Modeling and solving the Euclidean Steiner Problem

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Modeling and solving the Euclidean Steiner Problem Marcia Fampa Work done in collaboration with: Kurt Anstreicher, Claudia D'Ambrosio, Jon Lee, Nelson Maculan, Wendel Melo, Stefan Vigerske DIMACS Workshop DGTA July 26-29, 2016 1 / 32

Problem Denition Given p points in R n. Euclidean Steiner Tree Problem Find a tree with minimal Euclidean length that spans these points using or not extra points, which are called Steiner points. DIMACS Workshop DGTA July 26-29, 2016 2 / 32

Problem Denition Given p points in R n. Euclidean Steiner Tree Problem Find a tree with minimal Euclidean length that spans these points using or not extra points, which are called Steiner points. DIMACS Workshop DGTA July 26-29, 2016 2 / 32

Problem De nition Example in R3 Terminals are the 12 vertices of an icosahedron 0.5 z 0.0-0.5 0.0 y - 0.5 0.5 0.0 0.5 x DIMACS Workshop - 0.5 DGTA July 26-29, 2016 3 / 32

Problem Denition Determine the Steiner Minimal Tree (SMT): The number of Steiner points to be used on the minimal tree. The edges of the tree. Geometric position of the Steiner points. DIMACS Workshop DGTA July 26-29, 2016 4 / 32

The Euclidean Steiner Problem for a Given Topology a 1 a 2 a 1 a 2 x 6 x 5 x 6 x 5 a 4 a 3 a 4 a 3 Minimize a 1 x 5 + a 2 x 5 + x 5 x 6 + a 3 x 5 + a 4 x 6 subject to: x 5, x 6 R n The shortest tree for a given topology T is called a Relatively Minimal Tree (RMT) for T The minimum spanning tree DIMACS Workshop DGTA July 26-29, 2016 5 / 32

The Steiner Minimal Tree Properties: In a SMT for p given terminals: No angle between edges is < 120 degrees. Each terminal has degree between 1 and 3. Each Steiner point has degree equal to 3. The Steiner point and its 3 adjacent nodes lie in a plane. The number of Steiner points is no more than p 2. If the number of Steiner points is exactly p 2, each terminal has degree equal to 1. Denition: A Full Steiner Topology (FST) for p terminals is a topology with p 2 Steiner points, where each Steiner point has degree 3 and each terminal has degree 1. DIMACS Workshop DGTA July 26-29, 2016 6 / 32

Degenerate Steiner Topologies A topology is called a degeneracy of another if the former can be obtained from the latter by shrinking edges. degenerated Steiner points 2 7 2 1 3 1 7 8 8 3 6 10 9 4 6 9 4 5 10 5 DIMACS Workshop DGTA July 26-29, 2016 7 / 32

Degenerate Steiner Topologies A topology is called a degeneracy of another if the former can be obtained from the latter by shrinking edges. Fact: Given an SMT, an FST always exists such that the SMT is an RMT for it. degenerated Steiner points 2 7 2 1 3 1 7 8 8 3 6 10 9 4 6 9 4 5 10 5 DIMACS Workshop DGTA July 26-29, 2016 8 / 32

Exact Algorithms (general n-space) Gilbert and Pollak (1968) Find all the full Steiner topologies on the p given terminals. For each topology optimize the coordinates of the Steiner points. Output: the shortest tree found. E. N. Gilbert and H. O. Pollak, Steiner minimal trees, SIAM J. Applied Math., vol. 16, pp. 1-29, 1968. DIMACS Workshop DGTA July 26-29, 2016 9 / 32

Exact Algorithms (general n-space) Gilbert and Pollak (1968) Find all the full Steiner topologies on the p given terminals. For each topology optimize the coordinates of the Steiner points. Output: the shortest tree found. E. N. Gilbert and H. O. Pollak, Steiner minimal trees, SIAM J. Applied Math., vol. 16, pp. 1-29, 1968. The number of full Steiner topologies for p terminals is t(p) := 1 3 5 7... (2p 5) = (2p 5)!! t(2) = 1, t(4) = 3, t(6) = 105, t(8) = 10395, t(10) = 2, 027, 025, t(12) = 654, 729, 075 DIMACS Workshop DGTA July 26-29, 2016 9 / 32

Exact Algorithms (general n-space) Smith (1992) An implicit enumeration scheme to generate full Steiner topologies. W. D. Smith, How to nd Steiner minimal trees in Euclidean d-space, Algorithmica, vol. 7, pp. 137-177,1992. DIMACS Workshop DGTA July 26-29, 2016 10 / 32

Terminals: Steiner Points: Upper bound=11 SMT=13 DIMACS Workshop DGTA July 26-29, 2016 11 / 32

Improvements to Smith's B&B algorithm Use of a conic formulation for the subproblems of locating the Steiner points Implementation of a strong branching strategy New order in which the terminals are added M. Fampa and Kurt M. Anstreicher, An improved algorithm for computing Steiner minimal trees in Euclidean d-space, Discrete Optimization 5(2), 530-540, 2008. DIMACS Workshop DGTA July 26-29, 2016 12 / 32

Remarks on Smith's procedure Good: Enumeration scheme. Bad: The pruning criterion. Lower bounds for the subproblems are given by the length of the RMTs corresponding to FSTs on only a subset of the terminals - weak bounds. Few nodes are pruned until deep down in the B&B tree. Growth of tree is super-exponential with depth. Subproblems get larger at deeper levels. DIMACS Workshop DGTA July 26-29, 2016 13 / 32

1 6 2 7 8 10 5 9 3 4 MINLP Formulations for the Euclidean Steiner Problem Maculan, Michelon, Xavier (2000) Given p points in R n (a 1,..., a n ), let G = (V, E) be the graph where P := {1, 2,..., p 1, p} is the set of indices associated with p terminals; S := {p + 1, p + 2,..., 2p 3, 2p 2} is the set of indices associated with p 2 Steiner points; V := P S; E 1 := {[i, j] i P, j S}; E 2 := {[i, j] i S, j S}; E := E 1 E 2. DIMACS Workshop DGTA July 26-29, 2016 14 / 32

MINLP Formulations for the Euclidean Steiner Problem Maculan, Michelon, Xavier (2000) (MMX): y ij + i P Minimize [i,j] E1 ai x j y ij + [i,j] E2 xi x j y ij subject to k<j,k S y kj + y ij = 1, i P j S y jk = 3, j S k>j,k S y kj = 1, j S {p + 1} k<j,k S x i R n, i S, y ij {0, 1}, [i, j] E N. Maculan, P. Michelon and A. E. Xavier, The Euclidean Steiner problem in R n : A mathematical programming formulation, Annals of Operations Research, vol. 96, pp. 209-220, 2000. DIMACS Workshop DGTA July 26-29, 2016 15 / 32

MINLP Formulations for the Euclidean Steiner Problem G(V, E) for p = 6 6 terminals, 4 Steiner points, all possible edges; 2 1 3 7 8 10 9 6 4 5 DIMACS Workshop DGTA July 26-29, 2016 16 / 32

MINLP Formulations for the Euclidean Steiner Problem G(V, E) for p = 6 6 terminals, 4 Steiner points, all possible edges; a feasible solution; 2 1 3 7 8 10 9 6 4 5 DIMACS Workshop DGTA July 26-29, 2016 16 / 32

MINLP Formulations for the Euclidean Steiner Problem G(V, E) for p = 6 6 terminals, 4 Steiner points, all possible edges; a feasible solution; the optimal solution; degenerated Steiner points 2 7 2 1 3 1 7 8 8 3 6 10 9 4 6 9 4 5 10 5 DIMACS Workshop DGTA July 26-29, 2016 16 / 32

Drawbacks in applying a spatial branch-and-bound to MMX Weak lower bounds for subproblems. Nondierentiability at points where the FSTs degenerate - problem for algorithms that require the functions to be twice continuously dierentiable. C. D'Ambrosio, M. Fampa, J. Lee, and S. Vigerske. On a nonconvex MINLP formulation of the Euclidean Steiner tree problems in n-space, LNCS (SEA 2015), vol. 9125, pp. 122-133, 2015. DIMACS Workshop DGTA July 26-29, 2016 17 / 32

Non-combinatorial cuts Let η i be the distance from terminal a i to the nearest other terminal. Theorem y ik x k a i η i, for all i P, k S. Theorem (Extend to the case where x k is adjacent to fewer than two terminals) ) y ik y jk ( x k a i + x k a j 2 a i a j / 3, for all i, j P, i < j, k S. Theorem (Extend to n > 3) For n = 3 and i, j P, i < j, k, l S, k < l [ a i a j x k x l y ik y jk y kl det 1 1 1 1 ] = 0. DIMACS Workshop DGTA July 26-29, 2016 18 / 32

Combinatorial cuts Let η i be the distance from terminal a i to the nearest other terminal. T be a minimum-length spanning tree on the terminals. β ij be the length on the longest edge on the path between a i and a j in T. Theorem For i, j P, i < j, If a i a j > η i + η j, then y ik + y jk 1, for all k S. Lemma - well known An SMT contains no edge of length greater than β ij on the path between a i and a j, for all i, j P. Corollary For i, j P, i j, If a i a j > η i + η j + β ij, then y ik + y kl + y jl 2, for all k, l S, k < l. DIMACS Workshop DGTA July 26-29, 2016 19 / 32

Remarks on Nonconvex models/global solvers Good: Capacity and facility to formulate geometric cuts and other valid inequalities to the model. The valid cuts proposed were eective in reducing the number of nodes in the B&B tree and the computational time on small instances. Bad: With nonconvex constraints we lose a nice feature of the problem - it can be eciently solved once the topology (binary variables) is xed. DIMACS Workshop DGTA July 26-29, 2016 20 / 32

MINLP Formulations for the Euclidean Steiner Problem Fampa and Maculan (2004) (FM) : Minimize [i,j] E d ij subject to d ij a i x j M(1 y ij ), [i, j] E 1 d ij x i x j M(1 y ij ), [i, j] E 2 y ij = 1, i P j S y ij + y kj + y jk = 3, j S i P k<j,k S k>j,k S y kj = 1, j S {p + 1} i<j,i S x i R n, i S, y ij {0, 1}, [i, j] E, d ij R +, [i, j] E M. Fampa and N. Maculan, Using a conic formulation for nding Steiner minimal trees, Numerical Algorithms, vol. 35, pp. 315-330, 2004. DIMACS Workshop DGTA July 26-29, 2016 21 / 32

Drawbacks in applying MINLP solvers to FM Weak lower bounds for subproblems. Isomorphic subproblems. M. Fampa, J. Lee, and W. Melo. A specialized branch-and-bound algorithm for the Euclidean Steiner tree problem in n-space, Computational Optimization and Applications, vol. 63(2) DOI 10.1007/s10589-016-9835-z, 2016. DIMACS Workshop DGTA July 26-29, 2016 22 / 32

Dealing with isomorphism Create a set of representative FSTs, with one topology saved for each group of isomorphic FSTs. 1 3 1 2 1 3 5 6 5 6 5 6 2 4 3 4 4 2 Later, during the B&B execution, we only solve subproblems corresponding to these representative topologies, pruning all the others. DIMACS Workshop DGTA July 26-29, 2016 23 / 32

Number of Representative FSTs vs Number of FSTs in the feasible set of FM p Rep. FSTs FSTs in FM Reduction (%) 4 3 6 50 5 15 30 50 6 105 450 76 7 945 9450 90 8 10395 264600 96 9 135135 9525600 98 10 2027025 428652000 99 DIMACS Workshop DGTA July 26-29, 2016 24 / 32

Branch-and-Bound - Preprocessing Phase Discarding representative FSTs based on geometric conditions satised by SMTs Let η i be the minimum Euclidean distance between a i and all other terminals. T be a minimum-length spanning tree on the terminals. β ij be the length on the longest edge on the path between a i and a j in T. Then 1 Two terminals a i and a j may be connected to a common Steiner point only if a i a j η i + η j. 2 Two terminals a i and a j may be connected by two or fewer Steiner points only if a i a j η i + η j + β ij. DIMACS Workshop DGTA July 26-29, 2016 25 / 32

Branch-and-Bound B&B enumeration scheme starts, at its rst level, with one node corresponding to each representative of spanning trees of Steiner points only. p+3 p+1 p+2 p+3 p+1 p+2 p+4 p+4 DIMACS Workshop DGTA July 26-29, 2016 26 / 32

Branch-and-Bound Each node on the enumeration tree has at most p 2 children. A terminal is connected to a dierent Steiner point (with degree less than 3) at each child of the node. p+3 p+1 p+2 p+4 p+3 p+1 p+2 p+3 p+1 p+2 p+3 p+1 p+2 p+4 p+4 p+4 DIMACS Workshop DGTA July 26-29, 2016 27 / 32

Branch-and-Bound Pruning by isomorphism Prune nodes where the variables xed at 1 do not correspond to edges in any representative FST. Fixing variables Let the following representative FSTs, be on the set of descendants of a given node 1 3 1 2 1 3 5 6 5 6 5 6 2 4 3 4 4 2 Fix y 1,5 = 1, y 1,6 = 0 DIMACS Workshop DGTA July 26-29, 2016 28 / 32

Numerical results Tabela: Average results on 10 instances of each dimension Dimension Muriqui Branch-and-Bound SAMBA gap cpu time gap cpu time (%) (sec) (%) (sec) n = 3 84 14400.38 0 240.96 n = 4 84 14401.12 0 740.87 n = 5 82 14400.34 0 1301.54 Muriqui - standard B&B algorithm for convex MINLP SAMBA - Steiner Adaptations on Muriqui B&B Algorithm Time limit of 4 hours (3.60 GHz core i7-4790 CPU, 8 MB, 16 GB, Linux) DIMACS Workshop DGTA July 26-29, 2016 29 / 32

Convex MINLP models/specialized B&B Capacity of formulating valid inequalities to strengthen the model. Solution of non isomorphic subproblems only. DIMACS Workshop DGTA July 26-29, 2016 30 / 32

Bibliography 1 E. N. Gilbert and H. O. Pollak, Steiner minimal trees, SIAM J. Applied Math., vol. 16, pp. 1-29, 1968. 2 F. K. Hwang, D. S. Richards and P. Winter, The Steiner Tree Problem, volume 53 of Annals of Discrete Mathematics. North-Holland, Amsterdam, Netherlands, 1992. 3 W. D. Smith, How to nd Steiner minimal trees in Euclidean d-space, Algorithmica, vol. 7, pp. 137-177,1992. 4 N. Maculan, P. Michelon and A. E. Xavier, The Euclidean Steiner problem in R n : A mathematical programming formulation, Annals of Operations Research, vol. 96, pp. 209-220, 2000. 5 J. W. Van Laarhoven and K. M. Anstreicher. Geometric conditions for Euclidean Steiner trees in R d. Computational Geometry 46, 520-531, 2013. 6 R. Fonseca, M. Brazil, P. Winter, and M. Zachariasen. Faster exact algorithm for computing Steiner trees in higher dimensional Euclidean spaces. Presented at the 11th DIMACS Implementation Challenge Workshop, 2014. http://dimacs11.cs.princeton.edu/workshop/fonsecabrazilwinterzachariasen.pdf DIMACS Workshop DGTA July 26-29, 2016 31 / 32

Bibliography 1 M. Fampa and N. Maculan, Using a conic formulation for nding Steiner minimal trees, Numerical Algorithms, vol. 35, pp. 315-330, 2004. 2 M. Fampa and Kurt M. Anstreicher, An improved algorithm for computing Steiner minimal trees in Euclidean d-space, Discrete Optimization, pp. 530-540, 2008. 3 C. D'Ambrosio, M. Fampa, J. Lee, and S. Vigerske. On a nonconvex MINLP formulation of the Euclidean Steiner tree problems in n-space, LNCS (SEA 2015), vol. 9125, pp. 122-133, 2015. 4 M. Fampa, J. Lee, and W. Melo. A specialized branch-and-bound algorithm for the Euclidean Steiner tree problem in n-space, Computational Optimization and Applications, vol. 63(2) DOI 10.1007/s10589-016-9835-z, 2016. 5 M. Fampa, J. Lee, and N. Maculan. An overview of exact algorithms for the Euclidean Steiner tree problem in n-space, International Transactions in Operational Research, vol. 23, pp. 861-874, 2016. DIMACS Workshop DGTA July 26-29, 2016 32 / 32

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