1. 1 Steiner Tree on graphs of small treewidth Algorithms and Networks 2014/2015 Hans L. Bodlaender Johan M. M. van Rooij

2. 2 This lecture  Steiner Tree  Recap  Steiner Tree on Graphs of Small Treewidth  What do we store at tables  Computing on Nice Tree Decompositions  Improvement using Rank Based Approach  Based on work by:  B, Cygan, Kratsch, Nederlof (2013)  Fafianie, B, Nederlof (2013)

3. 3 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.

4. 4 Discussed results  Steiner Tree is NP-complete  Distance network  Steiner Tree, parameterized by number of terminals is FPT: with k terminals, can be solved in O(2 k p(n)) time  Today an algorithm using O(c k n) time for:  Given: Connected graph G, set of terminals W, tree decomposition of G of width at most k  Question: What is the minimum number of edges of a tree that connects all terminals?  And the weighted variant:  Given: Connected graph G, set of terminals W, tree decomposition of G of width at most k, weight for each edge  Question: What is the minimum total weight of a tree that connects all terminals?

5. 5 Nice tree decompositions of special type  Five types of nodes:  Leaf (as before)  Join (as before)  Forget (as before)  Introduce vertex (add a vertex, but without edges)  Introduce edge (add an edge between two terminals)  We assume in addition:  The root of the tree decomposition is a bag with one vertex: a terminal

6. 6 To each bag, we associate a subgraph G(i)  Bag i:  All vertices in bags in subtree rooted by i  All edges that are introduced in subtree rooted by i

7. 7 Leaf  G(i) is graph with one vertex and no edges: v v

8. 8 Join nodes  Identify the two subgraphs on the vertices in the bag + =

9. 9 Forget nodes  Same graph – one vertex is no longer ‘boundary’ v v

10. 10 Introduce vertex nodes  In an introduce vertex node, a new vertex (say v) is added to G(i): at this point, v has no incident edges (yet) v Degree 0 now

11. 11 Introduce edge node  Adds an edge between two terminals vv

12. 12 Partial solutions (1)  If we have a Steiner tree T in G, how does the part of T in G(i) look like? v v w w b b a a a, b terminals

13. 13 Partial solutions (2)  If we have a Steiner tree T in G, how does the part of T in G(i) look like?  Maybe like this: v v w w b b a a a, b terminals

14. 14 Partial solutions (3)  If we have a Steiner tree T in G, how does the part of T in G(i) look like?  Or this: v v w w b b a a a, b terminals

15. 15 Partial solutions (4)  If we have a Steiner tree T in G, how does the part of T in G(i) look like?  But not this:  We cannot connect a and b to other terminals outside G(i) v v w w b b a a a, b terminals

16. 16 Partial solutions (5)  A partial solution is a forest in G(i), such that  Each tree contains a vertex in the bag/boundary X(i)  Each terminal in G(i) belongs to a tree in the forest  Or, differently said: each terminal is connected to the boundary bag  Next step: characterize such forests with finite information

17. 17 Characteristic of partial solution  A characteristic of a partial solution for G(i) is:  A subset S of the vertices in X(i): the vertices in X(i) that belong to the forest  A partition of S Two vertices belong to the same partition, if and only if they are connected in the forest

18. 18 Example characterics (weighted example)  {{a,b,c}}, 10  {{a,b,c}}, 11  {{a},{b,c}}, 3  Table:  {{a},{b},{c}}, 0  {{a},{b,c}}, 3  {{b},{a,c}}, 9  {{c},{a,b}}, 7  {{a,b,c}}, 10 b b c c 8 3 6 1 1 2 8 3 6 1 1 2 a b b c c 4 8 3 6 1 1 a 2

19. 19 DP  For each bag, compute a table:  For each possible characteristic, what is the minimum weight of a partial solution (forest fulfilling criteria) with that characteristic  Compute these tables bottom up in the tree  Root bag gives the answer  Remember: the root bag had one vertex, a terminal  Only one possible characteristic in table for the root  Its value gives the answer

20. 20 Computing tables  Leaf: trivial  Forget: discard characteristics with the forgotten vertex as only element in a partition (becomes never connected)  Join: try all combinations of characteristics and compute resulting characteristics  Introduce vertex:  For each element of child bag table:  Add characteristic with introduced vertex as single element in partition  Add characteristic with introduced vertex not in forest  Introduce edge:  For each element of child bag table:  If both endpoints in forest: Add characteristic with introduced edge in the forest: this increases width and joins two forests  Add unchanged characteristic: edge not used in forest

21. 21 ? ? Improvement: intuition  It is sufficient to have something for all different futures  … if you have a set of insurance policies, such that each possible event in the future is covered by at least one of these, you do not need an additional insurance policy  The future is here: a forest in the “other part” 5 4 4 8 3 6 1 1 2 8 3 6 1 1 2

22. 22 Connectivity matrix  Rows and columns marked with partitions  Entry is 1 if: row+ column partition connect all a – b - cab - cac - bbc - aabc a – b - c00001 ab - c00111 ac - b01011 bc - a01101 abc11111

23. 23 The connectivity matrix 1 iff partitions joined connect all a – b - cab - cac - bbc - aabc a – b - c00001 ab - c00111 ac - b01011 bc - a01101 abc11111 Table: a – b – c: 0 ab – c: 7 ac – b: 9 bc – a: 7 abc: 10 Not needed!!!

24. 24 Linear combinations  If row i is a linear combination (when computing mod 2) of other rows, but is at least as expensive as these, then we do not need the solution of row i.  For any future that helps i, we have an entry that is as least as good, and is also helped by this future… a – b - cab - cac - bbc - aabc a – b - c00001 ab - c00111 ac - b01011 bc - a01101 abc11111 0 7 9 7 10

25. 25 Rank Theorem (without proof):  The connectivity matrix on r elements has rank 2 r-1  … and the partitions in at most 2 sets form a basis  Each column with partition p can be written as sum of the columns in 2 sets that it refines (except the trivial partition)

26. 26 The reduce step  This enables a reduce step: if a table has more than 2 |W| elements with the same vertices W in the forest, then we can delete some superfluous table entries:  Let W be the vertices in X(i) in the forest.  Build part of the connectivity matrix:  Rows are the entries in the current table with W as forest vertices  Columns are partitions in at most two sets of W  Perform Gaussian elimination:  Find rows that are linear combination of less or equally expensive rows, and delete the corresponding entries

27. 27 Wrapping up  Table size after reduction step is bounded by 3 tw.  Running time is dominated by Gaussian elimination step and join nodes: O(c tw n)  Implementation: works fast for real-life graphs  Proper representation, choice of data structures helps a lot