a total of 100 test instances with symmetric edge weights are generated and tested. The cost c assumes the randomly generated values over the range 1 to 500, i.e. 1 c 500. The exerimental results summarized in Table 5 concerns the mean results of each size of the instance by the roosed LSA. Table 5 also contains the standard deviation (SD) of CPU runtimes. The results given in Table 5 based on five indeendent runs. In Table 5, the roblems with 15 to 80 vertices with distinct cardinalities (k) could be solved in very less CPU runtimes. The CPU runtimes start rising when the roblems of size 90 or higher are tested. However, these comutational runtimes are fairly reasonable.

8 308 The overall results summarized in Table 4 and Table 5 reveal that roosed LSA could efficiently solve the k-mstpd for the small and medium size of instances. For the roblems of higher dimensions, the CPU runtimes may also be higher. Table 4 Results of the LSA for k-mstpd over benchmark instances Instance N E k D Best sol CPU Runtime (In seconds) Worst time Best time gr gr gr fri bayg dantzig gr Instance benchmark instances taken from TSPLIB; N number of vertices in the given grah; E number of edges in the given grah; k number of vertices in the resultant sanning tree; D degree limit on the vertices; Best sol. the otimal solution by the roosed LSA; Worst time worst comutational runtime (in CPU seconds) for the best found solution; Best time best comutational runtime (in CPU seconds) for the best found solution. Table 5 Mean results of the LSA for k-mstpd over random instances SN N E k D NPT CPU Runtime SD Min. Max. Avg SN roblem number; N number of vertices in the given grah; E number of edges in the given grah; k number of vertices in the resultant sanning tree; D degree limit on the vertices; NPT number of roblems tested on each dimension; Min. minimum CPU runtime (in seconds) to find best found solution for the ten runs; Max. maximum CPU runtime (in seconds) to find best found solution for the ten runs; Avg. mean CPU runtime (in seconds) to find best found solution for the ten runs; SD standard deviation of CPU runtimes. 6. Conclusions In this aer, a ractical variant of MST roblem reffered to as a degree constrained k-cardinality minimum sanning tree roblem (k-mstpd) has been taken into consideration and the model was then

9 T. J. Kumar and S. Purusotham / Decision Science Letters 7 (2018) 309 modeled using zero-one rogramming. The k-mstpd may have several ractical alications, articularly in the areas like designing of networks, electric circuits, and transortation networks. We have roosed an exact solution rocedure namely Lexi-search algorithm for k-mstpd. Through numerical exeriments for various benchmark and randomly generated instances, the erformance of the develoed LSA was measured. The exerimental results reveal that the roosed LSA for the k- MSTPD is found to be roductive to obtain the otimal solutions within considerable CPU runtimes. References Arya, S., & Ramesh, H. (1998). A 2.5-factor aroximation algorithm for the k-mst roblem. Information Processing Letters, 65(3), Blum, A., Ravi, R., & Vemala, S. (1996, July). A constant-factor aroximation algorithm for the k MST roblem. In Proceedings of the twenty-eighth annual ACM symosium on Theory of comuting ( ). ACM. Blum, C., & Blesa, M. J. (2005). New metaheuristic aroaches for the edge-weighted k- cardinality tree roblem. Comuters & Oerations Research, 32(6), Doan, M. N. (2007, Setember). An effective ant-based algorithm for the degree-constrained minimum sanning tree roblem. In Evolutionary Comutation, CEC IEEE Congress on ( ). IEEE. Garg, N., Hochbaum, D. (1997). An O(log k) aroximation algorithm for the k minimum sanning tree roblem in the lane. Algorithmica, 18(1), Hamacher, H.W., Jorsten, K., & Maffioli, F. (1991). Weighted k-cardinality trees. Technical Reort, , Politecnico di Milano, Diartimento di Elettronica, Italy. Hanr, L., & Wang, Y. (2006). A novel genetic algorithm for degree-constrained minimum sanning tree roblem. International Journal of comuter science and Network Security, 6(7A), Kar, R. M. (1972). Reducibility among combinatorial roblems. In Comlexity of comuter comutations ( ). Sringer US. Katagiri, H., Hayashida, T., Nishizaki, I., & Ishimatsu, J. (2010). An aroximate solution method based on tabu search for k-minimum sanning tree roblems. International Journal of Knowledge Engineering and Soft Data Paradigms, 2(3), Katagiri, H., Hayashida, T., Nishizaki, I., & Guo, Q. (2012). A hybrid algorithm based on tabu search and ant colony otimization for k-minimum sanning tree roblems. Exert Systems with Alications, 39(5), Katagiri, H., & Guo, Q. (2013). A Hybrid-Heuristics Algorithm for k-minimum Sanning Tree Problems. In IAENG Transactions on Engineering Technologies ( ). Sringer Netherlands. Ma, B., Hero, A., Gorman, J., & Michel, O. (2000). Image registration with minimum sanning tree algorithm. In Image Processing, Proceedings International Conference on (Vol. 1, ). IEEE. Martinez, L. C., & Da Cunha, A. S. (2014). The min-degree constrained minimum sanning tree roblem: Formulations and Branch-and-cut algorithm. Discrete Alied Mathematics, 164, Martins, P., & de Souza, M. C. (2009). VNS and second order heuristics for the min-degree constrained minimum sanning tree roblem. Comuters & Oerations Research, 36(11), Pandit, S. N. (1962). The loading roblem. Oerations Research, 10(5), Purusotham, S. & Sundara Murthy, M. (2012). A new aroach for solving the network roblems, OPSEARCH, 49(1), Philott, A. B., & Wormald, N. C. (2000). On the Otimal Extraction of Ore from an Oen Cast Mine. Deartment of Engineering Science, University of Auckland. Ravi, R., Sundaram, R., Marathe, M. V., Rosenkrantz, D. J., & Ravi, S. S. (1996). Sanning trees short or small. SIAM Journal on Discrete Mathematics, 9(2),

10 310 Sundara Murthy, M. (1979). Combinatorial rogramming roblems - A attern recognition aroach. Ph.D Thesis, REC, Warangal, India. Torkestani, J. A. (2013). Degree constrained minimum sanning tree roblem: a learning automata aroach. The Journal of Suercomuting, 64(1), TSBLIB: htts://github.com/drozdowski/tsplib.net/tree/master/tsplib95/ts 2018 by the authors; licensee Growing Science, Canada. This is an oen access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (htt://creativecommons.org/licenses/by/4.0/).