Applications of Tree Decompositions

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

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

3. l i j b d f k h a c e g j l abd acd cde def f i i j j k egh Example

4. 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. Standard graph optimization problems • Graph coloring • Graph bipartition • Max cut • Max stable set

6. Methods • 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

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

8. 123 345 348 378 234 1 5 6 2 8 3 9 4 7 Example: Vertex Coloring (1) 678 789 23 34 78 78 34 38

9. 234 378 348 34 38 Example: Vertex Coloring (2) • 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. 4 3 2 1 Example: Vertex Coloring (3) G=(V,E) S: vertex separating set

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

12. Frequency Assignment • Transmitters (= vertices) • Frequencies (= domain elements: numbers) • Interference (= edges with penalty matrices)

13. Constraint graph

14. Running time • Graph width = 10 • Number of frequencies per vertex = 40 • Total number of partial solutions 4010 • Needed: • Good upper bounds • Good processing methods such as reduction techniques and dominance relations • Or efficient way of storing solutions

15. 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. Problems easy on: Trees, Series-Parallel Graphs, Interval Graphs • Location problems • Steiner trees • Scheduling

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

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

19. Railway Line Planning • Given: • Paths: (“length  4”) • Costs for paths • Demands for commodities • Find: • Paths with capacities to satisfy all demands

20. 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. 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 5 4 t1 t2 2 4

22. Belgique 1 France Tarification

23. 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