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Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online http://mpawankumar.info
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€1000 €400 €700 Steal at most 2 items Greedy Algorithm €1000
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€400 €700 Steal at most 1 item Greedy Algorithm €1000 €1700
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€400 Steal at most 0 items Greedy Algorithm €1700 Success
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€1000 €400 €700 2 kg 1 kg 1.5 kg Steal at most 2.5 kg Greedy Algorithm (Most Expensive) €1000 2 kg
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€400 €700 1 kg 1.5 kg Steal at most 0.5 kg Greedy Algorithm (Most Expensive) €1000 2 kg Failure
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€1000 €400 €700 2 kg 1 kg 1.5 kg Steal at most 2.5 kg Greedy Algorithm (Best Ratio) €1000 2 kg
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€400 €700 1 kg 1.5 kg Steal at most 0.5 kg Greedy Algorithm (Best Ratio) €1000 2 kg Failure Why?
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Matroids Examples of Matroids Dual Matroid Outline
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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
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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
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Example Set S = {1,2,…,m} I = Set of all X ⊆ S such that |X| ≤ k Is (S, I ) a subset system? Yes
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Example Set S = {1,2,…,m}, w ≥ 0 I = Set of all X ⊆ S such that Σ s X w(s) ≤ W Is (S, I ) a subset system YesNot true if w can be negative
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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
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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
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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
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Example Set S = {1,2,…,m}, w ≥ 0 I = Set of all X ⊆ S such that Σ s X w(s) ≤ W Is M = (S, I ) a matroid?No Coincidence?No
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Matroids (S, I ) is a matroid (S, I ) admits an optimal greedy algorithm
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Matroids (S, I ) is a matroid (S, I ) admits an optimal greedy algorithm Why? We will find out by the end of the lecture
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Matroids –Connection to Linear Algebra –Connection to Graph Theory Examples of Matroids Dual Matroid Outline
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12341234 24682468 11111111 22222222 33333333 12121212 24242424 12121212 24242424 Matrix ASubset of columns {a 1,a 2,…,a k } Linearly independent (LI)? There exists no α ≠ 0 such that Σ i α i a i = 0 ✗
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12341234 24682468 11111111 22222222 33333333 12121212 24242424 12121212 24242424 Matrix ASubset of columns {a 1,a 2,…,a k } Linearly independent (LI)? There exists no α ≠ 0 such that Σ i α i a i = 0 ✓
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12341234 24682468 11111111 22222222 33333333 12121212 24242424 12121212 24242424 Matrix ASubset of columns {a 1,a 2,…,a k } Linearly independent (LI)? There exists no α ≠ 0 such that Σ i α i a i = 0 ✓
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12341234 24682468 11111111 22222222 33333333 12121212 24242424 12121212 24242424 Matrix ASubset of columns {a 1,a 2,…,a k } Linearly independent (LI)? There exists no α ≠ 0 such that Σ i α i a i = 0 ✓
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12341234 24682468 11111111 22222222 33333333 12121212 24242424 12121212 24242424 Matrix ASubset of columns {a 1,a 2,…,a k } Subset of LI columns are LI Define a subset system
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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?
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Answer Yes Matroids connected to Linear Algebra Inspires some naming conventions Linear Matroid
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Independent Set Matroid M = (S, I ) X ⊆ S is independent if X I X ⊆ S is dependent if X ∉ I
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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
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Independent Sets of Uniform Matroid X ⊆ S is independent if |X| ≤ k S = {1,2,…,m} X ⊆ S
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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
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Base of a Subset (Linear Matroid) 12341234 24682468 11111111 22222222 33333333 12121212 24242424 12121212 24242424 U Is X a base of U? ✗
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Base of a Subset (Linear Matroid) 12341234 24682468 11111111 22222222 33333333 12121212 24242424 12121212 24242424 U ✗ Is X a base of U?
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Base of a Subset (Linear Matroid) 12341234 24682468 11111111 22222222 33333333 12121212 24242424 12121212 24242424 U ✓ Is X a base of U?
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Base of a Subset (Linear Matroid) 12341234 24682468 11111111 22222222 33333333 12121212 24242424 12121212 24242424 U ✗ Is X a base of U?
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Base of a Subset (Linear Matroid) 12341234 24682468 11111111 22222222 33333333 12121212 24242424 12121212 24242424 U ✓ Is X a base of U?
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Base of a Subset (Linear Matroid) 12341234 24682468 11111111 22222222 33333333 12121212 24242424 12121212 24242424 U Is X a base of U? ✓
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Base of a Subset (Linear Matroid) 12341234 24682468 11111111 22222222 33333333 12121212 24242424 12121212 24242424 U Base of U?
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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
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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
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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?
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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?
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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
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Rank of a Subset Matroid M = (S, I ) U ⊆ S r M (U) = Size of a base of U
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Rank of a Subset (Linear Matroid) 12341234 24682468 11111111 22222222 33333333 12121212 24242424 12121212 24242424 U r M (U)? 2
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Rank of a Subset (Linear Matroid) 12341234 24682468 11111111 22222222 33333333 12121212 24242424 12121212 24242424 U r M (U)? 1
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Rank of a Subset (Linear Matroid) r M (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
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Rank of a Subset (Uniform Matroid) r M (U) is equal to min{|U|,k} S = {1,2,…,m} X ⊆ S I = Set of all X ⊆ S such that |X| ≤ k
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Base of a Matroid Matroid M = (S, I ) X is a base S
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Base of a Linear Matroid 12341234 24682468 11111111 22222222 33333333 12121212 24242424 12121212 24242424 Is X a base? ✗
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Base of a Linear Matroid 12341234 24682468 11111111 22222222 33333333 12121212 24242424 12121212 24242424 Is X a base? ✓
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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
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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
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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|
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Rank of a Matroid Matroid M = (S, I ) r M = Rank of S
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Rank of a Linear Matroid 12341234 24682468 11111111 22222222 33333333 12121212 24242424 12121212 24242424 rM?rM? 3
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r M 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
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Rank of a Uniform Matroid r M is equal to k S = {1,2,…,m} X ⊆ S I = Set of all X ⊆ S such that |X| ≤ k
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Spanning Subset Matroid M = (S, I ) U ⊆ S U is spanning if it contains a base of the matroid
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True or False A base is an inclusionwise minimal spanning subset TRUE
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Spanning Subsets of Linear Matroid 12341234 24682468 11111111 22222222 33333333 12121212 24242424 12121212 24242424 Is X a spanning subset? ✗
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Spanning Subsets of Linear Matroid 12341234 24682468 11111111 22222222 33333333 12121212 24242424 12121212 24242424 Is X a spanning subset? ✓
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Spanning Subsets of Linear Matroid 12341234 24682468 11111111 22222222 33333333 12121212 24242424 12121212 24242424 Is X a spanning subset? ✓
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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
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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
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Recap What is a subset system? Bases of a subset of a matroid? Rank r M (U) of a subset U? What is a matroid? Spanning subset?
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Matroids –Connection to Linear Algebra –Connection to Graph Theory Examples of Matroids Dual Matroid Outline
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Undirected Graph v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 G = (V, E) V = {v 1,…,v n } E = {e 1,…,e m } Parallel edgesLoop
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Walk G = (V, E) Sequence P = (v 0,e 1,v 1,…,e k,v k ), e i = (v i-1,v i ) v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 v 0, (v 0,v 4 ), v 4, (v 4,v 2 ), v 2, (v 2,v 5 ), v 5, (v 5,v 4 ), v 4 V = {v 1,…,v n } E = {e 1,…,e m }
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Path G = (V, E) Sequence P = (v 0,e 1,v 1,…,e k,v k ), e i = (v i-1,v i ) v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 Vertices v 0,v 1,…,v k are distinct V = {v 1,…,v n } E = {e 1,…,e m }
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Connected Graph v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 G = (V, E) V = {v 1,…,v n } E = {e 1,…,e m } There exists a walk from one vertex to another Connected?
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k-Vertex-Connected Graph v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 G = (V, E) V = {v 1,…,v n } E = {e 1,…,e m } Remove any i < k vertices. Graph is connected. 2-Vertex-Connected?3-Vertex-Connected?
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Circuit G = (V, E) V = {v 1,…,v n } E = {e 1,…,e m } v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 Circuit = (v 0,e 1,v 1,…,e k,v k ), e i = (v i-1,v i ) v 0 = v k Vertices v 0,v 1,…,v k-1 are distinct 1-circuit? 2-circuit?
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Forest v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 G = (V, E) V = {v 1,…,v n } E = {e 1,…,e m } Subset of edges that contain no circuit
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Forest v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 G = (V, E) V = {v 1,…,v n } E = {e 1,…,e m } Subset of edges that contain no circuit Forest?
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Forest v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 G = (V, E) V = {v 1,…,v n } E = {e 1,…,e m } Subset of edges that contain no circuit Forest?
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Forest v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 G = (V, E) V = {v 1,…,v n } E = {e 1,…,e m } Subset of edges that contain no circuit Forest?
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Forest v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 G = (V, E) V = {v 1,…,v n } E = {e 1,…,e m } Define a subset system on forests Subset of a forest is a forest
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Subset System v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 G = (V, E) V = {v 1,…,v n } E = {e 1,…,e m } S = ES = E X ⊆ S X ∈ I if X is a forest Is M = (S, I ) a matroid?
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Answer Yes Matroids connected to Graph Theory Inspires some naming conventions Cycle Matroid Graphic matroids (isomorphic to cycle matroid)
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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
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Circuit of a Graphic Matroid v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 Is this a circuit?
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Circuit of a Graphic Matroid v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 Is this a circuit?
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Circuit of a Graphic Matroid v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 Is this a circuit?
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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
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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
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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
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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
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Loop Matroid M = (S, I ) Element s ∈ S {s} is a circuit
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Loop of a Graphic Matroid v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 Any loops in the matroid?
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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
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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
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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
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Parallel Elements Matroid M = (S, I ) Elements s,t ∈ S {s,t} is a circuit
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v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 Any parallel elements? Parallel Elements of a Graphic Matroid
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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
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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
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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
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Recap What is a subset system? Bases of a subset of a matroid? Rank r M (U) of a subset U? What is a matroid? Spanning subset?
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Recap Circuit? Parallel elements? Loop?
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Matroids Examples of Matroids Dual Matroid Outline
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Uniform Matroid S = {1,2,…,m} X ⊆ S I = Set of all X ⊆ S such that |X| ≤ k
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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
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Graphic Matroid G = (V, E), S = E X ⊆ S X ∈ I if X is a forest
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Matroids Examples of Matroids –Partition Matroid –Transversal Matroid –Matching Matroid Dual Matroid Outline
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Partition Set S Non-empty subsets {S i } {1, 2, 3, 4, 5, 6, 7, 8, 9} Mutually exclusive S i ∩ S j = ϕ, for all i ≠ j Collectively exhaustive ∪ i S i = S {{1, 2, 3}, {4, 5, 6}, {7, 8}}? Partition {S i }
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Partition Set S Non-empty subsets {S i } {1, 2, 3, 4, 5, 6, 7, 8, 9} Mutually exclusive S i ∩ S j = ϕ, for all i ≠ j Collectively exhaustive ∪ i S i = S {{1, 2, 3}, {4, 5, 6, 7}, {7, 8, 9}}? Partition {S i }
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Partition Set S Non-empty subsets {S i } {1, 2, 3, 4, 5, 6, 7, 8, 9} Mutually exclusive S i ∩ S j = ϕ, for all i ≠ j Collectively exhaustive ∪ i S i = S {{1, 2, 3}, {4, 5, 6, 7}, {8, 9}}? Partition {S i }
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Partition Set S{1, 2, 3, 4, 5, 6, 7, 8, 9} {{1, 2, 3}, {4, 5, 6, 7}, {8, 9}} Partition {S i }
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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 {S i } Limits {l i } 3 2 1 Limited Subset (LS) X ⊆ S |X ∩ S i | ≤ l i, for all i {1, 2, 4, 5, 6, 8}?
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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 {S i } Limits {l i } 3 2 1 Limited Subset (LS) X ⊆ S |X ∩ S i | ≤ l i, for all i {1, 2, 4, 5, 8}?
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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 {S i } Limits {l i } 3 2 1 Limited Subset (LS) X ⊆ S {1, 2, 4, 5}? |X ∩ S i | ≤ l i, for all i
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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 {S i } Limits {l i } 3 2 1 Limited Subset (LS) X ⊆ S Subset of an LS is an LSSubset system |X ∩ S i | ≤ l i, for all i
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Subset System Set S {S i, i = 1, 2, …, n} is a partition {l 1,l 2,…,l n } are non-negative integers X ⊆ S ∈ I if X is a limited subset of partition
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Subset System {l 1,l 2,…,l n } are non-negative integers X ⊆ S ∈ I if |X ∩ S i | ≤ l i for all i ∈ {1,2,…,n} (S, I ) is a matroid? Partition Matroid Set S {S i, i = 1, 2, …, n} is a partition
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Matroids Examples of Matroids –Partition Matroid –Transversal Matroid –Matching Matroid Dual Matroid Outline
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Partial Transversal Set S S 1, S 2, …, S n ⊆ S (not necessarily disjoint) X ⊆ S is a partial transversal (PT) of {S i } {1, 2, 3, 4, 5, 6, 7, 8, 9} X = {x 1,…,x k }, each x j chosen from a distinct S i {{1, 2, 3}, {4, 5, 6, 7}, {7, 8, 9}} {S i } {1, 4, 7, 8}?
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Partial Transversal Set S S 1, S 2, …, S n ⊆ 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}} {S i } {1, 7, 8}? X = {x 1,…,x k }, each x j chosen from a distinct S i
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Partial Transversal Set S S 1, S 2, …, S n ⊆ 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}} {S i } {1, 7}? X = {x 1,…,x k }, each x j chosen from a distinct S i
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Partial Transversal Set S S 1, S 2, …, S n ⊆ 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}} {S i } {7}? X = {x 1,…,x k }, each x j chosen from a distinct S i
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Partial Transversal Set S S 1, S 2, …, S n ⊆ 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}} {S i } Subset of a PT is a PTSubset system X = {x 1,…,x k }, each x j chosen from a distinct S i
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Subset System Set S S 1, S 2, …, S n ⊆ S (not necessarily disjoint) X ⊆ S ∈ I if X is a partial transversal of {S i } (S, I ) is a matroid?Transversal Matroid
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Matroids Examples of Matroids –Partition Matroid –Transversal Matroid –Matching Matroid Dual Matroid Outline
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Matching v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 G = (V, E) Matching is a set of disjoint edges. No two edges in a matching share an endpoint.
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Matching v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 G = (V, E) Matching is a set of disjoint edges. No two edges in a matching share an endpoint. ✓
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Matching v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 G = (V, E) Matching is a set of disjoint edges. No two edges in a matching share an endpoint. ✗
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Matching Matroid v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 G = (V, E) X ⊆ S ∈ I if a matching covers X S = V (S, I ) is a matroid?Matching Matroid
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Matroids Examples of Matroids Dual Matroid Outline
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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
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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
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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
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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
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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
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Dual Matroid is a Subset System Proof?
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Dual Matroid is a Matroid Proof?
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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 MWhy? S\X contains a base of M
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Dual Matroid is a Matroid S\Y contains a base of MB S\X contains a base of M B\X ⊆ S\X B’ ⊆ Base B’ There exists s ∈ Y\X, s ∉ B’ Proof? By contradiction
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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
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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’
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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’
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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’
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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’
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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
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Dual of Dual Matroid is the Matroid Proof?
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Ranking Functions of M and M* M = (S, I )M* = (S, I *) r M* (U) = |U| + r M (S\U) - r M (S) Proof?
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Ranking Functions of M and M* M = (S, I )M* = (S, I *) r M* (U) = max{|U \ Y|, Y is a base of M} = |U| - min{|U ∩ Y|, Y is a base of M} = |U| + max{|Y\U|, Y is a base of M} - |Y| = |U| + r M (S \ U) - r M (S) = max{|U ∩ X|, X is a base of M*}
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Connected Matroid Matroid M = (S, I ) For all non-empty U ⊂ S r M (U) + r M (S\U) > r M (S) M is connected if and only if M* is connected
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