Utrecht, february 22, 2002 Applications of Tree Decompositions Stan van Hoesel KE-FdEWB Universiteit Maastricht

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

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

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

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

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

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

Utrecht, february 22, Example: Vertex Coloring (1)

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

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

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.

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

Utrecht, february 22, 2002 Constraint graph

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

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.

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

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.

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

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

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

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 t2t2 t1t1

Utrecht, february 22, 2002 Tarification

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