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Cambridge, Massachusetts
Srikumar Ramalingam
Introduction to Matroids and Applications
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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
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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
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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:
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Example: Camera network
Slide courtesy: Krause
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Greedy algorithm for matroids: Given: finite set V
Want: such that
Greedy algorithm:
Start with
While
Slide courtesy: Krause
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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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