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A Flow Model Based on Linking Systemswith Applications in Network Coding
Rico Zenklusen
Institute for Discrete Optimization, EPFL
Joint work with Michel Goemans and Satoru Iwata
Aussois Workshop 2010
Outline
1 Motivation (wireless information flow)
2 A flow model based on (poly-)linking systems
3 Conclusions
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Outline
1 Motivation (wireless information flow)
2 A flow model based on (poly-)linking systems
3 Conclusions
Wireless information flows
Features of wireless information flows
I Broadcasting (signal emitted by one transmitter is received by manynodes).
I Superposition of signal (interference).
⇒ This leads to complex signal interactions.
Classical model: Multiuser Gaussian Channel
I Unknown how the capacity of the network can be determined exceptfor simplest networks.
The ADT model [Avestimehr, Diggavi, and Tse, 2007a]
I A deterministic model to approximate Multiuser Gaussian Channels.
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Wireless information flows
Features of wireless information flows
I Broadcasting (signal emitted by one transmitter is received by manynodes).
I Superposition of signal (interference).
⇒ This leads to complex signal interactions.
Classical model: Multiuser Gaussian Channel
I Unknown how the capacity of the network can be determined exceptfor simplest networks.
The ADT model [Avestimehr, Diggavi, and Tse, 2007a]
I A deterministic model to approximate Multiuser Gaussian Channels.
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Wireless information flows
Features of wireless information flows
I Broadcasting (signal emitted by one transmitter is received by manynodes).
I Superposition of signal (interference).
⇒ This leads to complex signal interactions.
Classical model: Multiuser Gaussian Channel
I Unknown how the capacity of the network can be determined exceptfor simplest networks.
The ADT model [Avestimehr, Diggavi, and Tse, 2007a]
I A deterministic model to approximate Multiuser Gaussian Channels.
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The ADT information flow model
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The ADT information flow model
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The ADT information flow model
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The ADT information flow model
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The ADT information flow model
I Task: Send maximum number of signals from s to t.
I A signal is an element of F2.
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The ADT information flow model
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The ADT information flow model
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The ADT information flow model
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The ADT information flow model
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The ADT information flow model
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The ADT information flow model
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The ADT information flow model
I → Interference between the two signals!
I Interference is modelled as XOR.
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The ADT information flow model
I → Interference between the two signals!
I Interference is modelled as XOR.
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The ADT information flow model
I Receiver gets signals (x , x + y).
I Thanks to linear independence, received signals can be decoded toget original signals.
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The ADT information flow model
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The ADT information flow model
I Received signals are linearly dependent.
→ Receiver cannot properly decode.
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The ADT information flow model
GoalRoute maximum number of decodable (i.e., linearly independent) signalsfrom s to t.
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Another representation of ADT flows
An ADT flow can be represented by set of used vertices.
I Concerning linear independence, exact wiring does not matter.
I Linear independence ⇔ Adjacency matrix induced by used verticesin any two consecutive layers is full rank.
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Another representation of ADT flows
An ADT flow can be represented by set of used vertices.
I Concerning linear independence, exact wiring does not matter.
I Linear independence ⇔ Adjacency matrix induced by used verticesin any two consecutive layers is full rank.
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Another representation of ADT flows
An ADT flow can be represented by set of used vertices.
I Concerning linear independence, exact wiring does not matter.
I Linear independence ⇔ Adjacency matrix induced by used verticesin any two consecutive layers is full rank.
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Another representation of ADT flows
Propagation of signals from second to third layer:
(x , y) ·(
1 01 1
)︸ ︷︷ ︸
Induced adjacencymatrix
= (x + y , y).
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Results on ADT network flows
Theorem ([Avestimehr, Diggavi, and Tse, 2007b])
A notion of cut was introduced such that:Max ADT flow = Min ADT cut.
Theorem ([Amaudruz and Fragouli, 2009])
A maximum flow and a minimum cut can be found polynomial time.
In this talk: A more general flow model
I Max-flow min-cut theorem.
I Efficient optimization is possible (even with costs and capacities).
I Many other results can easily be deduced from matroid theory.
I Classical matroid algorithms can be used for optimization.
I We heavily use results from Lex Schrijver’s Ph.D. thesis (on linkingsystems and polylinking systems).
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Results on ADT network flows
Theorem ([Avestimehr, Diggavi, and Tse, 2007b])
A notion of cut was introduced such that:Max ADT flow = Min ADT cut.
Theorem ([Amaudruz and Fragouli, 2009])
A maximum flow and a minimum cut can be found polynomial time.
In this talk: A more general flow model
I Max-flow min-cut theorem.
I Efficient optimization is possible (even with costs and capacities).
I Many other results can easily be deduced from matroid theory.
I Classical matroid algorithms can be used for optimization.
I We heavily use results from Lex Schrijver’s Ph.D. thesis (on linkingsystems and polylinking systems).
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Outline
1 Motivation (wireless information flow)
2 A flow model based on (poly-)linking systems• Linking systems• Linking network• Optimization in linking networks• Linking flow polytope
3 Conclusions
Motivation of linking systems
IntuitionRelation between two finite sets V1,V2 that preserves matroid structure.
Induction of matroids (by a bipartite graph)
Let G = (V1 ∪ V2,E ) be a bipartite graph and let M = (V1,F) be amatroid.
{P2 ⊆ V2 | ∃P1 ∈ F such that G [P1 ∪ P2] contains a perfect matching}are independent sets of a matroid on V2.
→ Generalizations ?
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Motivation of linking systems
IntuitionRelation between two finite sets V1,V2 that preserves matroid structure.
Induction of matroids (by a bipartite graph)
Let G = (V1 ∪ V2,E ) be a bipartite graph and let M = (V1,F) be amatroid.
{P2 ⊆ V2 | ∃P1 ∈ F such that G [P1 ∪ P2] contains a perfect matching}are independent sets of a matroid on V2.
→ Generalizations ?
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Motivation of linking systems
IntuitionRelation between two finite sets V1,V2 that preserves matroid structure.
Induction of matroids (by a bipartite graph)
Let G = (V1 ∪ V2,E ) be a bipartite graph and let M = (V1,F) be amatroid.
{P2 ⊆ V2 | ∃P1 ∈ F such that G [P1 ∪ P2] contains a perfect matching}are independent sets of a matroid on V2.
→ Generalizations ?
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Linking systems [Schrijver, 1978]
Definition: Linking system
A linking system between V1 and V2 is a triple (V1,V2,Λ) with∅ 6= Λ ⊆ 2V1 × 2V2 and satisfying:
i) (P1,P2) ∈ Λ⇒ |P1| = |P2|,
ii) (P1,P2) ∈ Λ,Q1 ⊆ P1 ⇒ ∃Q2 ⊆ P2 with (Q1,Q2) ∈ Λ,
iii) (P1,P2) ∈ Λ,Q2 ⊆ P2 ⇒ ∃Q1 ⊆ P1 with (Q1,Q2) ∈ Λ,
iv) (P1,P2), (Q1,Q2) ∈ Λ⇒ ∃(R1,R2) ∈ Λ with P1 ⊆ R1 ⊆ P1 ∪ Q1,Q2 ⊆ R2 ⊆ P2 ∪ Q2.
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Linking systems [Schrijver, 1978]
Definition: Linking system
A linking system between V1 and V2 is a triple (V1,V2,Λ) with∅ 6= Λ ⊆ 2V1 × 2V2 and satisfying:
i) (P1,P2) ∈ Λ⇒ |P1| = |P2|,
ii) (P1,P2) ∈ Λ,Q1 ⊆ P1 ⇒ ∃Q2 ⊆ P2 with (Q1,Q2) ∈ Λ,
iii) (P1,P2) ∈ Λ,Q2 ⊆ P2 ⇒ ∃Q1 ⊆ P1 with (Q1,Q2) ∈ Λ,
iv) (P1,P2), (Q1,Q2) ∈ Λ⇒ ∃(R1,R2) ∈ Λ with P1 ⊆ R1 ⊆ P1 ∪ Q1,Q2 ⊆ R2 ⊆ P2 ∪ Q2.
ii)
iv)
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Linking systems [Schrijver, 1978]
Definition: Linking system
A linking system between V1 and V2 is a triple (V1,V2,Λ) with∅ 6= Λ ⊆ 2V1 × 2V2 and satisfying:
i) (P1,P2) ∈ Λ⇒ |P1| = |P2|,
ii) (P1,P2) ∈ Λ,Q1 ⊆ P1 ⇒ ∃Q2 ⊆ P2 with (Q1,Q2) ∈ Λ,
iii) (P1,P2) ∈ Λ,Q2 ⊆ P2 ⇒ ∃Q1 ⊆ P1 with (Q1,Q2) ∈ Λ,
iv) (P1,P2), (Q1,Q2) ∈ Λ⇒ ∃(R1,R2) ∈ Λ with P1 ⊆ R1 ⊆ P1 ∪ Q1,Q2 ⊆ R2 ⊆ P2 ∪ Q2.
ii)
iv)
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Linking systems [Schrijver, 1978]
Definition: Linking system
A linking system between V1 and V2 is a triple (V1,V2,Λ) with∅ 6= Λ ⊆ 2V1 × 2V2 and satisfying:
i) (P1,P2) ∈ Λ⇒ |P1| = |P2|,
ii) (P1,P2) ∈ Λ,Q1 ⊆ P1 ⇒ ∃Q2 ⊆ P2 with (Q1,Q2) ∈ Λ,
iii) (P1,P2) ∈ Λ,Q2 ⊆ P2 ⇒ ∃Q1 ⊆ P1 with (Q1,Q2) ∈ Λ,
iv) (P1,P2), (Q1,Q2) ∈ Λ⇒ ∃(R1,R2) ∈ Λ with P1 ⊆ R1 ⊆ P1 ∪ Q1,Q2 ⊆ R2 ⊆ P2 ∪ Q2.
ii) iv)
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Linking systems [Schrijver, 1978]
Definition: Linking system
A linking system between V1 and V2 is a triple (V1,V2,Λ) with∅ 6= Λ ⊆ 2V1 × 2V2 and satisfying:
i) (P1,P2) ∈ Λ⇒ |P1| = |P2|,
ii) (P1,P2) ∈ Λ,Q1 ⊆ P1 ⇒ ∃Q2 ⊆ P2 with (Q1,Q2) ∈ Λ,
iii) (P1,P2) ∈ Λ,Q2 ⊆ P2 ⇒ ∃Q1 ⊆ P1 with (Q1,Q2) ∈ Λ,
iv) (P1,P2), (Q1,Q2) ∈ Λ⇒ ∃(R1,R2) ∈ Λ with P1 ⊆ R1 ⊆ P1 ∪ Q1,Q2 ⊆ R2 ⊆ P2 ∪ Q2.
ii) iv)
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Linking systems: Examples I
Induced by bipartite graph
Let G = (V1 ∪ V2,E ) be a bipartite graph. Then (V1,V2,Λ) is a linkingsystem where
Λ = {(P1,P2) ∈ 2V1 × 2V2 | ∃ perfect matching in G [P1 ∪ P2]}.
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Linking systems: Examples II
Induced by matrix
Let A ∈ Rn×m where V1 resp. V2 are the sets of row and column indices.Then (V1,V2,Λ) is a linking system where
Λ = {(P1,P2) ∈ 2V1 × 2V2 | A[P1,P2] is full rank}.1 2 5 0 100 0 3 3 70 1 2 1 42 0 7 2 8
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Linking function (bisubmodular functions)
Definition of linking function
λ(P1,P2) = max{|Q1| | (Q1,Q2) ∈ Λ,Q1 ⊆ P1,Q2 ⊆ P2}.
Linking function determines linking system
(P1,P2) ∈ Λ⇔ λ(P1,P2) = |P1| = |P2|.
Characterization of linking functions
i) 0 ≤ λ(P1,P2) ≤ min{|P1|, |P2|},
ii) Q1 ⊆ P1,Q2 ⊆ P2 ⇒ λ(Q1,Q2) ≤ λ(P1,P2),
iii) λ(P1 ∩Q1,P2 ∪Q2) + λ(P1 ∪Q1,P2 ∩Q2) ≤ λ(P1,P2) + λ(Q1,Q2).
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Linking function (bisubmodular functions)
Definition of linking function
λ(P1,P2) = max{|Q1| | (Q1,Q2) ∈ Λ,Q1 ⊆ P1,Q2 ⊆ P2}.
Linking function determines linking system
(P1,P2) ∈ Λ⇔ λ(P1,P2) = |P1| = |P2|.
Characterization of linking functions
i) 0 ≤ λ(P1,P2) ≤ min{|P1|, |P2|},
ii) Q1 ⊆ P1,Q2 ⊆ P2 ⇒ λ(Q1,Q2) ≤ λ(P1,P2),
iii) λ(P1 ∩Q1,P2 ∪Q2) + λ(P1 ∪Q1,P2 ∩Q2) ≤ λ(P1,P2) + λ(Q1,Q2).
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Linking function (bisubmodular functions)
Definition of linking function
λ(P1,P2) = max{|Q1| | (Q1,Q2) ∈ Λ,Q1 ⊆ P1,Q2 ⊆ P2}.
Linking function determines linking system
(P1,P2) ∈ Λ⇔ λ(P1,P2) = |P1| = |P2|.
Characterization of linking functions
i) 0 ≤ λ(P1,P2) ≤ min{|P1|, |P2|},
ii) Q1 ⊆ P1,Q2 ⊆ P2 ⇒ λ(Q1,Q2) ≤ λ(P1,P2),
iii) λ(P1 ∩Q1,P2 ∪Q2) + λ(P1 ∪Q1,P2 ∩Q2) ≤ λ(P1,P2) + λ(Q1,Q2).
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A matroidal property
TheoremLet (V1,V2,Λ) be a linking system.
BΛ = {P1 ∪ (V2 \ P2) | (P1,P2) ∈ Λ}
forms the set of bases of a matroid. We denote this matroid byMΛ = (V1 ∪ V2,FΛ).
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The product of linking systems
linking system ? linking system → linking system.
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The product of linking systems
linking system ? linking system → linking system.
Linking system ? linking system
Let (V1,V2,Λ1), (V2,V3,Λ2) be two linking systems with linkingfunctions λ1, λ2 and let
Λ1 ? Λ2 = {(P1,P3) ∈ 2V1 × 2V3 | ∃P2 ⊆ V2 with (P1,P2) ∈ Λ1,(P2,P3) ∈ Λ2}.
Then (V1,V3,Λ1 ? Λ2) is a linking system with linking function
(λ1 ? λ2)(P1,P3) = minP2⊆V2
(λ1(P1,P2) + λ2(V2 \ P2,P3)).
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Linking network (A flow model based on linking systems)
Definition: Linking network
Let V1, . . . ,Vr be finite disjoint sets and let (Vi ,Vi+1,Λi ) be a linkingsystem for i ∈ {1, . . . , r − 1}. Then G = (V ,Λ) is a linking networkwhere V = (V1, . . . ,Vr ), Λ = (Λ1, . . . ,Λr−1).
Definition: Linking flow
Tuple F = (F1, . . . ,Fr ) where (Fi ,Fi+1) ∈ Λi for i ∈ {1, . . . , r − 1}.
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Linking network (A flow model based on linking systems)
Definition: Linking network
Let V1, . . . ,Vr be finite disjoint sets and let (Vi ,Vi+1,Λi ) be a linkingsystem for i ∈ {1, . . . , r − 1}. Then G = (V ,Λ) is a linking networkwhere V = (V1, . . . ,Vr ), Λ = (Λ1, . . . ,Λr−1).
Definition: Linking flow
Tuple F = (F1, . . . ,Fr ) where (Fi ,Fi+1) ∈ Λi for i ∈ {1, . . . , r − 1}.
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ADT flow is a linking flow
I In every node we add a complete bipartite graph.
I The linking systems alternate between:I Linking system induced by adjacency matrix.I Linking system induced by bipartite graph.
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ADT flow is a linking flow
I In every node we add a complete bipartite graph.
I The linking systems alternate between:I Linking system induced by adjacency matrix.I Linking system induced by bipartite graph.
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Source-destination cuts in linking networks
Definition: CutTuple C = (C1, . . . ,Cr ) with Ci ⊆ Vi ∀i ∈ {1, . . . , r}, C1 = V1, Cr = ∅.
Definition: Value of a cut
φ(C ) =r−1∑i=1
λi (Ci ,Vi+1 \ Ci+1).
Min cut ≥ Max flow.
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Source-destination cuts in linking networks
Definition: CutTuple C = (C1, . . . ,Cr ) with Ci ⊆ Vi ∀i ∈ {1, . . . , r}, C1 = V1, Cr = ∅.
Definition: Value of a cut
φ(C ) =r−1∑i=1
λi (Ci ,Vi+1 \ Ci+1).
Min cut ≥ Max flow.
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Source-destination cuts in linking networks
Definition: CutTuple C = (C1, . . . ,Cr ) with Ci ⊆ Vi ∀i ∈ {1, . . . , r}, C1 = V1, Cr = ∅.
Definition: Value of a cut
φ(C ) =r−1∑i=1
λi (Ci ,Vi+1 \ Ci+1).
Min cut ≥ Max flow.
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Max-flow min-cut theorem in linking networks
Theorem: Max-flow min-cut
Value of max-flow = Value of min-cut
Proof.
I Let Λ = Λ1 ? · · · ? Λr−1 with corresponding linking function λ.
I Value of max flow = λ(V1,Vr ).
I Recall: Linking function of two chained linking systems Λ1 ? Λ2:
(λ1 ? λ2)(P1,P3) = minP2⊆V2
(λ1(P1,P2) + λ2(V2 \ P2,P3)).
I By repeatedly applying the above formula we get
λ(V1,Vr ) = min
{φ(V1 ∪
r−1⋃i=2
Pi ) | P2 ⊆ V2, . . . ,Pr−1 ⊆ Vr−1
}.
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Max-flow min-cut theorem in linking networks
Theorem: Max-flow min-cut
Value of max-flow = Value of min-cut
Proof.
I Let Λ = Λ1 ? · · · ? Λr−1 with corresponding linking function λ.
I Value of max flow = λ(V1,Vr ).
I Recall: Linking function of two chained linking systems Λ1 ? Λ2:
(λ1 ? λ2)(P1,P3) = minP2⊆V2
(λ1(P1,P2) + λ2(V2 \ P2,P3)).
I By repeatedly applying the above formula we get
λ(V1,Vr ) = min
{φ(V1 ∪
r−1⋃i=2
Pi ) | P2 ⊆ V2, . . . ,Pr−1 ⊆ Vr−1
}.
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Some other properties
Submodularity of cut value
The value function of cuts φ(C ) =∑r−1
i=1 λi (Ci ,Vi+1 \ Ci+1) issubmodular.
Gammoid property
The set of attainable vertices in layer r (or any other fixed layerl ∈ {1, . . . , r}) form a matroid, i.e,
{Fr | (F1, . . . ,Fr ) linking flow}
are independent sets of a matroid on Vr .
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Some other properties
Submodularity of cut value
The value function of cuts φ(C ) =∑r−1
i=1 λi (Ci ,Vi+1 \ Ci+1) issubmodular.
Gammoid property
The set of attainable vertices in layer r (or any other fixed layerl ∈ {1, . . . , r}) form a matroid, i.e,
{Fr | (F1, . . . ,Fr ) linking flow}
are independent sets of a matroid on Vr .
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Finding flows through matroid union
Let MΛ = (∪ri=1Vi ,FΛ) be the union of the matroids MΛ1 , . . . ,MΛr−1 .
I For any flow F ,
F1 ∪ (r−1⋃i=2
Vi ) ∪ (Vr \ Fr ) ∈ FΛ.
I A maximum flow can be found by a matroid partitioning algorithm:Find a maximum independent set in MΛ with as many elements inV1 as possible (the number of elements in V1 is the value of theflow).
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Finding flows through matroid union
Let MΛ = (∪ri=1Vi ,FΛ) be the union of the matroids MΛ1 , . . . ,MΛr−1 .
I For any flow F ,
F1 ∪ (r−1⋃i=2
Vi ) ∪ (Vr \ Fr ) ∈ FΛ.
I A maximum flow can be found by a matroid partitioning algorithm:Find a maximum independent set in MΛ with as many elements inV1 as possible (the number of elements in V1 is the value of theflow).
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Finding flows through matroid union
Let MΛ = (∪ri=1Vi ,FΛ) be the union of the matroids MΛ1 , . . . ,MΛr−1 .
I For any flow F ,
F1 ∪ (r−1⋃i=2
Vi ) ∪ (Vr \ Fr ) ∈ FΛ.
I A maximum flow can be found by a matroid partitioning algorithm:Find a maximum independent set in MΛ with as many elements inV1 as possible (the number of elements in V1 is the value of theflow).
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Finding flows through matroid union
Let MΛ = (∪ri=1Vi ,FΛ) be the union of the matroids MΛ1 , . . . ,MΛr−1 .
I For any flow F ,
F1 ∪ (r−1⋃i=2
Vi ) ∪ (Vr \ Fr ) ∈ FΛ.
I A maximum flow can be found by a matroid partitioning algorithm:Find a maximum independent set in MΛ with as many elements inV1 as possible (the number of elements in V1 is the value of theflow).
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Finding a minimum cut
I Let M−Λ be the matroid MΛ restricted to ∪r−1i=1 Vi .
I Let ∪r−1i=1 Ii be a maximum cardinality independent set M−Λ with
∪r−1i=2 Vi ⊆ ∪r−1
i=1 Ii .
I By the Theorem of Nash-Williams we have
ρ−Λ (∪r−1i=1 Vi )︸ ︷︷ ︸
=|∪r−1i=1 Ii |
= minA⊆∪r−1
i=1 Vi
{|(∪r−1
i=1 Vi ) \ A|+r−1∑i=1
ρΛi (A)
}.
I Let A be a set attaining the above minimum (typically obtained asbyproduct of a matroid partitioning algorithm).
I Expanding the minimum in the Nash-Williams formula, it can beshown that (A ∩ V1, . . . ,A ∩ Vr−1,Vr ) is a minimum cut.
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Linking flow polytope
Linking flow polytope
Let G = (V ,Λ) be a linking network. Its linking flow polytope is definedby
LFP(G ) =
x(Pi )− x(Vi+1 \ Pi+1) ≤ λi (Pi ,Pi+1) ∀i ∈ {1, . . . , r − 1},∀Pi ⊆ Vi ,∀Pi+1 ⊆ Vi+1
x(Vi ) = x(Vi+1) ∀i ∈ {1, . . . , r − 1}
x ∈ RPr
i=1 |Vi |+ .
Theorem: Integrality of LFP(G )
LFP(G ) is integral and its vertices correspond to linking flows.
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Integrality of LFP(G): Sketch of proof
LFP(G ) is a projection of the following polytope.
x i (Pi )− x i (Vi+1 \ Pi+1) ≤ λ(Pi ,Pi+1) ∀i ∈ {1, . . . , r − 1},∀Pi ⊆ Vi ,Pi+1 ⊆ Vi+1
x i (Vi ) = x i (Vi+1) ∀i ∈ {1, . . . , r − 1}x i (v) = x i+1(v) ∀i ∈ {1, . . . , r − 1},∀v ∈ Vi+1
x i ∈ R|Vi |+ ∀i ∈ {1, . . . , r}
I It suffices to show that the above polytope is integral.
I Choose a vertex of above polytope → defined by a set of equalities.
I We can uncross the equalities of this type for i ∈ {1, . . . , r − 1}such that if for a given i we have equalities for the tuples(Pi,1,Pi+1,1), . . . , (Pi,m,Pi+1,m) then the family
{Pi,k ∪ (Vi+1 \ Pi+1,k) | k ∈ {1, . . . ,m}}is laminar.
I Obtained equation system is totally unimodular.
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Outline
1 Motivation (wireless information flow)
2 A flow model based on (poly-)linking systems
3 Conclusions
Conclusions and OutlookI Linking networks: A flow model based on linking systems and
generalizing the ADT model.
I Many nice properties:I Gammoid property.I Submodularity of cut-values.I Max-flow min-cut result.
I Efficient optimization is possible using standard matroid algorithms.
I Optimization with respect to costs is possible.
I Capacities can be incorporated by replacing linking systems withpolylinking systems.
I Generalization to more general model where the graph does notneed to be acyclic?
I How to adapt current matroid algorithms to exploit specialstructure of linking systems?
I Applications to other problems in network coding?
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Polylinking systems [Schrijver, 1978]
Definition: Polylinking system
A polylinking system between V1 and V2 is a triple (V1,V2, L) where∅ 6= L ⊆ RV1
+ × RV2+ is a compact set satisfying:
i) (x1, x2) ∈ L⇒ |x1| = |x2|,
ii) (x1, x2) ∈ L, 0 ≤ y1 ≤ x1 ⇒ ∃y2 ≤ x2 with (y1, y2) ∈ L,
iii) (x1, x2) ∈ L, 0 ≤ y2 ≤ x2 ⇒ ∃y1 ≤ x1 with (y1, y2) ∈ L,
iv) (x1, x2), (y1, y2) ∈ L⇒ ∃(z1, z2) ∈ L with x1 ≤ z1 ≤ x1 ∨ y1,y2 ≤ z2 ≤ x2 ∨ y2.
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References
A. Amaudruz and C. Fragouli. Combinatorial algorithms for wirelessinformation flow. In SODA ’09: Proceedings of the Twentieth AnnualACM-SIAM Symposium on Discrete Algorithms, 2009.
A. S. Avestimehr, S. N. Diggavi, and D. N. C. Tse. A deterministicapproach to wireless relay networks. In Proceedings of AllertonConference on Communication, Control, and Computing, September2007a. http://licos.epfl.ch/index.php?p=research projWNC.
A. S. Avestimehr, S. N. Diggavi, and D. N. C. Tse. Wireless networkinformation flow. In Proceedings of Allerton Conference onCommunication, Control, and Computing, September 2007b.http://licos.epfl.ch/index.php?p=research projWNC.
A. Schrijver. Matroids and Linking Systems. PhD thesis, MathematischCentrum, 1978.
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