Treewidth. Network Algorithms 2002/2003. R 1. R 2. R 1. R 2. Computing the Resistance With the Laws of Ohm. 1789-1854. Two resistors in series. Two resistors in parallel. Repeated use of the rules. 6. 6. 5. 2. 2. 1. 7. Has resistance 4. 1/6 + 1/2 = 1/(1.5) 1.5 + 1.5 + 5 = 8
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Treewidth
Network Algorithms 2002/2003
Netwerk Algorithms:Treewidth
R1
R2
R1
R2
Computing the Resistance With the Laws of Ohm
1789-1854
Two resistorsin series
Two resistorsin parallel
Netwerk Algorithms:Treewidth
Repeated use of the rules
6
6
5
2
2
1
7
Has resistance 4
1/6 + 1/2 = 1/(1.5)
1.5 + 1.5 + 5 = 8
1 + 7 = 8
1/8 + 1/8 = 1/4
Netwerk Algorithms:Treewidth
A tree structure
5
P
7
6
2
S
S
P
P
5
7
1
1
6
2
6
2
6
2
Netwerk Algorithms:Treewidth
5
7
6
2
4
1
6
2
Carry on!
¼ + ¼ = ½
Netwerk Algorithms:Treewidth
Using tree structures for solving hard problems on graphs 1
e.g.:
NP-complete
Linear / polynomial
time computable
Netwerk Algorithms:Treewidth
Netwerk Algorithms:Treewidth
A(v) = maximum weight of independent set S in T(v)
B(v) = maximum weight of independent set S in T(v), such that S does not contain v.
Netwerk Algorithms:Treewidth
A(v) = max{ w(v) + B(w1) + … + B(wr) , A(w1) + … A(wr) }
B(v) = A(w1) + … A(wr)
Netwerk Algorithms:Treewidth
Netwerk Algorithms:Treewidth
Netwerk Algorithms:Treewidth
Netwerk Algorithms:Treewidth
Netwerk Algorithms:Treewidth
Netwerk Algorithms:Treewidth
Netwerk Algorithms:Treewidth
s1
s1
s2
s2
s2=t1
t1
t2
t2
Netwerk Algorithms:Treewidth
s1
s2
s1
s1=s2
t1
t2
t1
t1=t2
Netwerk Algorithms:Treewidth
5
P
7
6
2
S
S
P
P
5
7
1
1
6
2
6
2
6
2
Netwerk Algorithms:Treewidth
5
6
2
P
S
S
6
2
P
P
5
7
1
6
2
6
2
Netwerk Algorithms:Treewidth
Netwerk Algorithms:Treewidth
Netwerk Algorithms:Treewidth
Netwerk Algorithms:Treewidth
Tree decomposition
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Netwerk Algorithms:Treewidth
Tree decomposition
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Netwerk Algorithms:Treewidth
Treewidth (definition)
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Netwerk Algorithms:Treewidth
Netwerk Algorithms:Treewidth
Netwerk Algorithms:Treewidth
Netwerk Algorithms:Treewidth
w
i
If both v and w not in Xi, then
v and w are not adjacent
v
Netwerk Algorithms:Treewidth
Netwerk Algorithms:Treewidth
Netwerk Algorithms:Treewidth
A heuristic for treewidth
Works often well
N(v)
N(v)
vN(v)
Netwerk Algorithms:Treewidth