Belo Horizonte, Minas Gerais, Brasil. Universidade Federal de Minas Gerais Belo Horizonte, Minas Gerais, Brasil

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1 Detailed computational results for the paper entitled The k-cardinality Tree Problem: Reformulations and Lagrangian Relaxation, under revision for Discrete Applied Mathematics Frederico P. Quintão a,b Alexandre Salles da Cunha b,,1 Geraldo R. Mateus b,2 Abilio Lucena c,3 a Google Engineering Belo Horizonte, Minas Gerais, Brasil b Departamento de Ciência da Computação Universidade Federal de Minas Gerais Belo Horizonte, Minas Gerais, Brasil c Departamento de Administração and Programa de Engenharia de Sistemas e Computação, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brasil 1 Introduction In this manuscript, we report detailed computational results for the paper The k-cardinality Tree Problem: Reformulations and Lagrangian Relaxation. The results presented here were summarized in the main text body of the paper. For the sake of completeness, we describe in Section 2 all instances in our test bed. Two complete sets of computational results are presented in this manuscript: in the first one (see Section 3), we evaluate the quality of the Linear Programming bounds of each proposed formulation, as well as the overall performance Corresponding author addresses: fred@dcc.ufmg.br (Frederico P. Quintão), acunha@dcc.ufmg.br (Alexandre Salles da Cunha), mateus@dcc.ufmg.br (Geraldo R. Mateus), abiliolucena@globo.com (Abilio Lucena ). 1 Alexandre Salles da Cunha was partially funded by CNPq grants / and / Geraldo Robson Mateus was partially funded by CNPq grant / Abilio Lucena was partially funded by CNPq grant / Preprint submitted to Elsevier 21 October 2008

2 of Branch-and-bound algorithms based on them. In the second set (Section 4), we indicate the main results attained by the Lagrangian Relaxation procedure proposed in the paper. 2 Instances Our test bed involves 3 sets of instances. The first set, denoted g, was proposed by Blesa and Xhafa [2] and contains instances corresponding to 4-regular graphs with n ranging from 25 to 400. For all instances in this set, the same cardinality k = 20 was imposed. The second test set, named d, was proposed by Blum and Blesa [4]. Instances in this set are organized in groups, according to the types of graphs they were generated from. The first group of instances in set d, whose names start with a prefix bb, corresponds to grid graphs with n = 225 and k {20, 40, 60, 80}. The second group consists of some instances in set g, where k assumes values other than k = 20. The third group comes from the set of benchmark instances for the Steiner Problem in Graphs [1] and have n {500, 1000} and various values of k. These instances start their names with prefix stein followed by c or d, depending, on the Steiner set of instances in [1] (c or d) they were generated from. The last group of instances in this set are Leighton graphs [4,1], having n = 450 and k {45, 135, 225, 405}. The third and last set of instances in our study, named NWG, consists of nodeweighted grid graphs generated as suggested in [5]. We considered instances with sizes varying from to with integer node weights, uniformly chosen at random from the interval [10, 1000]. For intances in this set, we have fixed k n, since our computational experience suggests these are hard values 2 for the tree cardinality (see Kataoka et al. [6] as well). All our computational testings were performed with a Pentium XEON machine running at 3.0 GHz and with 2 GBytes of RAM memory, under Linux operating system. We used the state of the art CPLEX package, version , under default settings as the MIP solver, to evaluate the two proposed reformulations. 3 Linear Programming Strength and Branch-and-bound computational results In this section, we present the detailed computational results summarized at section of the paper. In Tables 1-3, we present Linear Programming re- 2

3 sults for each set of instances tested here. In the first four columns of these Tables, we indicate each instance name, n, m and k. In the following two columns, we report on the LP bound given by the multicommodity flow reformulation (LP MCFR ) and the time taken (in seconds) to evaluate this bound. Similar entries are presented in the next two columns, for MTZR. In the last column of these Tables, we present the ratio between LP MCFR and LP MTZR, the LP bound given by MTZR. Detailed computational results obtained by MTZR based Branch-and-bound algorithms are presented in Tables 4-7. Due to the difficulties mentioned above to evaluate LP MCFR, we do not quote results for the BB algorithm based on this reformulation. Typically these algorithms involved substantially less nodes but much longer computing times (sometimes more than a week). Detailed results for set d are split in Tables 5 and 6. The first four columns in Tables 4-7 are the instance name, n, m and k. In the next four, we quote the number of nodes in the enumeration tree, the time taken to run the algorithm to completion (in seconds), the best upper bound found during the search (BestUB), and, finally the status of the BB algorithm when it ended. An status OPT and OFM respectively indicates that CPLEX solved the instance to proven optimality and that CPLEX ran out of memory before completing it. An indication (+) right after OPT and OFM means that the upper bound found by CPLEX improves on the best previously known. As it can be appreciated from the Tables, 67 new optimality certificates were given. In particular, 16 new best known upper bounds were presented for instances in set d. 4 Lagrangian Relaxation computational results In this Section, we present detailed computational results summarized in Section 3.6 of the paper. Detailed results obtained by the Lagrangian Relaxation procedure are presented in Tables Due to the large amount of data, results for set d are split in Tables 9 and 10. The first four columns in these Tables are the instance name, n, m and k. In the next four columns, we present specific results attained by the method: the best dual bound, z d, the best Lagrangian upper bound, z, the implied duality gap,, the proportion of edges we managed to price out and the time (in seconds) taken to run the algorithm, t(s). In the following three columns, we present the ratio between z d and LP MCFR, the upper bound implied by the Kruskal Dynamic Tree algorithm of [3] and, finally, the best known upper bounds (BKV). z z d z d 3

4 MCFR MTZR n m k LP MCF R t(s) LP MTZR t(s) LP MCF R LP MTZR g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g Table 1 Linear programming results - set g 4

5 References [1] J.E. Beasley. OR-Library: distributing test problems by electronic mail. Journal of the Operational Research Society, 41(11): , [2] M.J. Blesa and F. Xhafa. A C++ Implementation of Tabu Search for v- Cardinality Tree Problem based on Generic Programming and Component reuse. In Net Objective Days 2000, pages , [3] C. Blum. Revisiting dynamic programming for finding optimal subtrees in trees. European Journal of Operational Research, 177: , [4] C. Blum and M. J. Blesa. New Metaheuristic approaches for the edge-weighted k- cardinality tree problem. Computers and Operations Research, 32(6): , [5] J. Brimberg, D. Urosević, and N. Mladenović. Variable neighborhood search for the vertex weighted k-cardinality tree problem. European Journal of Operational Research, 171:74 84, [6] S. Kataoka, N. Araki, and T. Yamada. Upper and lower bounding procedures for minimum rooted k-subtree problem. European Journal of Operational Research, 122: ,

6 MCFR MTZR n m k LP MCF R t(s) LP MTZR t(s) LP MCF R LP MTZR bb15x bb15x bb45x bb45x g g steinc steinc steind le450 15a Table 2 Linear programming results - set d

7 MCFR MTZR n m k LP MCF R t(s) LP MTZR t(s) LP MCF R LP MTZR NWG NWG NWG NWG NWG NWG NWG NWG NWG NWG NWG NWG NWG NWG NWG Table 3 Linear programming results - set NWG 7

8 BB Results n m k nodes t(s) BestUB Status g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT g OPT Table 4 Branch-and-bound results, MTZR, set g. 8

9 BB Results n m k nodes t(s) BestUB Status bb15x OPT OPT OPT OPT bb15x OPT OPT OPT OPT bb45x OPT OPT OFM OFM bb45x OPT OPT OFM OFM bb33x OFM OFM OFM OFM bb33x OFM OFM OFM OFM g OPT OPT OPT OPT g OPT+ Table 5 Branch-and-bound results, MTZR, set d, part I OPT OPT OPT OPT 9

10 BB Results n m k nodes t(s) BestUB Status g OPT OFM OFM OFM OFM OFM g OPT OFM OFM OFM OFM OFM+ steinc OPT OPT OPT OPT OFM steinc OPT OPT OPT OPT OPT+ steind OFM OFM OFM OFM steind OPT OPT OFM OFM OFM le450 15a OPT Table 6 Branch-and-bound results, MTZR, set d, part II OPT OPT OFM 10

11 BB Results n m k nodes t(s) BestUB Status NWG OPT NWG OPT NWG OPT NWG OPT NWG OPT NWG OPT NWG OFM NWG OFM NWG OFM NWG OFM NWG OFM NWG OFM NWG OFM NWG OFM NWG OFM Table 7 Branch-and-bound results, MTZR, set NWG. 11

12 Lagrangian Relaxation n m k z d z gap (%) Fix (%) t(s) z d LP MCF R z DP BKV g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g Table 8 Lagrangian Relaxation results, set g. 12

13 Lagrangian Relaxation n m k z d z gap (%) Fix (%) t(s) z d LP MCF R z DP BKV bb15x bb15x bb45x bb45x bb33x bb33x bb100x bb100x Table 9 Lagrangian Relaxation results - set d, part I

14 Lagrangian Relaxation n m k z d z gap (%) Fix (%) t(s) z d LP MCF R z DP BKV g g g g steinc steinc steind steind le450 15a Table Lagrangian Relaxation results - set d, part II

15 Lagrangian Relaxation n m k z d z gap (%) Fix (%) t(s) z d LP MCF R z DP BKV NWG NWG NWG NWG NWG NWG NWG NWG NWG NWG NWG NWG NWG NWG NWG NWG NWG NWG NWG NWG NWG NWG NWG NWG Table 11 Lagrangian Relaxation results - set NWG. 15