discrete optimization lecture 4 – part 1 m. pawan kumar [email protected] slides available online
TRANSCRIPT
![Page 1: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/1.jpg)
Discrete OptimizationLecture 4 – Part 1
M. Pawan Kumar
Slides available online http://mpawankumar.info
![Page 2: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/2.jpg)
€1000
€400
€700
Steal at most 2 items
Greedy Algorithm
€1000
![Page 3: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/3.jpg)
€400
€700
Steal at most 1 item
Greedy Algorithm
€1000€1700
![Page 4: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/4.jpg)
€400
Steal at most 0 items
Greedy Algorithm
€1700
Success
![Page 5: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/5.jpg)
€1000
€400
€700
2 kg
1 kg
1.5 kg
Steal at most 2.5 kg
Greedy Algorithm (Most Expensive)
€1000
2 kg
![Page 6: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/6.jpg)
€400
€700
1 kg
1.5 kg
Steal at most 0.5 kg
Greedy Algorithm (Most Expensive)
€1000
2 kg
Failure
![Page 7: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/7.jpg)
€1000
€400
€700
2 kg
1 kg
1.5 kg
Steal at most 2.5 kg
Greedy Algorithm (Best Ratio)
€1000
2 kg
![Page 8: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/8.jpg)
€400
€700
1 kg
1.5 kg
Steal at most 0.5 kg
Greedy Algorithm (Best Ratio)
€1000
2 kg
Failure
Why?
![Page 9: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/9.jpg)
• Matroids
• Examples of Matroids
• Dual Matroid
Outline
![Page 10: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/10.jpg)
Subset System
Set S
Non-empty collection of subsets I
Property: If X I and Y X, then Y ⊆ I
(S, I) is a subset system
![Page 11: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/11.jpg)
Hereditary Property
Set S
Non-empty collection of subsets I
Property: If X I and Y X, then Y ⊆ I
(S, I) is a subset system
![Page 12: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/12.jpg)
Example
Set S = {1,2,…,m}
I = Set of all X S such that |X| ≤ k ⊆
Is (S, I) a subset system?
Yes
![Page 13: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/13.jpg)
Example
Set S = {1,2,…,m}, w ≥ 0
I = Set of all X S such that Σ⊆ sX w(s) ≤ W
Is (S, I) a subset system
Yes Not true if w can be negative
![Page 14: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/14.jpg)
Matroid
Subset system (S, I)
Property: If X, Y I and |X| < |Y| then
there exists a s Y\X
M = (S, I) is a matroid
such that X {s} ∪ I
![Page 15: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/15.jpg)
Augmentation/Exchange Property
Subset system (S, I)
Property: If X, Y I and |X| < |Y| then
there exists a s Y\X
M = (S, I) is a matroid
such that X {s} ∪ I
![Page 16: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/16.jpg)
Example
Set S = {1,2,…,m}
I = Set of all X S such that |X| ≤ k ⊆
Is M = (S, I) a matroid? Yes
Uniform matroid
![Page 17: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/17.jpg)
Example
Set S = {1,2,…,m}, w ≥ 0
I = Set of all X S such that Σ⊆ sX w(s) ≤ W
Is M = (S, I) a matroid? No
Coincidence? No
![Page 18: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/18.jpg)
Matroids
(S, I) is a matroid
⟹(S, I) admits an optimal greedy algorithm
![Page 19: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/19.jpg)
Matroids
(S, I) is a matroid
⟹(S, I) admits an optimal greedy algorithm
Why?
We will find out by the end of the lecture
![Page 20: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/20.jpg)
• Matroids– Connection to Linear Algebra– Connection to Graph Theory
• Examples of Matroids
• Dual Matroid
Outline
![Page 21: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/21.jpg)
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Matrix A Subset of columns {a1,a2,…,ak}
Linearly independent (LI)?
There exists no α ≠ 0 such that Σi αi ai = 0
✗
![Page 22: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/22.jpg)
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Matrix A Subset of columns {a1,a2,…,ak}
Linearly independent (LI)?
There exists no α ≠ 0 such that Σi αi ai = 0
✓
![Page 23: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/23.jpg)
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Matrix A Subset of columns {a1,a2,…,ak}
Linearly independent (LI)?
There exists no α ≠ 0 such that Σi αi ai = 0
✓
![Page 24: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/24.jpg)
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Matrix A Subset of columns {a1,a2,…,ak}
Linearly independent (LI)?
There exists no α ≠ 0 such that Σi αi ai = 0
✓
![Page 25: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/25.jpg)
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Matrix A Subset of columns {a1,a2,…,ak}
Subset of LI columns are LI
Define a subset system
![Page 26: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/26.jpg)
Subset System
Matrix A of size n x m, S = {1,2,…,m}
X S, A(X) = set of columns of A indexed by X⊆
X I if and only if A(X) are linearly independent
Is M = (S, I) a matroid?
![Page 27: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/27.jpg)
Answer
Yes
Matroids connected to Linear Algebra
Inspires some naming conventions
Linear Matroid
![Page 28: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/28.jpg)
Independent Set
Matroid M = (S, I)
X S is independent if X ⊆ I
X S is dependent if X ⊆ ∉ I
![Page 29: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/29.jpg)
Independent Sets of Linear Matroid
X S is independent if⊆
column vectors A(X) are linearly independent
Matrix A of size n x m, S = {1,2,…,m}
X S, A(X) = set of columns of A indexed by X⊆
![Page 30: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/30.jpg)
Independent Sets of Uniform Matroid
X S is independent if⊆
|X| ≤ k
S = {1,2,…,m}
X S⊆
![Page 31: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/31.jpg)
Base of a Subset
Matroid M = (S, I)
X is a base of U S if it satisfies three properties⊆
(i) X U⊆ (ii) X ∈ I
(iii) There exists no U’∈I, such that X U’ U⊂ ⊆
subset of Uindependent
Inclusionwise maximal
![Page 32: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/32.jpg)
Base of a Subset (Linear Matroid)
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![Page 33: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/33.jpg)
Base of a Subset (Linear Matroid)
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![Page 34: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/34.jpg)
Base of a Subset (Linear Matroid)
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✓Is X a base of U?
![Page 35: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/35.jpg)
Base of a Subset (Linear Matroid)
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![Page 36: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/36.jpg)
Base of a Subset (Linear Matroid)
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![Page 37: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/37.jpg)
Base of a Subset (Linear Matroid)
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Is X a base of U? ✓
![Page 38: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/38.jpg)
Base of a Subset (Linear Matroid)
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Base of U?
![Page 39: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/39.jpg)
Base of a Subset (Linear Matroid)
X S is base of U if⊆
A(X) is a base of A(U)
Matrix A of size n x m, S = {1,2,…,m}
X S, A(X) = set of columns of A indexed by X⊆
X I if and only if A(X) are linearly independent
![Page 40: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/40.jpg)
Base of a Subset (Uniform Matroid)
X S is base of U if⊆
X U and |X| = min{|U|,k}⊆
S = {1,2,…,m}
X S⊆
I = Set of all X S such that |X| ≤ k ⊆
![Page 41: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/41.jpg)
An Interesting Property
M = (S, I) is a subset system
M is a matroid
For all U S, all bases of U have same size⊆⟹
Proof?
![Page 42: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/42.jpg)
An Interesting Property
M = (S, I) is a subset system
M is a matroid
For all U S, all bases of U have same size⊆⟹
Proof?
![Page 43: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/43.jpg)
An Interesting Property
M = (S, I) is a subset system
M is a matroid
For all U S, all bases of U have same size⊆
An alternate definition for matroids
⟺
![Page 44: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/44.jpg)
Rank of a Subset
Matroid M = (S, I)
U S⊆
rM(U) = Size of a base of U
![Page 45: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/45.jpg)
Rank of a Subset (Linear Matroid)
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rM(U)? 2
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Rank of a Subset (Linear Matroid)
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rM(U)? 1
![Page 47: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/47.jpg)
Rank of a Subset (Linear Matroid)
rM(U) is equal to
rank of the matrix with columns A(U)
Matrix A of size n x m, S = {1,2,…,m}
X S, A(X) = set of columns of A indexed by X⊆
X I if and only if A(X) are linearly independent
![Page 48: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/48.jpg)
Rank of a Subset (Uniform Matroid)
rM(U) is equal to
min{|U|,k}
S = {1,2,…,m}
X S⊆
I = Set of all X S such that |X| ≤ k ⊆
![Page 49: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/49.jpg)
Base of a Matroid
Matroid M = (S, I)
X is a base S
![Page 50: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/50.jpg)
Base of a Linear Matroid
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Is X a base? ✗
![Page 51: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/51.jpg)
Base of a Linear Matroid
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Is X a base? ✓
![Page 52: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/52.jpg)
Base of a Linear Matroid
X S is base of the matroid if⊆
A(X) is a base of A
Matrix A of size n x m, S = {1,2,…,m}
X S, A(X) = set of columns of A indexed by X⊆
X I if and only if A(X) are linearly independent
![Page 53: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/53.jpg)
Base of a Uniform Matroid
X S is a base of the matroid if⊆
|X| = min{|S|,k} Assume k ≤ |S|
S = {1,2,…,m}
X S⊆
I = Set of all X S such that |X| ≤ k ⊆
![Page 54: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/54.jpg)
Base of a Uniform Matroid
X S is a base of the matroid if⊆
|X| = k
S = {1,2,…,m}
X S⊆
I = Set of all X S such that |X| ≤ k ⊆
Assume k ≤ |S|
![Page 55: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/55.jpg)
Rank of a Matroid
Matroid M = (S, I)
rM = Rank of S
![Page 56: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/56.jpg)
Rank of a Linear Matroid
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rM? 3
![Page 57: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/57.jpg)
Rank of a Linear Matroid
rM is equal to
rank of the matrix A
Matrix A of size n x m, S = {1,2,…,m}
X S, A(X) = set of columns of A indexed by X⊆
X I if and only if A(X) are linearly independent
![Page 58: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/58.jpg)
Rank of a Uniform Matroid
rM is equal to
k
S = {1,2,…,m}
X S⊆
I = Set of all X S such that |X| ≤ k ⊆
![Page 59: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/59.jpg)
Spanning Subset
Matroid M = (S, I)
U S⊆
U is spanning if it contains a base of the matroid
![Page 60: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/60.jpg)
True or False
A base is an inclusionwise minimal spanning subset
TRUE
![Page 61: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/61.jpg)
Spanning Subsets of Linear Matroid
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Is X a spanning subset? ✗
![Page 62: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/62.jpg)
Spanning Subsets of Linear Matroid
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Is X a spanning subset? ✓
![Page 63: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/63.jpg)
Spanning Subsets of Linear Matroid
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Is X a spanning subset? ✓
![Page 64: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/64.jpg)
Spanning Subsets of Linear Matroid
U S is spanning subset of the matroid if⊆
A(U) spans A
Matrix A of size n x m, S = {1,2,…,m}
X S, A(X) = set of columns of A indexed by X⊆
X I if and only if A(X) are linearly independent
![Page 65: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/65.jpg)
Spanning Subsets of Uniform Matroid
U S is a spanning subset of the matroid if⊆
|X| ≥ k
S = {1,2,…,m}
X S⊆
I = Set of all X S such that |X| ≤ k ⊆
![Page 66: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/66.jpg)
Recap
What is a subset system?
Bases of a subset of a matroid?
Rank rM(U) of a subset U?
What is a matroid?
Spanning subset?
![Page 67: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/67.jpg)
• Matroids– Connection to Linear Algebra– Connection to Graph Theory
• Examples of Matroids
• Dual Matroid
Outline
![Page 68: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/68.jpg)
Undirected Graph
v1
v0
v2
v6
v4
v5
v3
G = (V, E)
V = {v1,…,vn}
E = {e1,…,em}
Parallel edges Loop
![Page 69: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/69.jpg)
Walk
G = (V, E)
Sequence P = (v0,e1,v1,…,ek,vk), ei = (vi-1,vi)
v1
v0
v2
v6
v4
v5
v3
v0, (v0,v4), v4, (v4,v2), v2, (v2,v5), v5, (v5,v4), v4
V = {v1,…,vn}
E = {e1,…,em}
![Page 70: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/70.jpg)
Path
G = (V, E)
Sequence P = (v0,e1,v1,…,ek,vk), ei = (vi-1,vi)
v1
v0
v2
v6
v4
v5
v3
Vertices v0,v1,…,vk are distinct
V = {v1,…,vn}
E = {e1,…,em}
![Page 71: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/71.jpg)
Connected Graph
v1
v0
v2
v6
v4
v5
v3
G = (V, E)
V = {v1,…,vn}
E = {e1,…,em}
There exists a walk from one vertex to another
Connected?
![Page 72: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/72.jpg)
k-Vertex-Connected Graph
v1
v0
v2
v6
v4
v5
v3
G = (V, E)
V = {v1,…,vn}
E = {e1,…,em}
Remove any i < k vertices. Graph is connected.
2-Vertex-Connected? 3-Vertex-Connected?
![Page 73: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/73.jpg)
Circuit
G = (V, E)
V = {v1,…,vn}
E = {e1,…,em}
v1
v0
v2
v6
v4
v5
v3
Circuit = (v0,e1,v1,…,ek,vk), ei = (vi-1,vi)
v0 = vk Vertices v0,v1,…,vk-1 are distinct
1-circuit? 2-circuit?
![Page 74: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/74.jpg)
Forest
v1
v0
v2
v6
v4
v5
v3
G = (V, E)
V = {v1,…,vn}
E = {e1,…,em}
Subset of edges that contain no circuit
![Page 75: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/75.jpg)
Forest
v1
v0
v2
v6
v4
v5
v3
G = (V, E)
V = {v1,…,vn}
E = {e1,…,em}
Subset of edges that contain no circuit
Forest?
![Page 76: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/76.jpg)
Forest
v1
v0
v2
v6
v4
v5
v3
G = (V, E)
V = {v1,…,vn}
E = {e1,…,em}
Subset of edges that contain no circuit
Forest?
![Page 77: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/77.jpg)
Forest
v1
v0
v2
v6
v4
v5
v3
G = (V, E)
V = {v1,…,vn}
E = {e1,…,em}
Subset of edges that contain no circuit
Forest?
![Page 78: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/78.jpg)
Forest
v1
v0
v2
v6
v4
v5
v3
G = (V, E)
V = {v1,…,vn}
E = {e1,…,em}
Define a subset system on forests
Subset of a forest is a forest
![Page 79: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/79.jpg)
Subset System
v1
v0
v2
v6
v4
v5
v3
G = (V, E)
V = {v1,…,vn}
E = {e1,…,em}
S = E X S⊆
X∈I if X is a forest
Is M = (S, I) a matroid?
![Page 80: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/80.jpg)
Answer
Yes
Matroids connected to Graph Theory
Inspires some naming conventions
Cycle Matroid
Graphic matroids (isomorphic to cycle matroid)
![Page 81: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/81.jpg)
Circuit
Matroid M = (S, I)
X is a circuit if it satisfies three properties
(i) X S⊆ (ii) X ∉ I
(iii) There exists no Y ∉ I, such that Y X⊂
subset of Sdependent
Inclusionwise minimal
![Page 82: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/82.jpg)
Circuit of a Graphic Matroid
v1
v0
v2
v6
v4
v5
v3
Is this a circuit?
![Page 83: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/83.jpg)
Circuit of a Graphic Matroid
v1
v0
v2
v6
v4
v5
v3
Is this a circuit?
![Page 84: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/84.jpg)
Circuit of a Graphic Matroid
v1
v0
v2
v6
v4
v5
v3
Is this a circuit?
![Page 85: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/85.jpg)
Circuit of a Graphic Matroid
G = (V, E), S = E
X S⊆
X ∈ I if X is a forest
X S is a circuit if⊆
X is a circuit of G
![Page 86: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/86.jpg)
Circuit of a Uniform Matroid
X S is a circuit if⊆
|X| = k+1
S = {1,2,…,m}
X S⊆
I = Set of all X S such that |X| ≤ k ⊆
![Page 87: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/87.jpg)
Circuit of a Linear Matroid
X S is a circuit if⊆
A(X) = {a base of A } {any other column of A}∪
Matrix A of size n x m, S = {1,2,…,m}
X S, A(X) = set of columns of A indexed by X⊆
X I if and only if A(X) are linearly independent
![Page 88: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/88.jpg)
Circuit of a Linear Matroid
X S is a circuit if⊆
A(X) = two linearly dependent columns
Matrix A of size n x m, S = {1,2,…,m}
X S, A(X) = set of columns of A indexed by X⊆
X I if and only if A(X) are linearly independent
![Page 89: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/89.jpg)
Loop
Matroid M = (S, I)
Element s S∈
{s} is a circuit
![Page 90: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/90.jpg)
Loop of a Graphic Matroid
v1
v0
v2
v6
v4
v5
v3
Any loops in the matroid?
![Page 91: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/91.jpg)
Loop of a Graphic Matroid
G = (V, E), S = E
X S⊆
X ∈ I if X is a forest
s S is a loop if∈
{s} is a loop of G
![Page 92: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/92.jpg)
Loop of a Uniform Matroid
S = {1,2,…,m}
X S⊆
I = Set of all X S such that |X| ≤ k ⊆
s S is a loop if∈
k = 0
![Page 93: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/93.jpg)
Loop of a Linear Matroid
Matrix A of size n x m, S = {1,2,…,m}
X S, A(X) = set of columns of A indexed by X⊆
X I if and only if A(X) are linearly independent
s S is a loop if∈
A(s) = 0
![Page 94: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/94.jpg)
Parallel Elements
Matroid M = (S, I)
Elements s,t S∈
{s,t} is a circuit
![Page 95: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/95.jpg)
v1
v0
v2
v6
v4
v5
v3
Any parallel elements?
Parallel Elements of a Graphic Matroid
![Page 96: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/96.jpg)
Parallel Elements of a Graphic Matroid
G = (V, E), S = E
X S⊆
X ∈ I if X is a forest
s,t S are parallel if∈
{s,t} are parallel edges of G
![Page 97: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/97.jpg)
Parallel Elements of a Uniform Matroid
S = {1,2,…,m}
X S⊆
I = Set of all X S such that |X| ≤ k ⊆
s,t S are parallel elements if∈
k = 1
![Page 98: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/98.jpg)
Parallel Elements of a Linear Matroid
Matrix A of size n x m, S = {1,2,…,m}
X S, A(X) = set of columns of A indexed by X⊆
X I if and only if A(X) are linearly independent
s,t S are parallel elements if∈
A(s) and A(t) are linearly dependent
![Page 99: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/99.jpg)
Recap
What is a subset system?
Bases of a subset of a matroid?
Rank rM(U) of a subset U?
What is a matroid?
Spanning subset?
![Page 100: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/100.jpg)
Recap
Circuit?
Parallel elements?
Loop?
![Page 101: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/101.jpg)
• Matroids
• Examples of Matroids
• Dual Matroid
Outline
![Page 102: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/102.jpg)
Uniform Matroid
S = {1,2,…,m}
X S⊆
I = Set of all X S such that |X| ≤ k ⊆
![Page 103: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/103.jpg)
Linear Matroid
Matrix A of size n x m, S = {1,2,…,m}
X S, A(X) = set of columns of A indexed by X⊆
X I if and only if A(X) are linearly independent
![Page 104: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/104.jpg)
Graphic Matroid
G = (V, E), S = E
X S⊆
X ∈ I if X is a forest
![Page 105: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/105.jpg)
• Matroids
• Examples of Matroids– Partition Matroid– Transversal Matroid– Matching Matroid
• Dual Matroid
Outline
![Page 106: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/106.jpg)
Partition
Set S
Non-empty subsets {Si}
{1, 2, 3, 4, 5, 6, 7, 8, 9}
Mutually exclusive Si ∩ Sj = ϕ, for all i ≠ j
Collectively exhaustive ∪i Si = S
{{1, 2, 3}, {4, 5, 6}, {7, 8}}?
Partition
{Si}
![Page 107: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/107.jpg)
Partition
Set S
Non-empty subsets {Si}
{1, 2, 3, 4, 5, 6, 7, 8, 9}
Mutually exclusive Si ∩ Sj = ϕ, for all i ≠ j
Collectively exhaustive ∪i Si = S
{{1, 2, 3}, {4, 5, 6, 7}, {7, 8, 9}}?
Partition
{Si}
![Page 108: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/108.jpg)
Partition
Set S
Non-empty subsets {Si}
{1, 2, 3, 4, 5, 6, 7, 8, 9}
Mutually exclusive Si ∩ Sj = ϕ, for all i ≠ j
Collectively exhaustive ∪i Si = S
{{1, 2, 3}, {4, 5, 6, 7}, {8, 9}}?
Partition
{Si}
![Page 109: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/109.jpg)
Partition
Set S {1, 2, 3, 4, 5, 6, 7, 8, 9}
{{1, 2, 3}, {4, 5, 6, 7}, {8, 9}}Partition {Si}
![Page 110: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/110.jpg)
Limited Subset of Partition
Set S {1, 2, 3, 4, 5, 6, 7, 8, 9}
{{1, 2, 3}, {4, 5, 6, 7}, {8, 9}}Partition {Si}
Limits {li} 3 2 1
Limited Subset (LS) X S⊆
|X ∩ Si| ≤ li, for all i
{1, 2, 4, 5, 6, 8}?
![Page 111: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/111.jpg)
Limited Subset of Partition
Set S {1, 2, 3, 4, 5, 6, 7, 8, 9}
{{1, 2, 3}, {4, 5, 6, 7}, {8, 9}}Partition {Si}
Limits {li} 3 2 1
Limited Subset (LS) X S⊆
|X ∩ Si| ≤ li, for all i
{1, 2, 4, 5, 8}?
![Page 112: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/112.jpg)
Limited Subset of Partition
Set S {1, 2, 3, 4, 5, 6, 7, 8, 9}
{{1, 2, 3}, {4, 5, 6, 7}, {8, 9}}Partition {Si}
Limits {li} 3 2 1
Limited Subset (LS) X S⊆
{1, 2, 4, 5}?
|X ∩ Si| ≤ li, for all i
![Page 113: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/113.jpg)
Limited Subset of Partition
Set S {1, 2, 3, 4, 5, 6, 7, 8, 9}
{{1, 2, 3}, {4, 5, 6, 7}, {8, 9}}Partition {Si}
Limits {li} 3 2 1
Limited Subset (LS) X S⊆
Subset of an LS is an LS Subset system
|X ∩ Si| ≤ li, for all i
![Page 114: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/114.jpg)
Subset System
Set S
{Si, i = 1, 2, …, n} is a partition
{l1,l2,…,ln} are non-negative integers
X S⊆ ∈I if X is a limited subset of partition
![Page 115: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/115.jpg)
Subset System
{l1,l2,…,ln} are non-negative integers
X S⊆ ∈I if |X ∩ Si| ≤ li for all i {1,2,…,n}∈
(S, I) is a matroid? Partition Matroid
Set S
{Si, i = 1, 2, …, n} is a partition
![Page 116: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/116.jpg)
• Matroids
• Examples of Matroids– Partition Matroid– Transversal Matroid– Matching Matroid
• Dual Matroid
Outline
![Page 117: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/117.jpg)
Partial Transversal
Set S
S1, S2, …, Sn S (not necessarily disjoint) ⊆
X S is a partial transversal (PT) of {S⊆ i}
{1, 2, 3, 4, 5, 6, 7, 8, 9}
X = {x1,…,xk}, each xj chosen from a distinct Si
{{1, 2, 3}, {4, 5, 6, 7}, {7, 8, 9}}{Si}
{1, 4, 7, 8}?
![Page 118: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/118.jpg)
Partial Transversal
Set S
S1, S2, …, Sn S (not necessarily disjoint) ⊆
X S is a partial transversal (PT) of {S⊆ i}
{1, 2, 3, 4, 5, 6, 7, 8, 9}
{{1, 2, 3}, {4, 5, 6, 7}, {7, 8, 9}}{Si}
{1, 7, 8}?
X = {x1,…,xk}, each xj chosen from a distinct Si
![Page 119: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/119.jpg)
Partial Transversal
Set S
S1, S2, …, Sn S (not necessarily disjoint) ⊆
X S is a partial transversal (PT) of {S⊆ i}
{1, 2, 3, 4, 5, 6, 7, 8, 9}
{{1, 2, 3}, {4, 5, 6, 7}, {7, 8, 9}}{Si}
{1, 7}?
X = {x1,…,xk}, each xj chosen from a distinct Si
![Page 120: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/120.jpg)
Partial Transversal
Set S
S1, S2, …, Sn S (not necessarily disjoint) ⊆
X S is a partial transversal (PT) of {S⊆ i}
{1, 2, 3, 4, 5, 6, 7, 8, 9}
{{1, 2, 3}, {4, 5, 6, 7}, {7, 8, 9}}{Si}
{7}?
X = {x1,…,xk}, each xj chosen from a distinct Si
![Page 121: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/121.jpg)
Partial Transversal
Set S
S1, S2, …, Sn S (not necessarily disjoint) ⊆
X S is a partial transversal (PT) of {S⊆ i}
{1, 2, 3, 4, 5, 6, 7, 8, 9}
{{1, 2, 3}, {4, 5, 6, 7}, {7, 8, 9}}{Si}
Subset of a PT is a PT Subset system
X = {x1,…,xk}, each xj chosen from a distinct Si
![Page 122: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/122.jpg)
Subset System
Set S
S1, S2, …, Sn S (not necessarily disjoint) ⊆
X S⊆ ∈I if X is a partial transversal of {Si}
(S, I) is a matroid? Transversal Matroid
![Page 123: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/123.jpg)
• Matroids
• Examples of Matroids– Partition Matroid– Transversal Matroid– Matching Matroid
• Dual Matroid
Outline
![Page 124: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/124.jpg)
Matching
v1
v0
v2
v6
v4
v5
v3
G = (V, E)
Matching is a set of disjoint edges.
No two edges in a matching share an endpoint.
![Page 125: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/125.jpg)
Matching
v1
v0
v2
v6
v4
v5
v3
G = (V, E)
Matching is a set of disjoint edges.
No two edges in a matching share an endpoint.
✓
![Page 126: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/126.jpg)
Matching
v1
v0
v2
v6
v4
v5
v3
G = (V, E)
Matching is a set of disjoint edges.
No two edges in a matching share an endpoint.
✗
![Page 127: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/127.jpg)
Matching Matroid
v1
v0
v2
v6
v4
v5
v3
G = (V, E)
X S ⊆ ∈I if a matching covers X
S = V
(S, I) is a matroid? Matching Matroid
![Page 128: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/128.jpg)
• Matroids
• Examples of Matroids
• Dual Matroid
Outline
![Page 129: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/129.jpg)
Dual Matroid
M = (S, I) M* = (S, I*)
X ∈I* if two conditions are satisfied
(i) X S⊆
(ii) S\X is a spanning set of M
Bases of M, M* are complements of each other
If M* is also a matroid then
![Page 130: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/130.jpg)
Dual of Graphic Matroid
G = (V, E), S = E
X S⊆
X ∈ I if X is a forest
Y ∈ I* if
E\Y contains a maximal forest of G
![Page 131: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/131.jpg)
Dual of Graphic Matroid
G = (V, E), S = E
X S⊆
X ∈ I if X is a forest
Y ∈ I* if, after removing Y,
number of connected components don’t change
Cographic Matroid
![Page 132: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/132.jpg)
Dual of Uniform Matroid
S = {1,2,…,m}
X S⊆
I = Set of all X S such that |X| ≤ k ⊆
Y ∈ I* if
|Y| ≤ m-k
![Page 133: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/133.jpg)
Dual of Linear Matroid
Matrix A of size m x n, S = {1,2,…,m}
X S, A(X) = set of columns of A indexed by X⊆
X I if and only if A(X) are linearly independent
Y ∈ I* if
A(S\Y) spans A
![Page 134: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/134.jpg)
Dual Matroid is a Subset System
Proof?
![Page 135: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/135.jpg)
Dual Matroid is a Matroid
Proof?
![Page 136: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/136.jpg)
Dual Matroid is a Matroid
M = (S, I) M* = (S, I*)
Let X ∈ I* and Y ∈ I*, such that |X| < |Y|
There should exist s Y\X, X {s} ∈ ∪ ∈ I*
S\Y contains a base of M Why?
S\X contains a base of M
![Page 137: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/137.jpg)
Dual Matroid is a Matroid
S\Y contains a base of M B
S\X contains a base of M
B\X S\X⊆ B’ ⊆ Base B’
There exists s Y\X, s B’∈ ∉
Proof? By contradiction
![Page 138: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/138.jpg)
Dual Matroid is a Matroid
B\X S\X⊆ B’ ⊆ Base B’
There exists s Y\X, s B’∈ ∉
|B| = |B ∩ X| + |B \ X|
≤ |X \ Y| + |B \ X| Why?
Because B is disjoint from Y
![Page 139: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/139.jpg)
Dual Matroid is a Matroid
B\X S\X⊆ B’ ⊆ Base B’
|B| = |B ∩ X| + |B \ X|
≤ |X \ Y| + |B \ X|
< |Y \ X| + |B \ X| Why?
Because |X| < |Y|
There exists s Y\X, s B’∈ ∉
![Page 140: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/140.jpg)
Dual Matroid is a Matroid
B\X S\X⊆ B’ ⊆ Base B’
|B| = |B ∩ X| + |B \ X|
≤ |X \ Y| + |B \ X|
< |Y \ X| + |B \ X|
Why?
Because Y\X B’⊆
≤ |B’|
B\X B’⊆
B ∩ Y = ϕ
There exists s Y\X, s B’∈ ∉
![Page 141: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/141.jpg)
Dual Matroid is a Matroid
B\X S\X⊆ B’ ⊆ Base B’
|B| = |B ∩ X| + |B \ X|
≤ |X \ Y| + |B \ X|
< |Y \ X| + |B \ X|
Contradiction≤ |B’|
There exists s Y\X, s B’∈ ∉
![Page 142: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/142.jpg)
Dual Matroid is a Matroid
B\X S\X⊆ B’ ⊆ Base B’
There exists s Y\X, X {s} ∈ ∪ ∈ I*
Hence proved.
There exists s Y\X, s B’∈ ∉
![Page 143: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/143.jpg)
Dual Matroid is a Matroid
Circuits of M* are called cocircuits of M
Loops of M* are called coloops of M
Parallel elements in M* are coparallel in M
![Page 144: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/144.jpg)
Dual of Dual Matroid is the Matroid
Proof?
![Page 145: Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022070401/56649f1e5503460f94c35cc5/html5/thumbnails/145.jpg)
Ranking Functions of M and M*
M = (S, I) M* = (S, I*)
rM*(U) = |U| + rM(S\U) - rM(S)
Proof?