1. 1 Steiner Tree Algorithms and Networks 2014/2015 Hans L. Bodlaender Johan M. M. van Rooij

2. 2 The Steiner Tree Problem  Let G = (V,E) be an undirected graph, and let N µ V be a subset of the terminals.  A Steiner tree is a tree T = (V’,E’) in G connecting all terminals in N  V’ µ V, E’ µ E, N µ V’  We use k=|N|.  Streiner tree problem:  Given: an undirected graph G = (V,E), a terminal set N µ V, and an integer t.  Question: is there a Steiner tree consisting of at most t edges in G.

3. 3 My Last Lecture  Steiner Tree.  Interesting problem that we have not seen yet.  Introduction  Variants / applications  NP-Completeness  Polynomial time solvable special cases.  Distance network.  Solving Steiner tree with k-terminals in O*(2 k )-time.  Uses inclusion/exclusion.  Algorithm invented by one of our former students.

4. 4 INTRODUCTION Steiner Tree – Algorithms and Networks

5. 5 Variants and Applications  Applications:  Wire routing of VLSI.  Customer’s bill for renting communication networks.  Other network design and facility location problems.  Some variants:  Vertices are points in the plane.  Vertex weights / edge weights vs unit weights.  Different variants for directed graphs.

6. 6 Steiner Tree is NP-Complete  Steiner Tree is NP-Complete.  Membership of NP: certificate is a subset of the edges.  NP-Hard: reduction from Vertex Cover.  Take an instance of Vertex Cover, G=(V,E), integer k.  Build G’=(V’,E’) by subdividing each edge.  Set N = set of newly introduced vertices.  All edges length 1.  Add one superterminal connected to all vertices.  G’ has Steiner Tree with |E|+k edges, if and only if, G has vertex cover with k vertices. = terminal

7. 7 POLYNOMIAL-TIME SOLVABLE SPECIAL CASES Steiner Tree – Algorithms and Networks

8. 8 Special Cases of Steiner Tree  k = 1: trivial.  k = 2: shortest path.  k = n: minimum spanning tree.  k = c = O(1):constant number of terminals, polynomial- time solvable (next slides).

9. 9 Distance Networks  Distance network D(X) of G=(V,E) (induced by the set X).  Take complete graph with vertex set X.  Cost of edge {v,w} in distance network is length shortest path from v to w in G. Observations:  Let W be the set of vertices of degree larger than two for an optimal Steiner tree T in G with terminal set N.  The Steiner tree T consists of a series of shortest paths between vertices in N [ W.  The cost of T equals the cost of the minimum spanning tree in D(N [ W).  The cost of the optimal Steiner tree in D(V) equals the cost of T.

10. 10 Steiner Tree with O(1) Terminals  Suppose |N|= k is constant c.  Compute distance network D(V).  There is a minimum cost Steiner tree in D(V) that contains at most k – 2 non-terminals.  Any Steiner tree that has one that is no longer without non- terminal vertices of degree 1 and 2.  A tree with r leaves and internal vertices of degree at least 3 has at most r – 2 internal vertices.  Polynomial time algorithm for k = O(1) terminals:  Enumerate all sets W of at most k – 2 non-terminals in G.  For each W, find a minimum spanning tree in the distance network D(NW).  Take the best over all these solutions  Takes polynomial time for fixed k = O(1).

11. 11 O*(2 K ) ALGORITHM BY INCLUSION/EXCLUSION Steiner Tree – Algorithms and Networks

12. 12 Some background on the algorithm  Algorithm invented by Jesper Nederlof.  Just after he finished his Master thesis supervised by Hans (and a little bit by me).  Master thesis on Inclusion/Exclusion algorithms.

13. 13 A Recap: Inclusion/Exclusion Formula  General form of the Inclusion/Exclusion formula:  Let N be a collection of objects (anything).  Let 1,2,...,n be a series of requirements on objects.  Finally, let for a subset W µ {1,2,...,n}, N(W) be the number of objects in N that do not satisfy the requirements in W.  Then, the number of objects X that satisfy all requirements is:

14. 14 The Inclusion/Exclusion formula: Alternative proofs  Various ways to prove the formula. 1. See the formula as a branching algorithm branching on a requirement: required = optional – forbidden 2. If an object satisfies all requirements, it is counted in N(). If an object does not satisfy all requirements, say all but those in a set W’, then it is counted in all W µ W’  With a +1 if W is even, and a -1 if W is odd.  W’ has equally many even as odd subsets: total contribution is 0.

15. 15 Using the Inclusion/Exclusion Formula for Steiner Tree (problematic version)  One possible approach:  Objects: trees in the graph G.  Requirements: contain every terminal.  Then we need to compute 2 k times the number of trees in a subgraph of G.  For each W µ N, compute trees in G[V\W].  However, counting trees is difficult:  Hard to keep track of which vertices are already in the tree.  Compare to Hamiltonian Cycle:  We want something that looks like a walk, so that we do not need to remember where we have been.

16. 16 Branching Walks  Definition: Branching walk in G=(V,E) is a tuple (T, Á ):  Ordered tree T.  Mapping Á from nodes of T to nodes of G, s.t. for any edge {u,v} in the tree T we have that { Á (u), Á (v)} 2 E.  The length of a branching walk is the number of edges in T.  When r is the root of T, we say that the branching walk starts in Á (r) 2 V.  For any n 2 T, we say that the branching walk visits all vertices Á (n) 2 V.  Some examples on the blackboard...

17. 17 Branching Walks and Steiner Tree  Definition: Branching walk in G=(V,E) is a tuple (T, Á ):  Ordered tree T.  Mapping Á from nodes of T to nodes of G, s.t. for any edge {u,v} in the tree T we have that { Á (u), Á (v)} 2 E.  Lemma: Let s 2 N a terminal. There exists a Steiner tree T in G with at most c edges, if and only if, there exists a branching walk of length at most c starting in s visiting all terminals N.

18. 18 Using the Inclusion/Exclusion Formula for Steiner Tree  Approach:  Objects: branching walks from some s 2 N of length c in the graph G.  Requirements: contain every terminal in N\{s}.  We need to compute 2 k-1 times the number of branching walks of length c in a subgraph of G.  For each W µ N\{s}, compute branching walks from s in G[V\W].  Next: how do we count branching walks?  Dynamic programming (similar to ordinary walks).

19. 19 Counting Branching Walks  Let B W (v,j) be the number of branching walks of length j starting in v in G[W].  B W (v,0) = 1 for any vertex v.  B W (v,j) =  u 2 (N(v) Å W)  j 1 + j 2 = j-1 B W (u,j 1 ) B W (v,j 2 )  j 2 = 0 covers the case where we do not branch / split up and walk to vertex u.  Otherwise, a subtree of size j 1 is created from neighbour u, while a new tree of size j 2 is added starting in v. This splits off one branch, and can be repeated to split of more branches.  We can compute B W (v,j) for j = 0,1,2,....,t.  All in polynomial time.

20. 20 Putting It All Together Algorithm:  Choose any s 2 N.  For t = 1, 2, …  Use the inclusion/exclusion formula to count the number of branching walks from s of length t visiting all terminals N.  This results in 2 k-1 times counting branching walks from s of length c in G[V\W].  If this number is non-zero: stop the algorithm and output that the smallest Steiner tree has size t.

21. 21 THAT’S ALL FOLKS… Steiner Tree – Algorithms and Networks