1. Utrecht, february 22, 2002 Applications of Tree Decompositions Stan van Hoesel KE-FdEWB Universiteit Maastricht 043-3883727 s.vanhoesel@ke.unimaas.nl

2. Utrecht, february 22, 2002 For G=(V,E) a tree decomposition (X,T) is a tree T=(I,F), and a subset family of V: X={X i | i  I} s.t.  i  I X i = V (follows almost from 2)  For all {v,w}  E: there is an i  I with {v,w}  X i.  For all i,j,k  I with j on the path between i and k in T: if v  X i and v  X k, then v  X j Definitions The (tree) width of a decomposition (X,T) is max i  I |X i |-1

3. Utrecht, february 22, 2002 bdf ij l k h g eca abdacdcdedeff ii j j l j k egh Example

4. Utrecht, february 22, 2002 Problems Standard graph problems (Coloring: illustration of techniques) Partial Constraint Satisfaction Problems (Binary) Graph problems easy on trees Problems from “practice”; problems with a “natural” tree decomposition with small width Probabilistic Networks: Linda

5. Utrecht, february 22, 2002 Standard graph optimization problems Graph coloring Graph bipartition Max cut Max stable set

6. Utrecht, february 22, 2002 Three techniques of using tree width for solving (practical) combinatorial optimization problems (Bodlaender, 1997): Computing tables of characterizations of partial solutions (dynamic programming) Graph reduction Monadic second order logic Methods

7. Utrecht, february 22, 2002 Important property of tree decompositions Let i,j  I be vertices of the tree T, such that {i,j}  F. If X i  X i  X j  X j, then X i  X j is a vertex cut-set of V

8. Utrecht, february 22, 2002 3 9 2 6 8 4 5 1 7 123345 678789 234378348 78 3834 23 Example: Vertex Coloring (1)

9. Utrecht, february 22, 2002 Example: Vertex Coloring (2) 234378 348 3834 List of colorings of 34 with number of colors used for partial solution 12345 List of colorings of 38 with number of colors used for partial solution 36789 Create list of colorings of 348 with minimum colors used for solution 123456789 How long are the lists? Depends on the method used

10. Utrecht, february 22, 2002 Example: Vertex Coloring (3) S: vertex separating set G=(V,E) 1 2 3 4

11. Utrecht, february 22, 2002 Partial Constraint Satisfaction Problems (binary) Input: –Graph G=(V,E) –For each v  V : D v ={1,2,…,|D v |} –For each {v,w}  E : a |D v |x |D w | matrix of penalties. Frequency Assignment Satisfiability (MAX-SAT) Output: –An assignment of domain elements to vertices, that minimizes the total penalty incurred.

12. Utrecht, february 22, 2002 Frequency Assignment Transmitters (= vertices) Frequencies (= domain elements: numbers) Interference (= edges with penalty matrices)

13. Utrecht, february 22, 2002 Constraint graph

14. Utrecht, february 22, 2002 Running time Graph width = 10 Number of frequencies per vertex = 40 Total number of partial solutions 40 10 Needed: –Good upper bounds –Good processing methods such as reduction techniques and dominance relations –Or efficient way of storing solutions

15. Utrecht, february 22, 2002 Partial Constraint Satisfaction Problems (general) Combinations of assignments to more than 2 vertices can be penalized. This results in constraint hypergraphs. Thus, hypergraph tree decompositions necessary.

16. Utrecht, february 22, 2002 Problems easy on: Trees, Series-Parallel Graphs, Interval Graphs Location problems Steiner trees Scheduling

17. Utrecht, february 22, 2002 Location problems Select a set of vertices of size k such that the total (or maximum) distance to the closest nodes is minimized.

18. Utrecht, february 22, 2002 Problems from “practice” Railway network line planning Tarification Capacity planning in networks, Synthesis of trees Generalized subgraphs (Corinne Feremans)

19. Utrecht, february 22, 2002 Railway Line Planning Given: –Paths: (“length  4”) –Costs for paths –Demands for commodities Find: –Paths with capacities to satisfy all demands

20. Utrecht, february 22, 2002 Capacity Planning Given a telecom network: –Commodities with demands –Different capacity sizes –Costs for capacity sizes Find at minimum cost: –Routing of demands –Capacity of edges

21. Utrecht, february 22, 2002 Tarification Given: –Tariff arcs besides other arcs –Demands for commodities –Each commodity selects a shortest path Find: –Tariffs on tariff arcs, such that the total usage of tariff by commodities is maximized 2 4 5 4 t2t2 t1t1

22. Utrecht, february 22, 2002 Tarification

23. Utrecht, february 22, 2002 Conclusion Where do we start? And how do we proceed? Where do networks with small tree width naturally arise? Use of tree decomposition in heuristics. –Travelling salesman problem What about use of other decompositions? –Branch decomposition