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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

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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 2010 1 / 21

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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 2010 2 / 21

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The Minimum Spanning Caterpillar Problem

What is a caterpillar ?

Cid de Souza (IC) MCSP Aussois, January 2010 3 / 21

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The Minimum Spanning Caterpillar Problem

Caterpillar trees

Path

Cid de Souza (IC) MCSP Aussois, January 2010 4 / 21

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The Minimum Spanning Caterpillar Problem

Caterpillar trees

Path + edges (extra edges)

Cid de Souza (IC) MCSP Aussois, January 2010 4 / 21

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The Minimum Spanning Caterpillar Problem

Caterpillar trees

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 2010 4 / 21

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The Minimum Spanning Caterpillar Problem

The Minimum Spanning Caterpillar Problem (MSCP)

Given:

◮ Graph G = (V , E) ◮ Cost le ≥ 0 for each edge e ∈ E (extra edge) ◮ Cost ce ≥ 0 for each edge e ∈ E (central path)

Find: least cost spanning Caterpillar tree T of G

Cid de Souza (IC) MCSP Aussois, January 2010 5 / 21

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The Minimum Spanning Caterpillar Problem

The Minimum Spanning Caterpillar Problem (MSCP)

Given:

◮ Graph G = (V , E) ◮ Cost le ≥ 0 for each edge e ∈ E (extra edge) ◮ Cost ce ≥ 0 for each edge e ∈ E (central path)

Find: least cost spanning Caterpillar tree T of G

Cid de Souza (IC) MCSP Aussois, January 2010 5 / 21

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The Minimum Spanning Caterpillar Problem

The Minimum Spanning Caterpillar Problem

Cost variation effect

◮ When le ≫ ce, ∀e ∈ E

MCSP = ⇒

  • Spanning tree with few leaves

Minimum Hamiltonian path

◮ When le ≪ ce, ∀e ∈ E

MCSP = ⇒

  • Spanning tree with many leaves

Minimum star

Cid de Souza (IC) MCSP Aussois, January 2010 6 / 21

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The Minimum Ring–Star Problem

A closely related problem

The Minimum Ring–Star Problem (MRSP)

◮ Input: graph G = (V , E), le, ce ≥ 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 2010 7 / 21

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The Minimum Ring–Star Problem

Relation between the MRSP and the MSCP

Original Graph New Graph

Cid de Souza (IC) MCSP Aussois, January 2010 8 / 21

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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 2010 8 / 21

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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 2010 9 / 21

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Solving the MCSP exactly

The Layered Graph

Graph GN = (VN, AN)

VN = {0} ∪ {(i, h) : 1 ≤ h ≤ 2, i ∈ V } R = {(i, 2) : i ∈ V } (terminals) AN = {(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 2010 10 / 21

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Solving the MCSP exactly

The Layered Graph

Graph GN = (VN, AN)

VN = {0} ∪ {(i, h) : 1 ≤ h ≤ 2, i ∈ V } R = {(i, 2) : i ∈ V } (terminals) AN = {(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 }. c0j1 = C → ∞ (only use one arc (0, (j, 1)))

Cid de Souza (IC) MCSP Aussois, January 2010 10 / 21

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Solving the MCSP exactly

The Layered Graph

Graph GN = (VN, AN)

VN = {0} ∪ {(i, h) : 1 ≤ h ≤ 2, i ∈ V } R = {(i, 2) : i ∈ V } (terminals) AN = {(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 }. ci1j1 = cij

Cid de Souza (IC) MCSP Aussois, January 2010 10 / 21

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Solving the MCSP exactly

The Layered Graph

Graph GN = (VN, AN)

VN = {0} ∪ {(i, h) : 1 ≤ h ≤ 2, i ∈ V } R = {(i, 2) : i ∈ V } (terminals) AN = {(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 }. ci1j2 = lij

Cid de Souza (IC) MCSP Aussois, January 2010 10 / 21

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Solving the MCSP exactly

The Layered Graph

Graph GN = (VN, AN)

VN = {0} ∪ {(i, h) : 1 ≤ h ≤ 2, i ∈ V } R = {(i, 2) : i ∈ V } (terminals) AN = {(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 }. ci1i2 = 0

Cid de Souza (IC) MCSP Aussois, January 2010 10 / 21

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Solving the MCSP exactly

The Layered Graph

Graph GN = (VN, AN)

VN = {0} ∪ {(i, h) : 1 ≤ h ≤ 2, i ∈ V } R = {(i, 2) : i ∈ V } (terminals) AN = {(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 2010 10 / 21

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Solving the MCSP exactly

The IP Formulation

min C X

j∈V

X0j + X

(i,j)∈A

cijX 1

ij +

X

(i,j)∈A

lijX 2

ij

s.t. Xjj + X

(i,j)∈A

X 2

ij = 1

j ∈ V X[VN \ S, S] ≥ 1 0 / ∈ S, S ∩ R = {∅}, |S| ≥ 2 X

(i,j)∈A

X 1

ij ≤ 1

i ∈ V Xa ∈ {0, 1} a ∈ AN.

Cid de Souza (IC) MCSP Aussois, January 2010 11 / 21

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Solving the MCSP exactly

The IP Formulation

min C X

j∈V

X0j + X

(i,j)∈A

cijX 1

ij +

X

(i,j)∈A

lijX 2

ij

s.t. Xjj + X

(i,j)∈A

X 2

ij = 1

j ∈ V X[VN \ S, S] ≥ 1 0 / ∈ S, S ∩ R = {∅}, |S| ≥ 2 X

(i,j)∈A

X 1

ij ≤ 1

i ∈ V Xa ∈ {0, 1} a ∈ AN.

Cid de Souza (IC) MCSP Aussois, January 2010 11 / 21

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SLIDE 22

Solving the MCSP exactly

The IP Formulation

min C X

j∈V

X0j + X

(i,j)∈A

cijX 1

ij +

X

(i,j)∈A

lijX 2

ij

s.t. Xjj + X

(i,j)∈A

X 2

ij = 1

j ∈ V X[VN \ S, S] ≥ 1 0 / ∈ S, S ∩ R = {∅}, |S| ≥ 2 X

(i,j)∈A

X 1

ij ≤ 1

i ∈ V Xa ∈ {0, 1} a ∈ AN.

Cid de Souza (IC) MCSP Aussois, January 2010 11 / 21

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Solving the MCSP exactly

The IP Formulation

min C X

j∈V

X0j + X

(i,j)∈A

cijX 1

ij +

X

(i,j)∈A

lijX 2

ij

s.t. Xjj + X

(i,j)∈A

X 2

ij = 1

j ∈ V X[VN \ S, S] ≥ 1 0 / ∈ S, S ∩ R = {∅}, |S| ≥ 2 X

(i,j)∈A

X 1

ij ≤ 1

i ∈ V Xa ∈ {0, 1} a ∈ AN.

Cid de Souza (IC) MCSP Aussois, January 2010 11 / 21

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Solving the MCSP exactly

Optimal solutions × Central path constraints

◮ Additional constraint

X0i + X

(k,i)∈A

X 1

ki ≥

X

(i,j)∈A

X 1

ij

i ∈ V

◮ Implicit constraint

X0i + X

(k,i)∈A

X 1

ki = Xii

i ∈ V

◮ New constraint

X

(i,j)∈A

X 1

ij ≤ Xii

i ∈ V Stronger than the original inequality X

(i,j)∈A

X 1

ij ≤ 1

i ∈ V

Cid de Souza (IC) MCSP Aussois, January 2010 12 / 21

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Solving the MCSP exactly

Optimal solutions × Central path constraints

◮ Additional constraint

X0i + X

(k,i)∈A

X 1

ki ≥

X

(i,j)∈A

X 1

ij

i ∈ V

◮ Implicit constraint

X0i + X

(k,i)∈A

X 1

ki = Xii

i ∈ V

◮ New constraint

X

(i,j)∈A

X 1

ij ≤ Xii

i ∈ V Stronger than the original inequality X

(i,j)∈A

X 1

ij ≤ 1

i ∈ V

Cid de Souza (IC) MCSP Aussois, January 2010 12 / 21

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SLIDE 26

Solving the MCSP exactly

Optimal solutions × Central path constraints

◮ Additional constraint

X0i + X

(k,i)∈A

X 1

ki ≥

X

(i,j)∈A

X 1

ij

i ∈ V

◮ Implicit constraint

X0i + X

(k,i)∈A

X 1

ki = Xii

i ∈ V

◮ New constraint

X

(i,j)∈A

X 1

ij ≤ Xii

i ∈ V Stronger than the original inequality X

(i,j)∈A

X 1

ij ≤ 1

i ∈ V

Cid de Souza (IC) MCSP Aussois, January 2010 12 / 21

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Solving the MCSP exactly

A note on the MRSP model

The Layered Graph

The new constraints are: X

(k,j)∈A

X 1

kj = Xkk for all k ∈ V

and X 2

1112 = X 2 1′

11′ 2 = 1 Cid de Souza (IC) MCSP Aussois, January 2010 13 / 21

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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 2010 14 / 21

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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)

S = { ∅ } f = [ ∞, ∞, ∞, ∞, ∞, ∞, ∞, ∞, ∞ ]

Local Search (basic operations)

Cid de Souza (IC) MCSP Aussois, January 2010 15 / 21

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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)

S = { 4 } f = [ 5, ∞, ∞, 0, 4, ∞, ∞, 7, ∞ ]

Local Search (basic operations)

Cid de Souza (IC) MCSP Aussois, January 2010 15 / 21

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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)

S = { 4, 5 } f = [ 5, 5, ∞, 0, 0, 5, ∞, 4, ∞ ]

Local Search (basic operations)

Cid de Souza (IC) MCSP Aussois, January 2010 15 / 21

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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)

S = { 4, 5, 6 } f = [ 5, 2, ∞, 0, 0, 0, 5, 4, 6 ]

Local Search (basic operations)

Cid de Souza (IC) MCSP Aussois, January 2010 15 / 21

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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)

S = { 4, 5, 6, 7 } f = [ 5, 2, 3, 0, 0, 0, 0, 4, 4 ]

Local Search (basic operations)

Cid de Souza (IC) MCSP Aussois, January 2010 15 / 21

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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 2010 15 / 21

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SLIDE 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 2010 15 / 21

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SLIDE 36

Computational Results

Experiments

Environment

◮ IP solver: xpress version 2008A.1, optimizer version 19.00.04 ◮ 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 ce = α × de for each edge e ∈ E (central path) ◮ Cost le = (10 − α) × de for each edge e ∈ E (extra edge) ◮ α = {3, 5, 7, 9} ◮ |V | = {26, . . . , 200}

Cid de Souza (IC) MCSP Aussois, January 2010 16 / 21

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SLIDE 37

Computational Results

Experiments

MRSP results

◮ Compared to those reported in [Labb´

e et al (2004)] between computational environments

  • M. Labb´

e, G. Laporte, I.R. Mart´ ın and J.J.S. Gonz´ alez. The Ring Star Problem: Polyhedral Analysis and Exact Algorithm Networks, 43:177-189, 2004.

◮ “undirected (natural)” graph model ◮ branch-and-cut

◮ CPU times adjusted to reflect the difference between computational

environments

Cid de Souza (IC) MCSP Aussois, January 2010 17 / 21

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SLIDE 38

Computational Results

Experiments

MRSP results

◮ Compared to those reported in [Labb´

e et al (2004)] between computational environments

  • M. Labb´

e, G. Laporte, I.R. Mart´ ın and J.J.S. Gonz´ alez. The Ring Star Problem: Polyhedral Analysis and Exact Algorithm Networks, 43:177-189, 2004.

◮ “undirected (natural)” graph model ◮ branch-and-cut

◮ CPU times adjusted to reflect the difference between computational

environments

Cid de Souza (IC) MCSP Aussois, January 2010 17 / 21

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SLIDE 39

Computational Results

Experiments

MRSP results

◮ Compared to those reported in [Labb´

e et al (2004)] between computational environments

  • M. Labb´

e, G. Laporte, I.R. Mart´ ın and J.J.S. Gonz´ alez. The Ring Star Problem: Polyhedral Analysis and Exact Algorithm Networks, 43:177-189, 2004.

◮ “undirected (natural)” graph model ◮ branch-and-cut

◮ CPU times adjusted to reflect the difference between computational

environments

Cid de Souza (IC) MCSP Aussois, January 2010 17 / 21

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SLIDE 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 MAX 6.25% 4.16 MIN 0% 0.12

  • N. OPT

Total 44 124

Model

LP Nodes T(s) Avr 0.10% 108.1 330.73 MAX 0.69% 3644 7200 MIN 0% 1 0.39 LP = OPT Total 64 124

Cid de Souza (IC) MCSP Aussois, January 2010 18 / 21

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SLIDE 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 MAX 6.25% 4.16 MIN 0% 0.12

  • N. OPT

Total 44 124

Model

LP Nodes T(s) Avr 0.10% 108.1 330.73 MAX 0.69% 3644 7200 MIN 0% 1 0.39 LP = OPT Total 64 124

Cid de Souza (IC) MCSP Aussois, January 2010 18 / 21

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SLIDE 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 MAX 6.25% 4.16 MIN 0% 0.12

  • N. OPT

Total 44 124

Model

LP Nodes T(s) Avr 0.10% 108.1 330.73 MAX 0.69% 3644 7200 MIN 0% 1 0.39 LP = OPT Total 64 124

Cid de Souza (IC) MCSP Aussois, January 2010 18 / 21

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SLIDE 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 MAX 6.25% 4.16 MIN 0% 0.12

  • N. OPT

Total 44 124

Model

LP Nodes T(s) Avr 0.10% 108.1 330.73 MAX 0.69% 3644 7200 MIN 0% 1 0.39 LP = OPT Total 64 124

Cid de Souza (IC) MCSP Aussois, January 2010 18 / 21

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SLIDE 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 MAX 6.25% 4.16 MIN 0% 0.12

  • N. OPT

Total 44 124

Model

LP Nodes T(s) Avr 0.10% 108.1 330.73 MAX 0.69% 3644 7200 MIN 0% 1 0.39 LP = OPT Total 64 124

Cid de Souza (IC) MCSP Aussois, January 2010 18 / 21

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SLIDE 45

Computational Results

Computational Results - MRSP

grasp

GAP T(s) Avr 0.29% 2.78 Max 2.65% 10.14 Min 0% 0.07

  • N. OPT

Total 60 124

Cid de Souza (IC) MCSP Aussois, January 2010 19 / 21

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SLIDE 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 748.12 7111 327.87 5250 35.09 736 1 1 LLMG 59.74 727 33.32 323 7.32 59 1 1 T(s) MSCP 1497.21 7200 1295.63 7200 329.91 7200 403.66 5375 LLMG 336.44 7200 654.74 7200 442.44 7200 1210.08 7200 α Total 3 5 7 9 Solved MCSP 28 27 30 31 116 LLMG 30 29 30 27 116 Wins MCSP 4 20 30 28 82 LLMG 26 9 3 38

Cid de Souza (IC) MCSP Aussois, January 2010 19 / 21

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SLIDE 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 2010 20 / 21

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SLIDE 48

Conclusions

Conclusions Questions ? Remarks ?

Cid de Souza (IC) MCSP Aussois, January 2010 21 / 21

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SLIDE 49

Conclusions

Conclusions Thank you !

This research is supported by

Cid de Souza (IC) MCSP Aussois, January 2010 21 / 21