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
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
The Minimum Spanning Caterpillar Problem What is a caterpillar ? Cid de Souza (IC) MCSP Aussois, January 2010 3 / 21
The Minimum Spanning Caterpillar Problem Caterpillar trees Path Cid de Souza (IC) MCSP Aussois, January 2010 4 / 21
The Minimum Spanning Caterpillar Problem Caterpillar trees Path + edges (extra edges) Cid de Souza (IC) MCSP Aussois, January 2010 4 / 21
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
The Minimum Spanning Caterpillar Problem The Minimum Spanning Caterpillar Problem (MSCP) ◮ Graph G = ( V , E ) Given: ◮ 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 2010 5 / 21
The Minimum Spanning Caterpillar Problem The Minimum Spanning Caterpillar Problem (MSCP) ◮ Graph G = ( V , E ) Given: ◮ 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 2010 5 / 21
The Minimum Spanning Caterpillar Problem The Minimum Spanning Caterpillar Problem Cost variation effect ◮ When l e ≫ c e , ∀ e ∈ E � Spanning tree with few leaves MCSP = ⇒ 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 2010 6 / 21
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 2010 7 / 21
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
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
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
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 2010 10 / 21
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 0 j 1 = C → ∞ (only use one arc (0 , ( j , 1))) Cid de Souza (IC) MCSP Aussois, January 2010 10 / 21
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 i 1 j 1 = c ij Cid de Souza (IC) MCSP Aussois, January 2010 10 / 21
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 i 1 j 2 = l ij Cid de Souza (IC) MCSP Aussois, January 2010 10 / 21
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 i 1 i 2 = 0 Cid de Souza (IC) MCSP Aussois, January 2010 10 / 21
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 2010 10 / 21
Solving the MCSP exactly The IP Formulation c ij X 1 l ij X 2 X X X min C X 0 j + ij + ij j ∈ V ( i , j ) ∈ A ( i , j ) ∈ A X 2 X s.t. X jj + ij = 1 j ∈ V ( i , j ) ∈ A X [ V N \ S , S ] ≥ 1 0 / ∈ S , S ∩ R � = {∅} , | S | ≥ 2 X 1 X ij ≤ 1 i ∈ V ( i , j ) ∈ A X a ∈ { 0 , 1 } a ∈ A N . Cid de Souza (IC) MCSP Aussois, January 2010 11 / 21
Solving the MCSP exactly The IP Formulation c ij X 1 l ij X 2 X X X min C X 0 j + ij + ij j ∈ V ( i , j ) ∈ A ( i , j ) ∈ A X 2 X s.t. X jj + ij = 1 j ∈ V ( i , j ) ∈ A X [ V N \ S , S ] ≥ 1 0 / ∈ S , S ∩ R � = {∅} , | S | ≥ 2 X 1 X ij ≤ 1 i ∈ V ( i , j ) ∈ A X a ∈ { 0 , 1 } a ∈ A N . Cid de Souza (IC) MCSP Aussois, January 2010 11 / 21
Solving the MCSP exactly The IP Formulation c ij X 1 l ij X 2 X X X min C X 0 j + ij + ij j ∈ V ( i , j ) ∈ A ( i , j ) ∈ A X 2 X s.t. X jj + ij = 1 j ∈ V ( i , j ) ∈ A X [ V N \ S , S ] ≥ 1 0 / ∈ S , S ∩ R � = {∅} , | S | ≥ 2 X 1 X ij ≤ 1 i ∈ V ( i , j ) ∈ A X a ∈ { 0 , 1 } a ∈ A N . Cid de Souza (IC) MCSP Aussois, January 2010 11 / 21
Solving the MCSP exactly The IP Formulation c ij X 1 l ij X 2 X X X min C X 0 j + ij + ij j ∈ V ( i , j ) ∈ A ( i , j ) ∈ A X 2 X s.t. X jj + ij = 1 j ∈ V ( i , j ) ∈ A X [ V N \ S , S ] ≥ 1 0 / ∈ S , S ∩ R � = {∅} , | S | ≥ 2 X 1 X ij ≤ 1 i ∈ V ( i , j ) ∈ A X a ∈ { 0 , 1 } a ∈ A N . Cid de Souza (IC) MCSP Aussois, January 2010 11 / 21
Solving the MCSP exactly Optimal solutions × Central path constraints ◮ Additional constraint X X 1 X X 1 X 0 i + ki ≥ i ∈ V ij ( k , i ) ∈ A ( i , j ) ∈ A ◮ Implicit constraint X X 1 i ∈ V X 0 i + ki = X ii ( k , i ) ∈ A ◮ New constraint X 1 X ij ≤ X ii i ∈ V ( i , j ) ∈ A Stronger than the original inequality X 1 X ij ≤ 1 i ∈ V ( i , j ) ∈ A Cid de Souza (IC) MCSP Aussois, January 2010 12 / 21
Solving the MCSP exactly Optimal solutions × Central path constraints ◮ Additional constraint X X 1 X X 1 X 0 i + ki ≥ i ∈ V ij ( k , i ) ∈ A ( i , j ) ∈ A ◮ Implicit constraint X X 1 i ∈ V X 0 i + ki = X ii ( k , i ) ∈ A ◮ New constraint X 1 X ij ≤ X ii i ∈ V ( i , j ) ∈ A Stronger than the original inequality X 1 X ij ≤ 1 i ∈ V ( i , j ) ∈ A Cid de Souza (IC) MCSP Aussois, January 2010 12 / 21
Solving the MCSP exactly Optimal solutions × Central path constraints ◮ Additional constraint X X 1 X X 1 X 0 i + ki ≥ i ∈ V ij ( k , i ) ∈ A ( i , j ) ∈ A ◮ Implicit constraint X X 1 i ∈ V X 0 i + ki = X ii ( k , i ) ∈ A ◮ New constraint X 1 X ij ≤ X ii i ∈ V ( i , j ) ∈ A Stronger than the original inequality X 1 X ij ≤ 1 i ∈ V ( i , j ) ∈ A Cid de Souza (IC) MCSP Aussois, January 2010 12 / 21
Solving the MCSP exactly A note on the MRSP model The Layered Graph The new constraints are: X X 1 X 2 1 1 1 2 = X 2 kj = X kk for all k ∈ V 2 = 1 and 1 ′ 1 1 ′ ( k , j ) ∈ A Cid de Souza (IC) MCSP Aussois, January 2010 13 / 21
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
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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