# Minimum Caterpillar Trees and Ring-Stars: a branch-and-cut algorithm

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1 Minimum Caterpillar Trees and Ring-Stars: a branch-and-cut algorithm Luidi G. Simonetti Yuri A. M. Frota Cid C. de Souza Institute of Computing University of Campinas Brazil Aussois, January 2010 Cid de Souza (IC) MCSP Aussois, January / 21

2 Outline The Minimum Spanning Caterpillar Problem The Minimum Ring Star Problem Solving the MCSP exactly A Primal Heuristic Computational Results Conclusions Cid de Souza (IC) MCSP Aussois, January / 21

3 The Minimum Spanning Caterpillar Problem What is a caterpillar? Cid de Souza (IC) MCSP Aussois, January / 21

4 Caterpillar trees The Minimum Spanning Caterpillar Problem Path Cid de Souza (IC) MCSP Aussois, January / 21

5 Caterpillar trees The Minimum Spanning Caterpillar Problem Path + edges (extra edges) Cid de Souza (IC) MCSP Aussois, January / 21

6 Caterpillar trees The Minimum Spanning Caterpillar Problem Path + edges (extra edges) A tree T is said to be a caterpillar if the remaining subgraph after removing all the leaves from T is a path Cid de Souza (IC) MCSP Aussois, January / 21

7 The Minimum Spanning Caterpillar Problem The Minimum Spanning Caterpillar Problem (MSCP) Given: Graph G = (V, E) Cost l e 0 for each edge e E (extra edge) Cost c e 0 for each edge e E (central path) Find: least cost spanning Caterpillar tree T of G Cid de Souza (IC) MCSP Aussois, January / 21

8 The Minimum Spanning Caterpillar Problem The Minimum Spanning Caterpillar Problem (MSCP) Given: Graph G = (V, E) Cost l e 0 for each edge e E (extra edge) Cost c e 0 for each edge e E (central path) Find: least cost spanning Caterpillar tree T of G Cid de Souza (IC) MCSP Aussois, January / 21

9 The Minimum Spanning Caterpillar Problem The Minimum Spanning Caterpillar Problem Cost variation effect When l e c e, e E MCSP = { Spanning tree with few leaves Minimum Hamiltonian path When l e c e, e E { Spanning tree with many leaves MCSP = Minimum star Cid de Souza (IC) MCSP Aussois, January / 21

10 The Minimum Ring Star Problem A closely related problem The Minimum Ring Star Problem (MRSP) Input: graph G = (V, E), l e, c e 0 e E, a special vertex (the depot) Solution: a ring (cycle) with a set of leaves hanging from it (the star) and spanning all the vertices Cid de Souza (IC) MCSP Aussois, January / 21

11 The Minimum Ring Star Problem Relation between the MRSP and the MSCP Original Graph New Graph Cid de Souza (IC) MCSP Aussois, January / 21

12 The Minimum Ring Star Problem Relation between the MRSP and the MSCP MRSP MSCP the depot and its replica are the start and end vertices of the path, respectively Cid de Souza (IC) MCSP Aussois, January / 21

13 Solving the MCSP exactly A solution method for the MCSP Key idea: Reduction to the Minimum Steiner Arborescence Problem Construct the Layered Graph Fix the root of the arborescence (0) Define the set of terminals (R) Impose some side constraints Cid de Souza (IC) MCSP Aussois, January / 21

14 Solving the MCSP exactly The Layered Graph Graph G N = (V N, A N ) V N = {0} {(i, h) : 1 h 2, i V } R = {(i, 2) : i V } (terminals) A N = {(0, (j, 1)) : j V } {((i, 1), (j, 1)) : (i, j) A} {((i, 1), (j, 2)) : (i, j) A} {((i, 1), (i, 2)) : i V }. Cid de Souza (IC) MCSP Aussois, January / 21

15 Solving the MCSP exactly The Layered Graph Graph G N = (V N, A N ) V N = {0} {(i, h) : 1 h 2, i V } R = {(i, 2) : i V } (terminals) A N = {(0, (j, 1)) : j V } {((i, 1), (j, 1)) : (i, j) A} {((i, 1), (j, 2)) : (i, j) A} {((i, 1), (i, 2)) : i V }. c 0j1 = C (only use one arc (0, (j, 1))) Cid de Souza (IC) MCSP Aussois, January / 21

16 Solving the MCSP exactly The Layered Graph Graph G N = (V N, A N ) V N = {0} {(i, h) : 1 h 2, i V } R = {(i, 2) : i V } (terminals) A N = {(0, (j, 1)) : j V } {((i, 1), (j, 1)) : (i, j) A} {((i, 1), (j, 2)) : (i, j) A} {((i, 1), (i, 2)) : i V }. c i1 j 1 = c ij Cid de Souza (IC) MCSP Aussois, January / 21

17 Solving the MCSP exactly The Layered Graph Graph G N = (V N, A N ) V N = {0} {(i, h) : 1 h 2, i V } R = {(i, 2) : i V } (terminals) A N = {(0, (j, 1)) : j V } {((i, 1), (j, 1)) : (i, j) A} {((i, 1), (j, 2)) : (i, j) A} {((i, 1), (i, 2)) : i V }. c i1 j 2 = l ij Cid de Souza (IC) MCSP Aussois, January / 21

18 Solving the MCSP exactly The Layered Graph Graph G N = (V N, A N ) V N = {0} {(i, h) : 1 h 2, i V } R = {(i, 2) : i V } (terminals) A N = {(0, (j, 1)) : j V } {((i, 1), (j, 1)) : (i, j) A} {((i, 1), (j, 2)) : (i, j) A} {((i, 1), (i, 2)) : i V }. c i1 i 2 = 0 Cid de Souza (IC) MCSP Aussois, January / 21

19 Solving the MCSP exactly The Layered Graph Graph G N = (V N, A N ) V N = {0} {(i, h) : 1 h 2, i V } R = {(i, 2) : i V } (terminals) A N = {(0, (j, 1)) : j V } {((i, 1), (j, 1)) : (i, j) A} {((i, 1), (j, 2)) : (i, j) A} {((i, 1), (i, 2)) : i V }. Cid de Souza (IC) MCSP Aussois, January / 21

20 The IP Formulation Solving the MCSP exactly min C j V 0j + c ij ij 1 + l ij ij 2 s.t. jj + ij 2 = 1 j V [V N \ S, S] 1 0 / S, S R { }, S 2 ij 1 1 i V a {0, 1} a A N. Cid de Souza (IC) MCSP Aussois, January / 21

21 The IP Formulation Solving the MCSP exactly min C j V 0j + c ij ij 1 + l ij ij 2 s.t. jj + ij 2 = 1 j V [V N \ S, S] 1 0 / S, S R { }, S 2 ij 1 1 i V a {0, 1} a A N. Cid de Souza (IC) MCSP Aussois, January / 21

22 The IP Formulation Solving the MCSP exactly min C j V 0j + c ij ij 1 + l ij ij 2 s.t. jj + ij 2 = 1 j V [V N \ S, S] 1 0 / S, S R { }, S 2 ij 1 1 i V a {0, 1} a A N. Cid de Souza (IC) MCSP Aussois, January / 21

23 The IP Formulation Solving the MCSP exactly min C j V 0j + c ij ij 1 + l ij ij 2 s.t. jj + ij 2 = 1 j V [V N \ S, S] 1 0 / S, S R { }, S 2 ij 1 1 i V a {0, 1} a A N. Cid de Souza (IC) MCSP Aussois, January / 21

24 Solving the MCSP exactly Optimal solutions Central path constraints Additional constraint Implicit constraint 0i + 0i + (k,i) A (k,i) A 1 ki 1 ki = ii 1 ij i V i V New constraint 1 ij ii i V Stronger than the original inequality 1 ij 1 i V Cid de Souza (IC) MCSP Aussois, January / 21

25 Solving the MCSP exactly Optimal solutions Central path constraints Additional constraint Implicit constraint 0i + 0i + (k,i) A (k,i) A 1 ki 1 ki = ii 1 ij i V i V New constraint 1 ij ii i V Stronger than the original inequality 1 ij 1 i V Cid de Souza (IC) MCSP Aussois, January / 21

26 Solving the MCSP exactly Optimal solutions Central path constraints Additional constraint Implicit constraint 0i + 0i + (k,i) A (k,i) A 1 ki 1 ki = ii 1 ij i V i V New constraint 1 ij ii i V Stronger than the original inequality 1 ij 1 i V Cid de Souza (IC) MCSP Aussois, January / 21

27 Solving the MCSP exactly A note on the MRSP model The Layered Graph The new constraints are: (k,j) A kj 1 = kk for all k V and = 2 1 = Cid de Souza (IC) MCSP Aussois, January / 21

28 Solving the MCSP exactly Improving the LP-relaxation Additional constraints from the (generalized) STSP (original graph) symmetric 2-matching One can also add constraints from the (generalized) ATSP (layered graph) assymmetric 2-matching D + k and D k inequalities... Cid de Souza (IC) MCSP Aussois, January / 21

29 A Primal Heuristic Local Cid Search de Souza (IC) (basic operations) MCSP Aussois, January / 21 Primal Heuristic: grasp Construction phase Builds the central path Iteratively adds the vertex that minimizes the cost of the solution (central path + extra edges) S = { } f = [,,,,,,,, ]

30 A Primal Heuristic Local Cid Search de Souza (IC) (basic operations) MCSP Aussois, January / 21 Primal Heuristic: grasp Construction phase Builds the central path Iteratively adds the vertex that minimizes the cost of the solution (central path + extra edges) S = { 4 } f = [ 5,,, 0, 4,,, 7, ]

31 A Primal Heuristic Local Cid Search de Souza (IC) (basic operations) MCSP Aussois, January / 21 Primal Heuristic: grasp Construction phase Builds the central path Iteratively adds the vertex that minimizes the cost of the solution (central path + extra edges) S = { 4, 5 } f = [ 5, 5,, 0, 0, 5,, 4, ]

32 A Primal Heuristic Local Cid Search de Souza (IC) (basic operations) MCSP Aussois, January / 21 Primal Heuristic: grasp Construction phase Builds the central path Iteratively adds the vertex that minimizes the cost of the solution (central path + extra edges) S = { 4, 5, 6 } f = [ 5, 2,, 0, 0, 0, 5, 4, 6 ]

33 A Primal Heuristic Local Cid Search de Souza (IC) (basic operations) MCSP Aussois, January / 21 Primal Heuristic: grasp Construction phase Builds the central path Iteratively adds the vertex that minimizes the cost of the solution (central path + extra edges) S = { 4, 5, 6, 7 } f = [ 5, 2, 3, 0, 0, 0, 0, 4, 4 ]

34 A Primal Heuristic Primal Heuristic: grasp Construction phase Builds the central path Iteratively adds the vertex that minimizes the cost of the solution (central path + extra edges) Local Search (basic operations) Add vertices to the central path Remove one vertex from the central path Change order of a pair of vertices of the central path (2-exchange) Path relinking Cid de Souza (IC) MCSP Aussois, January / 21

35 A Primal Heuristic Primal Heuristic: grasp Construction phase Builds the central path Iteratively adds the vertex that minimizes the cost of the solution (central path + extra edges) Local Search (basic operations) Add vertices to the central path Remove one vertex from the central path Change order of a pair of vertices of the central path (2-exchange) Path relinking Cid de Souza (IC) MCSP Aussois, January / 21

36 Computational Results Experiments Environment IP solver: xpress version 2008A.1, optimizer version Machine: Intel Core2 Quad, 2.83GHz and 8Gb of RAM Processing times were limited to two hours Instances from the TSPLib (124 in total) Cost c e = α d e for each edge e E (central path) Cost l e = (10 α) d e for each edge e E (extra edge) α = {3, 5, 7, 9} V = {26,..., 200} Cid de Souza (IC) MCSP Aussois, January / 21

37 Experiments Computational Results MRSP results Compared to those reported in [Labbé et al (2004)] between computational environments M. Labbé, G. Laporte, I.R. Martín and J.J.S. González. The Ring Star Problem: Polyhedral Analysis and Exact Algorithm Networks, 43: , undirected (natural) graph model branch-and-cut CPU times adjusted to reflect the difference between computational environments Cid de Souza (IC) MCSP Aussois, January / 21

38 Experiments Computational Results MRSP results Compared to those reported in [Labbé et al (2004)] between computational environments M. Labbé, G. Laporte, I.R. Martín and J.J.S. González. The Ring Star Problem: Polyhedral Analysis and Exact Algorithm Networks, 43: , undirected (natural) graph model branch-and-cut CPU times adjusted to reflect the difference between computational environments Cid de Souza (IC) MCSP Aussois, January / 21

39 Experiments Computational Results MRSP results Compared to those reported in [Labbé et al (2004)] between computational environments M. Labbé, G. Laporte, I.R. Martín and J.J.S. González. The Ring Star Problem: Polyhedral Analysis and Exact Algorithm Networks, 43: , undirected (natural) graph model branch-and-cut CPU times adjusted to reflect the difference between computational environments Cid de Souza (IC) MCSP Aussois, January / 21

40 Computational Results Computational Results - MSCP Summary 123 instances solved to optimality All but one instance were computed in less than 30 min and the average time remained below 6 min Performance improves as α increases grasp GAP T(s) Avr 0.56% 3.33 MA 6.25% 4.16 MIN 0% 0.12 Model LP Nodes T(s) Avr 0.10% MA 0.69% MIN 0% N. OPT Total LP = OPT Total Cid de Souza (IC) MCSP Aussois, January / 21

41 Computational Results Computational Results - MSCP Summary 123 instances solved to optimality All but one instance were computed in less than 30 min and the average time remained below 6 min Performance improves as α increases grasp GAP T(s) Avr 0.56% 3.33 MA 6.25% 4.16 MIN 0% 0.12 Model LP Nodes T(s) Avr 0.10% MA 0.69% MIN 0% N. OPT Total LP = OPT Total Cid de Souza (IC) MCSP Aussois, January / 21

42 Computational Results Computational Results - MSCP Summary 123 instances solved to optimality All but one instance were computed in less than 30 min and the average time remained below 6 min Performance improves as α increases grasp GAP T(s) Avr 0.56% 3.33 MA 6.25% 4.16 MIN 0% 0.12 Model LP Nodes T(s) Avr 0.10% MA 0.69% MIN 0% N. OPT Total LP = OPT Total Cid de Souza (IC) MCSP Aussois, January / 21

43 Computational Results Computational Results - MSCP Summary 123 instances solved to optimality All but one instance were computed in less than 30 min and the average time remained below 6 min Performance improves as α increases grasp GAP T(s) Avr 0.56% 3.33 MA 6.25% 4.16 MIN 0% 0.12 Model LP Nodes T(s) Avr 0.10% MA 0.69% MIN 0% N. OPT Total LP = OPT Total Cid de Souza (IC) MCSP Aussois, January / 21

44 Computational Results Computational Results - MSCP Summary 123 instances solved to optimality All but one instance were computed in less than 30 min and the average time remained below 6 min Performance improves as α increases grasp GAP T(s) Avr 0.56% 3.33 MA 6.25% 4.16 MIN 0% 0.12 Model LP Nodes T(s) Avr 0.10% MA 0.69% MIN 0% N. OPT Total LP = OPT Total Cid de Souza (IC) MCSP Aussois, January / 21

45 Computational Results Computational Results - MRSP grasp GAP T(s) Avr 0.29% 2.78 Max 2.65% Min 0% 0.07 N. OPT Total Cid de Souza (IC) MCSP Aussois, January / 21

46 Computational Results Computational Results - MRSP IP Model α = 3 α = 5 α = 7 α = 9 Avr Max Avr Max Avr Max Avr Max GAP MSCP 0.46% 2.00% 0.19% 0.87% 0.05% 0.51% 0.00% 0.00% LLMG 0.29% 1.41% 0.13% 1.01% 0.09% 1.68% 0.21% 1.99% Nodes MSCP LLMG T(s) MSCP LLMG α Total Solved MCSP LLMG Wins MCSP LLMG Cid de Souza (IC) MCSP Aussois, January / 21

47 Conclusions Conclusions MSCP Capable to solve to optimality instances with up to 200 vertices in reasonable time Strong LP bounds MRSP Stronger LP bounds than the previous undirected formulation (proved) Faster in 3 (of the 4) types of instances tested. α = {5, 7, 9} These instances are somewhat closer to real situations where the cost of the backbone (path or ring) is usually more expensive Cid de Souza (IC) MCSP Aussois, January / 21

48 Conclusions Conclusions Questions? Remarks? Cid de Souza (IC) MCSP Aussois, January / 21

49 Conclusions Conclusions Thank you! This research is supported by Cid de Souza (IC) MCSP Aussois, January / 21

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