Utrecht, february 22,

2002

Applications of

Tree Decompositions

Stan van Hoesel

KE-FdEWB

Universiteit Maastricht

043-3883727

s.vanhoesel@ke.unimaas.nl

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2002

For G=(V,E) a tree decomposition (X,T) is a tree T=(I,F), and a subset family of V: X={Xi | iI} s.t.

iI Xi = V (follows almost from 2)

For all {v,w}E: there is an iI with {v,w} Xi.

For all i,j,kI with j on the path between i and k in T: if vXi and vXk , then vXj

Definitions

The (tree) width of a decomposition (X,T) is maxiI |Xi|-1

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2002

b d fi j

l

kh

g

eca

abd acd cde def f i i j

j l

j k

egh

Example

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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

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2002

Standard graph optimization problems

• Graph coloring

• Graph bipartition

• Max cut

• Max stable set

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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

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2002

Important property of tree decompositions

Let i,jI be vertices of the tree T, such that {i,j}F. If XiXiXjXj , then

XiXj is a vertex cut-set of V

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2002

3

9

2

6

8

4

5 1

7

123 345 678 789

234 378

348

78 78

3834

3423

Example: Vertex Coloring (1)

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2002

Example: Vertex Coloring (2)

234 378

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

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2002

Example: Vertex Coloring (3)

S: vertex separating set

G=(V,E)

G V[ ]1G V[ ]2

1

2

3

4

# colors1 2 3 4 G[V2]+S

1,4 2,3 2,3,4,…1,4 2 3 3,4,5,…1 2,3 4 3,4,5,…1 2 3 4 4,5,6,…

Sets

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2002

Partial Constraint Satisfaction Problems (binary)

• Input:– Graph G=(V,E)

– For each vV : Dv={1,2,…,|Dv|}

– For each {v,w}E : a |Dv|x |Dw| matrix of penalties.

• Frequency Assignment

• Satisfiability (MAX-SAT)

• Output:– An assignment of

domain elements to vertices, that minimizes the total penalty incurred.

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2002

Frequency Assignment

• Transmitters (= vertices)

• Frequencies (= domain elements: numbers)

• Interference (= edges with penalty matrices)

41000514100401410300141200014154321

0penalty then 2|| if 1penalty then 1|| if 4penalty then 0|| if

}5,4,3,2,1{

wv

wv

wv

wv

ffffff

DD

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2002

Constraint graph

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2002

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

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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.

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Problems easy on: Trees, Series-Parallel Graphs,

Interval Graphs

• Location problems

• Steiner trees

• Scheduling

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Location problems

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

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Problems from “practice”

• Railway network line planning • Tarification • Capacity planning in networks, Synthesis of trees• Generalized subgraphs (Corinne Feremans)

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Railway Line Planning

• Given: – Paths: (“length 4”)– Costs for paths– Demands for commodities

• Find: – Paths with capacities to

satisfy all demands

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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

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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

t2t1

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2002

Tarification

1

Belgique

France

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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