Download G=(V,E)

Applications
of
Tree Decompositions
Utrecht,
february 22,
2002
Stan van Hoesel
KE-FdEWB
Universiteit Maastricht
043-3883727
[email protected]
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
Utrecht,
february 22,
2002
Example
b
a
d
i
f
c
l
j
k
h
e
g
jl
abd
acd
cde
def
fi
ij
jk
egh
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
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
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 XiXiXjXj , then
XiXj is a vertex cut-set of V
Utrecht,
february 22,
2002
Example: Vertex Coloring (1)
1
5
123
345
678
789
2
4
3
8
9
34
78
234
6
7
23
378
34
38
348
Utrecht,
february 22,
2002
78
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
Utrecht,
february 22,
2002
234
378
34
38
348
Example: Vertex Coloring (3)
G=(V,E)
4
G[V1 ]
3
G[V2 ]
2
1
1
Sets
2
3
4
# colors
G[V2]+S
4
2,3,4,…
3,4,5,…
3,4,5,…
4,5,6,…
S: vertex separating set
Utrecht,
february 22,
2002
1,4
1,4
1
1
2,3
2
2,3
2
3
4
3
Partial Constraint Satisfaction
Problems (binary)
• Frequency Assignment
• Satisfiability (MAX-SAT)
• 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.
Utrecht,
february 22,
2002
• Output:
– An assignment of
domain elements to
vertices, that
minimizes the total
penalty incurred.
Frequency Assignment
• Transmitters (= vertices)
• Frequencies (= domain elements: numbers)
• Interference (= edges with penalty matrices)
1
2
3
4
5
Utrecht,
february 22,
2002
1
4
1
0
0
0
2
1
4
1
0
0
3
0
1
4
1
0
4
0
0
1
4
1
5
0
0
0
1
4
Dv  Dw  {1,2,3,4,5}
if | f v  f w | 0 then penalty 4
if | f v  f w | 1 then penalty 1
if | f v  f w | 2 then penalty 0
Constraint graph
Utrecht,
february 22,
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
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
5
4
t1
t2
2
• Find:
– Tariffs on tariff arcs, such
that the total usage of tariff
by commodities is
maximized
Utrecht,
february 22,
2002
4
Tarification
Belgique
1
France
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
Utrecht,
february 22,
2002