MITSUBISHI ELECTRIC RESEARCH LABORATORIES Cambridge, Massachusetts … · 2015-06-18 · © MERL...

© MERL

MITSUBISHI ELECTRIC RESEARCH LABORATORIES

Cambridge, Massachusetts

Srikumar Ramalingam

Introduction to Matroids and Applications

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© MERL


Linear Algebra

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)0,0,1(a

)0,1,0(b

)1,0,0(c)1,1,0(d

• Identify subsets of linearly independent

vectors.

Linear independence in vectors:

For all non-trivial

we have

nvvv ,...,, 21

nsss ,...,, 21

.0...2211 nnvsvsvs

a

b

c

a

c d

b

c

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Graph Theory

Choose subsets of edges without

cycles (also known as forests –

collection of trees).

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b

a

c

d

c

a

c

d

b

a

cb

a

d

b c

a

c

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Assignment of Jobs

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a

b

c

d

1

2

3

Applicants Jobs

• 1 Person takes 1 Job.

• Every job has only 1 opening.

Identify possible assignments

between applicants and jobs.

a

b

c

d

1

2

3

D E

a

b

c

d

1

2

3

D E

a

b

c

d

1

2

3

D E

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Budget constraints

• Identify subsets of the set such that a maximum

of 1 element is taken from subset and a maximum of

2 elements are taken from subset

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a

b

d

c

1 2

),,,( dcbaE

}{1 aE

}.,,{2 dcbE

a

b

1 2

a

bc

1 2

b

d

1 2

}{1 aE }.,,{2 dcbE

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Vertex disjoint paths

• Find all vertex disjoint paths from vertices in to vertices in

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S .T

a

b

c

d

123

4

5

S T

a

b

c

d

123

4

5

S T

a

b

c

d

123

4

5

S T

a

b

c

d

123

4

5

S T

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Common properties

• All five problems have the same solutions.

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acdabdabc

cdbdbcadacab

dcba

,,

,,,,,

,,,

,

• The size of the largest set: 3

• The empty set is always a solution.

• All subsets of a given solution is also a solution.

• All these scenarios can be represented using matroids!

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• Matroids are everywhere, if only we knew how to look.

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Matroid Definition (introduced by Whitney in 1935)

• A matroid is a pair where

– is a finite set.

– is a family of subsets of such that:

• (I1)

• (I2) If and then

• (I3) If and , then there exists

such that

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)Ι,(E

E

Ι E

Ι

ΙB Ι.ABA

ΙBA, |||| BA

Be

is called the “ground set” and is referred to as the

collection of independent sets.

E Ι

I. )( eA

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Graphic Matroid

• Let be a graph and let be a

collection of edge sets (subsets of )

without cycles, then is a matroid.

)EVG ,( Ι

E

)Ι,(E

b

a

c

d

},,,{ dcbaE

b

a

Independent sets

d

c

a

d

b

a

a

c

d

b

a

d

b

a

d

b

a

cb c

a

c

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Matroid Definition (using an Example)

• (I1) is and independent set.

• (2) Since is independent, then all

its subset are also

independent.

• (I3) If and there exists

such that

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b

a

c

d

acdabdabc

cdbdbcadacab

dcba

,,

,,,,,

,,,

,

abc

},,,,,,{ bcacabcba

abc ad cad

a

c

d

a

d

b

a

c

ΙJI , |||| JI

Je I. )( eI

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Linear Matroid

• Let be a finite set of vectors in a

vector space and let be the

collection of “linearly dependent

sets” in then is a matroid.

E

V

)Ι,(E

Ι

E

)0,0,1(a

)0,1,0(b

)1,0,0(c)1,1,0(da a

b

a

b

c

a

c

a

c d

b

c

},,,{ dcbaE

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Partition Matroid

• Let be a finite set of vectors and let

be disjoint subsets of

Given positive integers , let

be the collection of subsets of

each subset has atmost from

is a matroid.

E

Ι

.ENEEE ,...,, 21

N Nkk ,...,1

E

ik .iE

)Ι,(E

1, 11 kE

2, 22 kE

Independent sets

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Transversal Matroid

• Let be a bipartite graph with

bipartition Let be the

collection of subsets of which

can be matched to

is a matroid.

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)Ι,(E

G

).,( ED Ι

E

.D

a

b

c

d

1

2

3

D E

a

b

c

d

1

2

3

D E

a

b

c

d

1

2

3

D E

a

b

c

d

1

2

3

D E

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Basis of a Matroid

• A basis of a matroid is a maximal independent set.

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b

a

c

d

Example:

acdabdabc ,,Bases:

• All the bases of a matroid have the same size.

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Rank

• Let be a matroid. The rank of a subset of is given by

the size of the largest independent set contained in it.

• The rank of a set

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)Ι,(E

:EAA,

|| BIBA,Bargmaxrank(A)

E

b

a

c

d2d})c,rank({b,

2b})rank({a,

3d})c,b,rank({a,

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Dual of a matroid is a matroid

• If the dual matroid of is and then

has a base of .

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)Ι,(EM )*Ι,(* EM *ΙA

AE )Ι,(EM

b

a

c

d)Ι,(EM

acdabdabc

cdbdbcadacab

dcba

,,

,,,,,,

,,,,

,

)*Ι,(* EM

dcd ,,

,

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Greedy Algorithm

• Given a matroid and weights , find a basis of

minimum weight.

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)Ι,(E REw :

1. Start with

2. Add to the smallest s.t

3. Repeat until you have a basis.

{}.A

A e I. eA

(Greedy algorithm guarantees

an optimal soln.)

(The underlying structure

is matroid)

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Greedy Algorithm

• Given a matroid and weights , find a basis of

minimum weight.

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)Ι,(E REw :

I)M

I,

p ,(

)(||

..

min

E

A

ErankA

ts

ewAe

iiEA

i

Minimal spanning algorithm is very simple and useful!

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Matroids in other domains – physical realizability

They also appear in several geometry problems:

arrangements of hyperplanes, configurations of points, etc.

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The lifting procedure where the 3D points are computed

on the projection rays satisfying all the constraints from

projections and line labels.

• Sugihara’s approach lifts

line drawings to 3D space

for trihedral drawings.

• Check whether a line

drawing is physically

realizable or not.

• For general line drawings,

Whiteley extended

Sugihara’s work using

matroids in 1989.

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Line-Lifting

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Red, Blue and Green denote

lines in orthogonal directions

VRML model of the line

reconstruction

Image

Given a single image captured by your mobile phone or other devices:

[Ramalingam and Brand, 2013]

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Our Main Idea

Our main assumption is that the maximum cardinality subset that satisfies

all the constraints will consist of correct intersections.

Orthogonal input lines from a real

image

A B

C

D

E

F

G H

9 intersections by connecting nearby

lines. H is a wrong intersection

All intersections can not simultaneously satisfy camera projection,

orthogonality and parallelism constraints

7 intersections when we remove H

A B

C

D

E

F

G

Only 6 intersections when we include H

A B

D

E

F

H

A B

C

D

E

F

G H

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Minimal Spanning Tree (MST) for line-lifting

• Using gap between the line segments as the edge costs, we compute

MST to identify the least number of constraints to lift the lines to 3D

space.

all intersections Intersections in the

MST

Two perspective views of the line

reconstruction

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Qualitative Evaluation

image detected lines Two perspective views of the line

reconstruction

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Greedy Algorithms for submodular objective

functions under a matroid constraint

• We can also find a subset that maximizes a submodular

function under the constraint that the subset is an

independent set of a matroid.

• The solution comes with some optimality guarantees.

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submodular

[Nemhauser, Fisher & Wolsey ’78]

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Theorem: For monotonic submodular functions, greedy algorithm gives constant factor approximation

26

Maximize monotonic submodular functions under

one or more matroids

[Nemhauser, Fisher & Wolsey ’78]

)()( 21

optgreedy AFAF

)1(1

pp Greedy gives over intersection of matroids.

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Ground set

Configuration:

Sensing quality model

Configuration is feasible if no camera is pointed in

two directions at once

27

Example: Camera network

Slide courtesy: Krause

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Ground set

Configuration:

Sensing quality model

Configuration is feasible if no camera is pointed in

two directions at once

This is a partition matroid:

Independence:

28

Example: Camera network


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29

Greedy algorithm for matroids: Given: finite set V

Want: such that

Greedy algorithm:

Start with

While


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Pixel representation Superpixel representation

Superpixel Segmentation

max F( A )

subject to and A results in K clusters

an optimization problem on graph topology

Subset Selection Problem

Produces state-of-the-art results in superpixel segmentation

and clustering datasets. [Liu et al. 2011, 2013]

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References

• Oxley, Matroid Theory, 2011 (possibly the best material, but

time-consuming).

• The Coming of the Matroids", William Cunningham, 2012.

• Welsh, Matroid Theory, 1975.

Slides and Videos:

• Jeff Bilmes, Submodular Functions, Optimization,

and Applications to Machine Learning, 2014.

• Federico Ardila, Matroid Theory, 2007.

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