Edge-connectivity augmentation of graphs over symmetric parity … · 2009. 5. 20. · Z. Szigeti...
Transcript of Edge-connectivity augmentation of graphs over symmetric parity … · 2009. 5. 20. · Z. Szigeti...
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Edge-connectivity augmentation of graphsover symmetric parity families
Zoltan Szigeti
Laboratoire G-SCOPINP Grenoble, France
8 April 2009
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Outline
1 Edge-connectivity
2 T -cuts
3 Symmetric parity families
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 2 / 19
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Outline
1 Edge-connectivity1 Definitions2 Cut equivalent trees3 Edge-connectivity augmentation
2 T -cuts
3 Symmetric parity families
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 2 / 19
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Outline
1 Edge-connectivity1 Definitions2 Cut equivalent trees3 Edge-connectivity augmentation
2 T -cuts1 Definitions2 Minimum T -cut3 Augmentation of minimum T -cut
3 Symmetric parity families
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 2 / 19
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Outline
1 Edge-connectivity1 Definitions2 Cut equivalent trees3 Edge-connectivity augmentation
2 T -cuts1 Definitions2 Minimum T -cut3 Augmentation of minimum T -cut
3 Symmetric parity families1 Definition, Examples2 Minimum cut over a symmetric parity family3 Augmentation of minimum cut over a symmetric parity family
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 2 / 19
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Definitions
Global edge-connectivity
Given a graph G = (V ,E ) and an integer k, G is called k-edge-connectedif each cut contains at least k edges.
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Definitions
Global edge-connectivity
Given a graph G = (V ,E ) and an integer k, G is called k-edge-connectedif each cut contains at least k edges.
Local edge-connectivity
Given a graph G = (V ,E ) and u, v ∈ V , the local edge-connectivityλG (u, v) is defined as the minimum cardinality of a cut separating u and v .
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Cut equivalent tree
Theorem (Gomory-Hu)
For every graph G = (V ,E ), we can find, in polynomial time, a treeH = (V ,E ′) and c : E ′ → Z such that for all u, v ∈ V
1 the local edge-connectivity λG (u, v) is equal to the minimum valuec(e) of the edges e of the unique (u, v)-path in H,
2 if e achives this minimum, then a minimum cut of G separating u andv is given by the two connected components of H − e.
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 4 / 19
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Cut equivalent tree
Theorem (Gomory-Hu)
For every graph G = (V ,E ), we can find, in polynomial time, a treeH = (V ,E ′) and c : E ′ → Z such that for all u, v ∈ V
1 the local edge-connectivity λG (u, v) is equal to the minimum valuec(e) of the edges e of the unique (u, v)-path in H,
2 if e achives this minimum, then a minimum cut of G separating u andv is given by the two connected components of H − e.
3
33
4
2
3 4 2
Graph G = (V ,E ) Cut equivalent tree H = (V ,E ′)
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 4 / 19
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Cut equivalent tree
Theorem (Gomory-Hu)
For every graph G = (V ,E ), we can find, in polynomial time, a treeH = (V ,E ′) and c : E ′ → Z such that for all u, v ∈ V
1 the local edge-connectivity λG (u, v) is equal to the minimum valuec(e) of the edges e of the unique (u, v)-path in H,
2 if e achives this minimum, then a minimum cut of G separating u andv is given by the two connected components of H − e.
3
33
4
2
3 4 2
Graph G = (V ,E ) Cut equivalent tree H = (V ,E ′)
u
v
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 4 / 19
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Cut equivalent tree
Theorem (Gomory-Hu)
For every graph G = (V ,E ), we can find, in polynomial time, a treeH = (V ,E ′) and c : E ′ → Z such that for all u, v ∈ V
1 the local edge-connectivity λG (u, v) is equal to the minimum valuec(e) of the edges e of the unique (u, v)-path in H,
2 if e achives this minimum, then a minimum cut of G separating u andv is given by the two connected components of H − e.
3
33
4
2
3 4 2
Graph G = (V ,E ) Cut equivalent tree H = (V ,E ′)
u
v
u
v
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 4 / 19
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Cut equivalent tree
Theorem (Gomory-Hu)
For every graph G = (V ,E ), we can find, in polynomial time, a treeH = (V ,E ′) and c : E ′ → Z such that for all u, v ∈ V
1 the local edge-connectivity λG (u, v) is equal to the minimum valuec(e) of the edges e of the unique (u, v)-path in H,
2 if e achives this minimum, then a minimum cut of G separating u andv is given by the two connected components of H − e.
3
33
4
2
3 4 2
Graph G = (V ,E ) Cut equivalent tree H = (V ,E ′)
u
v
u
v
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 4 / 19
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Cut equivalent tree
Theorem (Gomory-Hu)
For every graph G = (V ,E ), we can find, in polynomial time, a treeH = (V ,E ′) and c : E ′ → Z such that for all u, v ∈ V
1 the local edge-connectivity λG (u, v) is equal to the minimum valuec(e) of the edges e of the unique (u, v)-path in H,
2 if e achives this minimum, then a minimum cut of G separating u andv is given by the two connected components of H − e.
3
33
4
2
3 4 2
Graph G = (V ,E ) Cut equivalent tree H = (V ,E ′)
u
v
u
v
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 4 / 19
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Cut equivalent tree
Theorem (Gomory-Hu)
For every graph G = (V ,E ), we can find, in polynomial time, a treeH = (V ,E ′) and c : E ′ → Z such that for all u, v ∈ V
1 the local edge-connectivity λG (u, v) is equal to the minimum valuec(e) of the edges e of the unique (u, v)-path in H,
2 if e achives this minimum, then a minimum cut of G separating u andv is given by the two connected components of H − e.
3
33
4
2
3 4 2
Graph G = (V ,E ) Cut equivalent tree H = (V ,E ′)
u
v
u
v
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 4 / 19
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Cut equivalent tree
Theorem (Gomory-Hu)
For every graph G = (V ,E ), we can find, in polynomial time, a treeH = (V ,E ′) and c : E ′ → Z such that for all u, v ∈ V
1 the local edge-connectivity λG (u, v) is equal to the minimum valuec(e) of the edges e of the unique (u, v)-path in H,
2 if e achives this minimum, then a minimum cut of G separating u andv is given by the two connected components of H − e.
3
3
4
2
3 4 2
Graph G = (V ,E ) Cut equivalent tree H = (V ,E ′)
u
v
u
v
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 4 / 19
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Cut equivalent tree
Theorem (Gomory-Hu)
For every graph G = (V ,E ), we can find, in polynomial time, a treeH = (V ,E ′) and c : E ′ → Z such that for all u, v ∈ V
1 the local edge-connectivity λG (u, v) is equal to the minimum valuec(e) of the edges e of the unique (u, v)-path in H,
2 if e achives this minimum, then a minimum cut of G separating u andv is given by the two connected components of H − e.
3
3
4
2
3 4 2
Graph G = (V ,E ) Cut equivalent tree H = (V ,E ′)
u
v
u
v
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 4 / 19
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Cut equivalent tree
Theorem (Gomory-Hu)
For every graph G = (V ,E ), we can find, in polynomial time, a treeH = (V ,E ′) and c : E ′ → Z such that for all u, v ∈ V
1 the local edge-connectivity λG (u, v) is equal to the minimum valuec(e) of the edges e of the unique (u, v)-path in H,
2 if e achives this minimum, then a minimum cut of G separating u andv is given by the two connected components of H − e.
3
3
4
2
3 4 2
Graph G = (V ,E ) Cut equivalent tree H = (V ,E ′)
u
v
u
v
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 4 / 19
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Cut equivalent tree
Theorem (Gomory-Hu)
For every graph G = (V ,E ), we can find, in polynomial time, a treeH = (V ,E ′) and c : E ′ → Z such that for all u, v ∈ V
1 the local edge-connectivity λG (u, v) is equal to the minimum valuec(e) of the edges e of the unique (u, v)-path in H,
2 if e achives this minimum, then a minimum cut of G separating u andv is given by the two connected components of H − e.
3
3
4
2
3 4 2
Graph G = (V ,E ) Cut equivalent tree H = (V ,E ′)
u
v
u
v
3
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 4 / 19
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Cut equivalent tree
Theorem (Gomory-Hu)
For every graph G = (V ,E ), we can find, in polynomial time, a treeH = (V ,E ′) and c : E ′ → Z such that for all u, v ∈ V
1 the local edge-connectivity λG (u, v) is equal to the minimum valuec(e) of the edges e of the unique (u, v)-path in H,
2 if e achives this minimum, then a minimum cut of G separating u andv is given by the two connected components of H − e.
3
33
4
2
3 4 2
Graph G = (V ,E ) Cut equivalent tree H = (V ,E ′)
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 4 / 19
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Edge-Connectivity Augmentation
Global edge-connectivity augmentation of a graph
Given a graph G = (V ,E ) and an integer k ≥ 2, what is the minimumnumber of new edges whose addition results in a k-edge-connected graph ?
1 Minimax theorem (Watanabe, Nakamura)
2 Polynomially solvable (Cai, Sun)
Graph G , k = 4
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Edge-Connectivity Augmentation
Global edge-connectivity augmentation of a graph
Given a graph G = (V ,E ) and an integer k ≥ 2, what is the minimumnumber of new edges whose addition results in a k-edge-connected graph ?
1 Minimax theorem (Watanabe, Nakamura)
2 Polynomially solvable (Cai, Sun)
1
Graph G , k = 4
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 5 / 19
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Edge-Connectivity Augmentation
Global edge-connectivity augmentation of a graph
Given a graph G = (V ,E ) and an integer k ≥ 2, what is the minimumnumber of new edges whose addition results in a k-edge-connected graph ?
1 Minimax theorem (Watanabe, Nakamura)
2 Polynomially solvable (Cai, Sun)
1 2
Graph G , k = 4
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 5 / 19
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Edge-Connectivity Augmentation
Global edge-connectivity augmentation of a graph
Given a graph G = (V ,E ) and an integer k ≥ 2, what is the minimumnumber of new edges whose addition results in a k-edge-connected graph ?
1 Minimax theorem (Watanabe, Nakamura)
2 Polynomially solvable (Cai, Sun)
1
1
2
Graph G , k = 4
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 5 / 19
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Edge-Connectivity Augmentation
Global edge-connectivity augmentation of a graph
Given a graph G = (V ,E ) and an integer k ≥ 2, what is the minimumnumber of new edges whose addition results in a k-edge-connected graph ?
1 Minimax theorem (Watanabe, Nakamura)
2 Polynomially solvable (Cai, Sun)
1
1
1
2
Graph G , k = 4
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 5 / 19
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Edge-Connectivity Augmentation
Global edge-connectivity augmentation of a graph
Given a graph G = (V ,E ) and an integer k ≥ 2, what is the minimumnumber of new edges whose addition results in a k-edge-connected graph ?
1 Minimax theorem (Watanabe, Nakamura)
2 Polynomially solvable (Cai, Sun)
1
1
1
2
Opt≥ ⌈52⌉ = 3
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 5 / 19
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Edge-Connectivity Augmentation
Global edge-connectivity augmentation of a graph
Given a graph G = (V ,E ) and an integer k ≥ 2, what is the minimumnumber of new edges whose addition results in a k-edge-connected graph ?
1 Minimax theorem (Watanabe, Nakamura)
2 Polynomially solvable (Cai, Sun)
1
1
1
2
Graph G + F is 4-edge-connected and |F | = 3
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 5 / 19
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Edge-Connectivity Augmentation
Global edge-connectivity augmentation of a graph
Given a graph G = (V ,E ) and an integer k ≥ 2, what is the minimumnumber of new edges whose addition results in a k-edge-connected graph ?
1 Minimax theorem (Watanabe, Nakamura)
2 Polynomially solvable (Cai, Sun)
1
1
1
2
Opt= ⌈12maximum deficiency of a subpartition of V ⌉
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Edge-Connectivity Augmentation
Global edge-connectivity augmentation of a graph
Given a graph G = (V ,E ) and an integer k ≥ 2, what is the minimumnumber of new edges whose addition results in a k-edge-connected graph ?
1 Minimax theorem (Watanabe, Nakamura)
2 Polynomially solvable (Cai, Sun)
1
1
1
2
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 5 / 19
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General method
Frank’s algorithm
1 Minimal extension,
(i) Add a new vertex s,(ii) Add a minimum number of new edges incident to s to satisfy the
edge-connectivity requirements,(iii) If the degree of s is odd, then add an arbitrary edge incident to s.
2 Complete splitting off.
G = (V , E )
w-
Extension
s
vMinimal
u
z
G ′ k-e-c in V
-
CompleteSplitting off v
u w
z
G ′′ k-e-c
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General method
Frank’s algorithm
1 Minimal extension,
(i) Add a new vertex s,(ii) Add a minimum number of new edges incident to s to satisfy the
edge-connectivity requirements,(iii) If the degree of s is odd, then add an arbitrary edge incident to s.
2 Complete splitting off.
G = (V , E )
w-
Extension
s
vMinimal
u
z
G ′ k-e-c in V
-
CompleteSplitting off v
u w
z
G ′′ k-e-c
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 6 / 19
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General method
Frank’s algorithm
1 Minimal extension,
(i) Add a new vertex s,(ii) Add a minimum number of new edges incident to s to satisfy the
edge-connectivity requirements,(iii) If the degree of s is odd, then add an arbitrary edge incident to s.
2 Complete splitting off.
G = (V , E )
w-
Extension
s
vMinimal
u
z
G ′ k-e-c in V
-
CompleteSplitting off v
u w
z
G ′′ k-e-c
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 6 / 19
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General method
Frank’s algorithm
1 Minimal extension,
(i) Add a new vertex s,(ii) Add a minimum number of new edges incident to s to satisfy the
edge-connectivity requirements,(iii) If the degree of s is odd, then add an arbitrary edge incident to s.
2 Complete splitting off.
G = (V , E )
w-
Extension
s
vMinimal
u
z
G ′ k-e-c in V
-
CompleteSplitting off v
u w
z
G ′′ k-e-c
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General method
Frank’s algorithm
1 Minimal extension,
(i) Add a new vertex s,(ii) Add a minimum number of new edges incident to s to satisfy the
edge-connectivity requirements,(iii) If the degree of s is odd, then add an arbitrary edge incident to s.
2 Complete splitting off.
G = (V , E )
w-
Extension
s
vMinimal
u
z
G ′ k-e-c in V
-
CompleteSplitting off v
u w
z
G ′′ k-e-c
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Minimal extension
Definition
A function p on 2V is called skew-supermodular if at least one of followinginequalities hold for all X ,Y ⊆ V :
p(X ) + p(Y ) ≤ p(X ∩ Y ) + p(X ∪ Y ),p(X ) + p(Y ) ≤ p(X − Y ) + p(Y − X ).
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Minimal extension
Definition
A function p on 2V is called skew-supermodular if at least one of followinginequalities hold for all X ,Y ⊆ V :
p(X ) + p(Y ) ≤ p(X ∩ Y ) + p(X ∪ Y ),p(X ) + p(Y ) ≤ p(X − Y ) + p(Y − X ).
Theorem (Frank)
Let p : 2V → Z ∪ {−∞} be a symmetric skew-supermodular function.
1 The minimum number of edges in an extension (d(X ) ≥ p(X ) for allX ⊆ V ) is equal to the maximum p-value of a subpartition of V .
2 An optimal extension can be found in polynomial time in the specialcases mentioned in this talk.
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Minimal extension
Definition
A function p on 2V is called skew-supermodular if at least one of followinginequalities hold for all X ,Y ⊆ V :
p(X ) + p(Y ) ≤ p(X ∩ Y ) + p(X ∪ Y ),p(X ) + p(Y ) ≤ p(X − Y ) + p(Y − X ).
Theorem (Frank)
Let p : 2V → Z ∪ {−∞} be a symmetric skew-supermodular function.
1 The minimum number of edges in an extension (d(X ) ≥ p(X ) for allX ⊆ V ) is equal to the maximum p-value of a subpartition of V .
2 An optimal extension can be found in polynomial time in the specialcases mentioned in this talk.
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Minimal extension
Definition
A function p on 2V is called skew-supermodular if at least one of followinginequalities hold for all X ,Y ⊆ V :
p(X ) + p(Y ) ≤ p(X ∩ Y ) + p(X ∪ Y ),p(X ) + p(Y ) ≤ p(X − Y ) + p(Y − X ).
Theorem (Frank)
Let p : 2V → Z ∪ {−∞} be a symmetric skew-supermodular function.
1 The minimum number of edges in an extension (d(X ) ≥ p(X ) for allX ⊆ V ) is equal to the maximum p-value of a subpartition of V .
2 An optimal extension can be found in polynomial time in the specialcases mentioned in this talk.
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Minimal extension
Definition
A function p on 2V is called skew-supermodular if at least one of followinginequalities hold for all X ,Y ⊆ V :
p(X ) + p(Y ) ≤ p(X ∩ Y ) + p(X ∪ Y ),p(X ) + p(Y ) ≤ p(X − Y ) + p(Y − X ).
Theorem (Frank)
Let p : 2V → Z ∪ {−∞} be a symmetric skew-supermodular function.
1 The minimum number of edges in an extension (d(X ) ≥ p(X ) for allX ⊆ V ) is equal to the maximum p-value of a subpartition of V .
2 An optimal extension can be found in polynomial time in the specialcases mentioned in this talk.
For global edge-connectivity augmentation p(X ) := k − dG (X ).
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Complete splitting off
Definitionss s
u u
v v
G′ G ′
uv
-
Splitting off
V V
s s
u u
v v
G′
G′′
-
Splitting offComplete
V V
w
z
wz
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Complete splitting off
Definitionss s
u u
v v
G′ G ′
uv
-
Splitting off
V V
s s
u u
v v
G′
G′′
-
Splitting offComplete
V V
w
z
wz
Theorem (Mader)
Let G ′ = (V + s,E ) be a graph so that d(s) is even and no cut edge isincident to s.
1 Then there exists a complete splitting off at s that preserves the localedge-connectivity between all pairs of vertices in V .
2 Such a complete splitting off can be found in polynomial time.
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Complete splitting off
Definitionss s
u u
v v
G′ G ′
uv
-
Splitting off
V V
s s
u u
v v
G′
G′′
-
Splitting offComplete
V V
w
z
wz
Theorem (Mader)
Let G ′ = (V + s,E ) be a graph so that d(s) is even and no cut edge isincident to s.
1 Then there exists a complete splitting off at s that preserves the localedge-connectivity between all pairs of vertices in V .
2 Such a complete splitting off can be found in polynomial time.
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Complete splitting off
Definitionss s
u u
v v
G′ G ′
uv
-
Splitting off
V V
s s
u u
v v
G′
G′′
-
Splitting offComplete
V V
w
z
wz
Theorem (Mader)
Let G ′ = (V + s,E ) be a graph so that d(s) is even and no cut edge isincident to s.
1 Then there exists a complete splitting off at s that preserves the localedge-connectivity between all pairs of vertices in V .
2 Such a complete splitting off can be found in polynomial time.
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Negative Result
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Negative Result
Definition
A graph H covers a function p : 2V (H) → Z ∪ {−∞} if each cut δH(X )contains at least p(X ) edges.
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Negative Result
Definition
A graph H covers a function p : 2V (H) → Z ∪ {−∞} if each cut δH(X )contains at least p(X ) edges.
Minimum Cover of a Symmetric Skew-Supermodular
Function by a Graph
Instance : p : 2V → Z symmetric skew-supermodular, γ ∈ Z+.
Question : Does there exist a graph on V with at most γ edges thatcovers p?
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Negative Result
Definition
A graph H covers a function p : 2V (H) → Z ∪ {−∞} if each cut δH(X )contains at least p(X ) edges.
Minimum Cover of a Symmetric Skew-Supermodular
Function by a Graph
Instance : p : 2V → Z symmetric skew-supermodular, γ ∈ Z+.
Question : Does there exist a graph on V with at most γ edges thatcovers p?
Theorem (Z. Kiraly, Z. Nutov)
The above problem is NP-complete.
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T -cut, T -join
Definitions
Given a connected graph G = (V ,E ) and T ⊆ V with |T | even.
1 A subset X of V is called T -odd if |X ∩ T | is odd.
2 A cut δ(X ) is called T -cut if X is T -odd.
3 A subset F of E is called T -join if T = {v ∈ V : dF (v) is odd}.
Examples :
(a) T = {u, v} : a (u, v)-path is a T -join.(b) T = V : a perfect matching is a T -join.
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T -cut, T -join
Definitions
Given a connected graph G = (V ,E ) and T ⊆ V with |T | even.
1 A subset X of V is called T -odd if |X ∩ T | is odd.
2 A cut δ(X ) is called T -cut if X is T -odd.
3 A subset F of E is called T -join if T = {v ∈ V : dF (v) is odd}.
Examples :
(a) T = {u, v} : a (u, v)-path is a T -join.(b) T = V : a perfect matching is a T -join.
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T -cut, T -join
Definitions
Given a connected graph G = (V ,E ) and T ⊆ V with |T | even.
1 A subset X of V is called T -odd if |X ∩ T | is odd.
2 A cut δ(X ) is called T -cut if X is T -odd.
3 A subset F of E is called T -join if T = {v ∈ V : dF (v) is odd}.
Examples :
(a) T = {u, v} : a (u, v)-path is a T -join.(b) T = V : a perfect matching is a T -join.
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T -cut, T -join
Definitions
Given a connected graph G = (V ,E ) and T ⊆ V with |T | even.
1 A subset X of V is called T -odd if |X ∩ T | is odd.
2 A cut δ(X ) is called T -cut if X is T -odd.
3 A subset F of E is called T -join if T = {v ∈ V : dF (v) is odd}.
Examples :
(a) T = {u, v} : a (u, v)-path is a T -join.(b) T = V : a perfect matching is a T -join.
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T -cut, T -join
Definitions
Given a connected graph G = (V ,E ) and T ⊆ V with |T | even.
1 A subset X of V is called T -odd if |X ∩ T | is odd.
2 A cut δ(X ) is called T -cut if X is T -odd.
3 A subset F of E is called T -join if T = {v ∈ V : dF (v) is odd}.
Examples :
(a) T = {u, v} : a (u, v)-path is a T -join.(b) T = V : a perfect matching is a T -join.
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T -cut, T -join
Definitions
Given a connected graph G = (V ,E ) and T ⊆ V with |T | even.
1 A subset X of V is called T -odd if |X ∩ T | is odd.
2 A cut δ(X ) is called T -cut if X is T -odd.
3 A subset F of E is called T -join if T = {v ∈ V : dF (v) is odd}.
Examples :
(a) T = {u, v} : a (u, v)-path is a T -join.(b) T = V : a perfect matching is a T -join.
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T -cut, T -join
Definitions
Given a connected graph G = (V ,E ) and T ⊆ V with |T | even.
1 A subset X of V is called T -odd if |X ∩ T | is odd.
2 A cut δ(X ) is called T -cut if X is T -odd.
3 A subset F of E is called T -join if T = {v ∈ V : dF (v) is odd}.
Examples :
(a) T = {u, v} : a (u, v)-path is a T -join.(b) T = V : a perfect matching is a T -join.
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T -cut, T -join
Definitions
Given a connected graph G = (V ,E ) and T ⊆ V with |T | even.
1 A subset X of V is called T -odd if |X ∩ T | is odd.
2 A cut δ(X ) is called T -cut if X is T -odd.
3 A subset F of E is called T -join if T = {v ∈ V : dF (v) is odd}.
Examples :
(a) T = {u, v} : a (u, v)-path is a T -join.(b) T = V : a perfect matching is a T -join.
Properties
1 If X ,Y are T -odd, then either X ∩ Y ,X ∪ Y or X − Y ,Y − X areT -odd.
2 A T -join and a T -cut always have an edge in common.
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T -cut, T -join
Definitions
Given a connected graph G = (V ,E ) and T ⊆ V with |T | even.
1 A subset X of V is called T -odd if |X ∩ T | is odd.
2 A cut δ(X ) is called T -cut if X is T -odd.
3 A subset F of E is called T -join if T = {v ∈ V : dF (v) is odd}.
Examples :
(a) T = {u, v} : a (u, v)-path is a T -join.(b) T = V : a perfect matching is a T -join.
Properties
1 If X ,Y are T -odd, then either X ∩ Y ,X ∪ Y or X − Y ,Y − X areT -odd.
2 A T -join and a T -cut always have an edge in common.
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T -cut, T -join
Definitions
Given a connected graph G = (V ,E ) and T ⊆ V with |T | even.
1 A subset X of V is called T -odd if |X ∩ T | is odd.
2 A cut δ(X ) is called T -cut if X is T -odd.
3 A subset F of E is called T -join if T = {v ∈ V : dF (v) is odd}.
Examples :
(a) T = {u, v} : a (u, v)-path is a T -join.(b) T = V : a perfect matching is a T -join.
Properties
1 If X ,Y are T -odd, then either X ∩ Y ,X ∪ Y or X − Y ,Y − X areT -odd.
2 A T -join and a T -cut always have an edge in common.
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How to find a minimum T -join ?
Theorem (Edmonds-Johnson)
A minimum T-join can be found in polynomial time using
1 shortest paths algorithm (Dijkstra) and
2 minimum weight perfect matching algorithm (Edmonds).
Graph G and vertex set T
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How to find a minimum T -join ?
Theorem (Edmonds-Johnson)
A minimum T-join can be found in polynomial time using
1 shortest paths algorithm (Dijkstra) and
2 minimum weight perfect matching algorithm (Edmonds).
Graph G and vertex set T
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How to find a minimum T -join ?
Theorem (Edmonds-Johnson)
A minimum T-join can be found in polynomial time using
1 shortest paths algorithm (Dijkstra) and
2 minimum weight perfect matching algorithm (Edmonds).
1
Graph G and vertex set T
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How to find a minimum T -join ?
Theorem (Edmonds-Johnson)
A minimum T-join can be found in polynomial time using
1 shortest paths algorithm (Dijkstra) and
2 minimum weight perfect matching algorithm (Edmonds).
1
2
Graph G and vertex set T
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How to find a minimum T -join ?
Theorem (Edmonds-Johnson)
A minimum T-join can be found in polynomial time using
1 shortest paths algorithm (Dijkstra) and
2 minimum weight perfect matching algorithm (Edmonds).
1
2
2
Graph G and vertex set T
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 11 / 19
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How to find a minimum T -join ?
Theorem (Edmonds-Johnson)
A minimum T-join can be found in polynomial time using
1 shortest paths algorithm (Dijkstra) and
2 minimum weight perfect matching algorithm (Edmonds).
1
2
2
1
Graph G and vertex set T
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How to find a minimum T -join ?
Theorem (Edmonds-Johnson)
A minimum T-join can be found in polynomial time using
1 shortest paths algorithm (Dijkstra) and
2 minimum weight perfect matching algorithm (Edmonds).
1
1
2
2
2
Graph G and vertex set T
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 11 / 19
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How to find a minimum T -join ?
Theorem (Edmonds-Johnson)
A minimum T-join can be found in polynomial time using
1 shortest paths algorithm (Dijkstra) and
2 minimum weight perfect matching algorithm (Edmonds).
1
1
2
2
2
2
Graph G and vertex set T
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 11 / 19
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How to find a minimum T -join ?
Theorem (Edmonds-Johnson)
A minimum T-join can be found in polynomial time using
1 shortest paths algorithm (Dijkstra) and
2 minimum weight perfect matching algorithm (Edmonds).
1
1
2
2
2
2
Graph G and vertex set T
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 11 / 19
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How to find a minimum T -join ?
Theorem (Edmonds-Johnson)
A minimum T-join can be found in polynomial time using
1 shortest paths algorithm (Dijkstra) and
2 minimum weight perfect matching algorithm (Edmonds).
1
1
2
2
2
2
Graph G and minimum T -join
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 11 / 19
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How to find a minimum T -cut ?
Theorem (Padberg-Rao)
A minimum T-cut can be found in polynomial time
1 using a cut equivalent tree H and
2 taking the set J(H) edges e of H for which the two connectedcomponents of H − e are T -odd,
3 taking the minimum value c(e∗) of an edge of J(H),
4 taking the cut defined by the two connected components of H − e∗.
Graph G and vertex set T
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How to find a minimum T -cut ?
Theorem (Padberg-Rao)
A minimum T-cut can be found in polynomial time
1 using a cut equivalent tree H and
2 taking the set J(H) edges e of H for which the two connectedcomponents of H − e are T -odd,
3 taking the minimum value c(e∗) of an edge of J(H),
4 taking the cut defined by the two connected components of H − e∗.
3
33
4
2
3 4 2
Graph G and vertex set T Cut equivalent tree H
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How to find a minimum T -cut ?
Theorem (Padberg-Rao)
A minimum T-cut can be found in polynomial time
1 using a cut equivalent tree H and
2 taking the set J(H) edges e of H for which the two connectedcomponents of H − e are T -odd,
3 taking the minimum value c(e∗) of an edge of J(H),
4 taking the cut defined by the two connected components of H − e∗.
3
33
4
2
3 4 2
Graph G and vertex set T Cut equivalent tree H
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 12 / 19
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How to find a minimum T -cut ?
Theorem (Padberg-Rao)
A minimum T-cut can be found in polynomial time
1 using a cut equivalent tree H and
2 taking the set J(H) edges e of H for which the two connectedcomponents of H − e are T -odd,
3 taking the minimum value c(e∗) of an edge of J(H),
4 taking the cut defined by the two connected components of H − e∗.
3
33
2
3 4 2
Graph G and vertex set T Cut equivalent tree H
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 12 / 19
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How to find a minimum T -cut ?
Theorem (Padberg-Rao)
A minimum T-cut can be found in polynomial time
1 using a cut equivalent tree H and
2 taking the set J(H) edges e of H for which the two connectedcomponents of H − e are T -odd,
3 taking the minimum value c(e∗) of an edge of J(H),
4 taking the cut defined by the two connected components of H − e∗.
3
33
4
2
3 4 2
Graph G and vertex set T Cut equivalent tree H
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How to find a minimum T -cut ?
Theorem (Padberg-Rao)
A minimum T-cut can be found in polynomial time
1 using a cut equivalent tree H and
2 taking the set J(H) edges e of H for which the two connectedcomponents of H − e are T -odd,
3 taking the minimum value c(e∗) of an edge of J(H),
4 taking the cut defined by the two connected components of H − e∗.
3
3
4
2
3 4 2
Graph G and vertex set T Cut equivalent tree H
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 12 / 19
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How to find a minimum T -cut ?
Theorem (Padberg-Rao)
A minimum T-cut can be found in polynomial time
1 using a cut equivalent tree H and
2 taking the set J(H) edges e of H for which the two connectedcomponents of H − e are T -odd,
3 taking the minimum value c(e∗) of an edge of J(H),
4 taking the cut defined by the two connected components of H − e∗.
3
33
4
2
3 4 2
Graph G and vertex set T Cut equivalent tree H
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 12 / 19
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How to find a minimum T -cut ?
Theorem (Padberg-Rao)
A minimum T-cut can be found in polynomial time
1 using a cut equivalent tree H and
2 taking the set J(H) edges e of H for which the two connectedcomponents of H − e are T -odd,
3 taking the minimum value c(e∗) of an edge of J(H),
4 taking the cut defined by the two connected components of H − e∗.
3
33
4
2
3 2
Graph G and vertex set T Cut equivalent tree H
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How to find a minimum T -cut ?
Theorem (Padberg-Rao)
A minimum T-cut can be found in polynomial time
1 using a cut equivalent tree H and
2 taking the set J(H) edges e of H for which the two connectedcomponents of H − e are T -odd,
3 taking the minimum value c(e∗) of an edge of J(H),
4 taking the cut defined by the two connected components of H − e∗.
3
33
4
2
3 4 2
Graph G and vertex set T Cut equivalent tree H and edge set J(H)
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How to find a minimum T -cut ?
Theorem (Padberg-Rao)
A minimum T-cut can be found in polynomial time
1 using a cut equivalent tree H and
2 taking the set J(H) edges e of H for which the two connectedcomponents of H − e are T -odd,
3 taking the minimum value c(e∗) of an edge of J(H),
4 taking the cut defined by the two connected components of H − e∗.
3
33
4
2
3 4 2
Graph G and vertex set T Cut equivalent tree H
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 12 / 19
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How to find a minimum T -cut ?
Theorem (Padberg-Rao)
A minimum T-cut can be found in polynomial time
1 using a cut equivalent tree H and
2 taking the set J(H) edges e of H for which the two connectedcomponents of H − e are T -odd,
3 taking the minimum value c(e∗) of an edge of J(H),
4 taking the cut defined by the two connected components of H − e∗.
3
3
4
2
3 4 2
Graph G and vertex set T Cut equivalent tree H
Z. Szigeti (G-SCOP, Grenoble) Edge-connectivity augmentation 8 April 2009 12 / 19
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How to find a minimum T -cut ?
Theorem (Padberg-Rao)
A minimum T-cut can be found in polynomial time
1 using a cut equivalent tree H and
2 taking the set J(H) edges e of H for which the two connectedcomponents of H − e are T -odd,
3 taking the minimum value c(e∗) of an edge of J(H),
4 taking the cut defined by the two connected components of H − e∗.
3
3
4
2
3 4 2
Minimum T -cut in G Cut equivalent tree H
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Proof
Lemma
For any T -cut δ(X ) there exist x ∈ X , y /∈ X such that λG (x , y) ≥ c(e∗).
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Proof
Lemma
For any T -cut δ(X ) there exist x ∈ X , y /∈ X such that λG (x , y) ≥ c(e∗).
Proof : J(H) is a T -join so there exists xy ∈ J(H) ∩ δH(X ) andλG (x , y) = c(xy) ≥ c(e∗).
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Proof
Lemma
For any T -cut δ(X ) there exist x ∈ X , y /∈ X such that λG (x , y) ≥ c(e∗).
Proof : J(H) is a T -join so there exists xy ∈ J(H) ∩ δH(X ) andλG (x , y) = c(xy) ≥ c(e∗).
Correctness of Padberg-Rao’s algorithm
Let δ(X ) be a minimum T -cut and δ(Y ) the T -cut defined by e∗.By the lemma, there exist x ∈ X , y /∈ X such that
c(e∗) = d(Y ) ≥ d(X ) ≥ λG (x , y) ≥ c(e∗).
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How to augment a minimum T -cut ?
Theorem (Z.Sz.)
Given a connected graph G = (V ,E ),T ⊆ V and k ∈ Z, the minimumnumber of edges whose addition results in a graph so that each T-cut is ofsize at least k is equal to ⌈1
2 maximum p-value of a subpartition of V ⌉. Anoptimal augmentation can be found in polynomial time using
1 Frank’s minimal extension and
2 Mader’s complete splitting off.
Proof1 works because p(X ) := k − dG (X ) if X is T -odd and −∞ otherwise
is symmetric skew-supermodular
(i) k − dG (X ) satisfies both inequalities,(ii) X , Y are T -odd =⇒ either X ∩ Y , X ∪ Y or X −Y , Y −X are T -odd.
2 works because for all T -odd sets, dG ′(X ) ≥ k and, by the abovelemma, k ≤ λG ′(x , y) = λG ′′(x , y) ≤ dG ′′(X ).
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How to augment a minimum T -cut ?
Theorem (Z.Sz.)
Given a connected graph G = (V ,E ),T ⊆ V and k ∈ Z, the minimumnumber of edges whose addition results in a graph so that each T-cut is ofsize at least k is equal to ⌈1
2 maximum p-value of a subpartition of V ⌉. Anoptimal augmentation can be found in polynomial time using
1 Frank’s minimal extension and
2 Mader’s complete splitting off.
Proof1 works because p(X ) := k − dG (X ) if X is T -odd and −∞ otherwise
is symmetric skew-supermodular
(i) k − dG (X ) satisfies both inequalities,(ii) X , Y are T -odd =⇒ either X ∩ Y , X ∪ Y or X −Y , Y −X are T -odd.
2 works because for all T -odd sets, dG ′(X ) ≥ k and, by the abovelemma, k ≤ λG ′(x , y) = λG ′′(x , y) ≤ dG ′′(X ).
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How to augment a minimum T -cut ?
Theorem (Z.Sz.)
Given a connected graph G = (V ,E ),T ⊆ V and k ∈ Z, the minimumnumber of edges whose addition results in a graph so that each T-cut is ofsize at least k is equal to ⌈1
2 maximum p-value of a subpartition of V ⌉. Anoptimal augmentation can be found in polynomial time using
1 Frank’s minimal extension and
2 Mader’s complete splitting off.
Proof1 works because p(X ) := k − dG (X ) if X is T -odd and −∞ otherwise
is symmetric skew-supermodular
(i) k − dG (X ) satisfies both inequalities,(ii) X , Y are T -odd =⇒ either X ∩ Y , X ∪ Y or X −Y , Y −X are T -odd.
2 works because for all T -odd sets, dG ′(X ) ≥ k and, by the abovelemma, k ≤ λG ′(x , y) = λG ′′(x , y) ≤ dG ′′(X ).
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How to augment a minimum T -cut ?
Theorem (Z.Sz.)
Given a connected graph G = (V ,E ),T ⊆ V and k ∈ Z, the minimumnumber of edges whose addition results in a graph so that each T-cut is ofsize at least k is equal to ⌈1
2 maximum p-value of a subpartition of V ⌉. Anoptimal augmentation can be found in polynomial time using
1 Frank’s minimal extension and
2 Mader’s complete splitting off.
Proof1 works because p(X ) := k − dG (X ) if X is T -odd and −∞ otherwise
is symmetric skew-supermodular
(i) k − dG (X ) satisfies both inequalities,(ii) X , Y are T -odd =⇒ either X ∩ Y , X ∪ Y or X −Y , Y −X are T -odd.
2 works because for all T -odd sets, dG ′(X ) ≥ k and, by the abovelemma, k ≤ λG ′(x , y) = λG ′′(x , y) ≤ dG ′′(X ).
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How to augment a minimum T -cut ?
Theorem (Z.Sz.)
Given a connected graph G = (V ,E ),T ⊆ V and k ∈ Z, the minimumnumber of edges whose addition results in a graph so that each T-cut is ofsize at least k is equal to ⌈1
2 maximum p-value of a subpartition of V ⌉. Anoptimal augmentation can be found in polynomial time using
1 Frank’s minimal extension and
2 Mader’s complete splitting off.
Proof1 works because p(X ) := k − dG (X ) if X is T -odd and −∞ otherwise
is symmetric skew-supermodular
(i) k − dG (X ) satisfies both inequalities,(ii) X , Y are T -odd =⇒ either X ∩ Y , X ∪ Y or X −Y , Y −X are T -odd.
2 works because for all T -odd sets, dG ′(X ) ≥ k and, by the abovelemma, k ≤ λG ′(x , y) = λG ′′(x , y) ≤ dG ′′(X ).
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How to augment a minimum T -cut ?
Theorem (Z.Sz.)
Given a connected graph G = (V ,E ),T ⊆ V and k ∈ Z, the minimumnumber of edges whose addition results in a graph so that each T-cut is ofsize at least k is equal to ⌈1
2 maximum p-value of a subpartition of V ⌉. Anoptimal augmentation can be found in polynomial time using
1 Frank’s minimal extension and
2 Mader’s complete splitting off.
Proof1 works because p(X ) := k − dG (X ) if X is T -odd and −∞ otherwise
is symmetric skew-supermodular
(i) k − dG (X ) satisfies both inequalities,(ii) X , Y are T -odd =⇒ either X ∩ Y , X ∪ Y or X −Y , Y −X are T -odd.
2 works because for all T -odd sets, dG ′(X ) ≥ k and, by the abovelemma, k ≤ λG ′(x , y) = λG ′′(x , y) ≤ dG ′′(X ).
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How to augment a minimum T -cut ?
Theorem (Z.Sz.)
Given a connected graph G = (V ,E ),T ⊆ V and k ∈ Z, the minimumnumber of edges whose addition results in a graph so that each T-cut is ofsize at least k is equal to ⌈1
2 maximum p-value of a subpartition of V ⌉. Anoptimal augmentation can be found in polynomial time using
1 Frank’s minimal extension and
2 Mader’s complete splitting off.
Proof1 works because p(X ) := k − dG (X ) if X is T -odd and −∞ otherwise
is symmetric skew-supermodular
(i) k − dG (X ) satisfies both inequalities,(ii) X , Y are T -odd =⇒ either X ∩ Y , X ∪ Y or X −Y , Y −X are T -odd.
2 works because for all T -odd sets, dG ′(X ) ≥ k and, by the abovelemma, k ≤ λG ′(x , y) = λG ′′(x , y) ≤ dG ′′(X ).
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How to augment a minimum T -cut ?
Theorem (Z.Sz.)
Given a connected graph G = (V ,E ),T ⊆ V and k ∈ Z, the minimumnumber of edges whose addition results in a graph so that each T-cut is ofsize at least k is equal to ⌈1
2 maximum p-value of a subpartition of V ⌉. Anoptimal augmentation can be found in polynomial time using
1 Frank’s minimal extension and
2 Mader’s complete splitting off.
Graph G , vertex set T and k = 4
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How to augment a minimum T -cut ?
Theorem (Z.Sz.)
Given a connected graph G = (V ,E ),T ⊆ V and k ∈ Z, the minimumnumber of edges whose addition results in a graph so that each T-cut is ofsize at least k is equal to ⌈1
2 maximum p-value of a subpartition of V ⌉. Anoptimal augmentation can be found in polynomial time using
1 Frank’s minimal extension and
2 Mader’s complete splitting off.
Minimum T -cut in G + F is 4
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Definition : symmetric parity family
Definition
A family F of subsets of V is called symmetric parity family if
1 ∅,V /∈ F ,
2 if A ∈ F , then V − A ∈ F ,
3 if A,B /∈ F and A ∩ B = ∅, then A ∪ B /∈ F .
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Definition : symmetric parity family
Definition
A family F of subsets of V is called symmetric parity family if
1 ∅,V /∈ F ,
2 if A ∈ F , then V − A ∈ F ,
3 if A,B /∈ F and A ∩ B = ∅, then A ∪ B /∈ F .
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Definition : symmetric parity family
Definition
A family F of subsets of V is called symmetric parity family if
1 ∅,V /∈ F ,
2 if A ∈ F , then V − A ∈ F ,
3 if A,B /∈ F and A ∩ B = ∅, then A ∪ B /∈ F .
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Definition : symmetric parity family
Definition
A family F of subsets of V is called symmetric parity family if
1 ∅,V /∈ F ,
2 if A ∈ F , then V − A ∈ F ,
3 if A,B /∈ F and A ∩ B = ∅, then A ∪ B /∈ F .
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Definition : symmetric parity family
Definition
A family F of subsets of V is called symmetric parity family if
1 ∅,V /∈ F ,
2 if A ∈ F , then V − A ∈ F ,
3 if A,B /∈ F and A ∩ B = ∅, then A ∪ B /∈ F .
Examples
The most important examples are :
1 F := 2V − {∅,V }
2 F := {X ⊂ V : X is T -odd} where T ⊆ V with |T | even.
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Definition : symmetric parity family
Definition
A family F of subsets of V is called symmetric parity family if
1 ∅,V /∈ F ,
2 if A ∈ F , then V − A ∈ F ,
3 if A,B /∈ F and A ∩ B = ∅, then A ∪ B /∈ F .
Examples
The most important examples are :
1 F := 2V − {∅,V }
2 F := {X ⊂ V : X is T -odd} where T ⊆ V with |T | even.
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Definition : symmetric parity family
Definition
A family F of subsets of V is called symmetric parity family if
1 ∅,V /∈ F ,
2 if A ∈ F , then V − A ∈ F ,
3 if A,B /∈ F and A ∩ B = ∅, then A ∪ B /∈ F .
Examples
The most important examples are :
1 F := 2V − {∅,V }
2 F := {X ⊂ V : X is T -odd} where T ⊆ V with |T | even.
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How to find a minimum F -cut ?
Theorem (Goemans-Ramakrishnan)
Given a connected graph G and a symmetric parity family F , a minimumF-cut, that is a minimum cut over F , can be found in polynomial time
1 using a cut equivalent tree H and
2 taking the set J(H) edges e of H for which the two connectedcomponents of H − e are in F ,
3 taking the minimum value c(e∗) of an edge of J(H),
4 taking the cut defined by the two connected components of H − e∗.
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How to find a minimum F -cut ?
Theorem (Goemans-Ramakrishnan)
Given a connected graph G and a symmetric parity family F , a minimumF-cut, that is a minimum cut over F , can be found in polynomial time
1 using a cut equivalent tree H and
2 taking the set J(H) edges e of H for which the two connectedcomponents of H − e are in F ,
3 taking the minimum value c(e∗) of an edge of J(H),
4 taking the cut defined by the two connected components of H − e∗.
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Proof
Lemma
For any X ∈ F there exist x ∈ X , y /∈ X such that λG (x , y) ≥ c(e∗).
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Proof
Lemma
For any X ∈ F there exist x ∈ X , y /∈ X such that λG (x , y) ≥ c(e∗).
Proof : Exercise : there exists an edge xy ∈ δJ(H)(X ).
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Proof
Lemma
For any X ∈ F there exist x ∈ X , y /∈ X such that λG (x , y) ≥ c(e∗).
Proof : Exercise : there exists an edge xy ∈ δJ(H)(X ).
Correctness of Goemans-Ramakrishnan’s algorithm
The same proof works as for Padberg-Rao’s algorithm.
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How to augment a minimum F -cut ?
Theorem (Z.Sz.)
Given a connected graph G , a symmetric parity family F and k ∈ Z, theminimum number of edges whose addition results in a graph so that eachF-cut is of size at least k is equal to ⌈1
2 maximum p-value of a subpartitionof V ⌉. An optimal augmentation can be found in polynomial time using
1 Frank’s minimal extension and
2 Mader’s complete splitting off.
Proof1 works because p(X ) := k − dG (X ) if X ∈ F and −∞ otherwise is
symmetric skew-supermodular
(i) k − dG (X ) satisfies both inequalities,(ii) If X , Y ∈ F , then either X ∩ Y , X ∪ Y ∈ F or X − Y , Y − X ∈ F .
2 works because for all X ∈ F , dG ′(X ) ≥ k and, by the above lemma,k ≤ λG ′(x , y) = λG ′′(x , y) ≤ dG ′′(X ).
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Conclusion
1 Special cases :1 Global edge-connectivity augmentation (Watanabe, Nakamura)2 Minimum T -cut augmentation
2 A new polynomial special case of the NP-complete problemMinimum Cover of a Symmetric Skew-Supermodular
Function by a Graph
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Conclusion
1 Special cases :1 Global edge-connectivity augmentation (Watanabe, Nakamura)2 Minimum T -cut augmentation
2 A new polynomial special case of the NP-complete problemMinimum Cover of a Symmetric Skew-Supermodular
Function by a Graph
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Conclusion
1 Special cases :1 Global edge-connectivity augmentation (Watanabe, Nakamura)2 Minimum T -cut augmentation
2 A new polynomial special case of the NP-complete problemMinimum Cover of a Symmetric Skew-Supermodular
Function by a Graph
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Conclusion
1 Special cases :1 Global edge-connectivity augmentation (Watanabe, Nakamura)2 Minimum T -cut augmentation
2 A new polynomial special case of the NP-complete problemMinimum Cover of a Symmetric Skew-Supermodular
Function by a Graph
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