A Flow Model Based on Linking Systems with Applications in ......A Flow Model Based on Linking Systems with Applications in Network Coding Rico Zenklusen Institute for Discrete Optimization,

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



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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The ADT information flow model

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I Task: Send maximum number of signals from s to t.

I A signal is an element of F2.

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I → Interference between the two signals!

I Interference is modelled as XOR.

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I → Interference between the two signals!

I Interference is modelled as XOR.

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I Receiver gets signals (x , x + y).

I Thanks to linear independence, received signals can be decoded toget original signals.

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I Received signals are linearly dependent.

→ Receiver cannot properly decode.

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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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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


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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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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i) (P1,P2) ∈ Λ⇒ |P1| = |P2|,




ii)

iv)

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i) (P1,P2) ∈ Λ⇒ |P1| = |P2|,




ii)

iv)

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i) (P1,P2) ∈ Λ⇒ |P1| = |P2|,




ii) iv)

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i) (P1,P2) ∈ Λ⇒ |P1| = |P2|,




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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λ(P1,P2) = max{|Q1| | (Q1,Q2) ∈ Λ,Q1 ⊆ P1,Q2 ⊆ P2}.


(P1,P2) ∈ Λ⇔ λ(P1,P2) = |P1| = |P2|.


i) 0 ≤ λ(P1,P2) ≤ min{|P1|, |P2|},

ii) Q1 ⊆ P1,Q2 ⊆ P2 ⇒ λ(Q1,Q2) ≤ λ(P1,P2),


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λ(P1,P2) = max{|Q1| | (Q1,Q2) ∈ Λ,Q1 ⊆ P1,Q2 ⊆ P2}.


(P1,P2) ∈ Λ⇔ λ(P1,P2) = |P1| = |P2|.


i) 0 ≤ λ(P1,P2) ≤ min{|P1|, |P2|},

ii) Q1 ⊆ P1,Q2 ⊆ P2 ⇒ λ(Q1,Q2) ≤ λ(P1,P2),


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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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φ(C ) =r−1∑i=1

λi (Ci ,Vi+1 \ Ci+1).


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φ(C ) =r−1∑i=1

λi (Ci ,Vi+1 \ Ci+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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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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I For any flow F ,

F1 ∪ (r−1⋃i=2

Vi ) ∪ (Vr \ Fr ) ∈ FΛ.


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I For any flow F ,

F1 ∪ (r−1⋃i=2

Vi ) ∪ (Vr \ Fr ) ∈ FΛ.


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I For any flow F ,

F1 ∪ (r−1⋃i=2

Vi ) ∪ (Vr \ Fr ) ∈ FΛ.


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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



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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A Flow Model Based on Linking Systems with Applications in ......A Flow Model Based on Linking Systems with Applications in Network Coding Rico Zenklusen Institute for Discrete Optimization,

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