Discrete Optimization Lecture 4 – Part 1 M. Pawan Kumar Slides available online

€1000 €400 €700 Steal at most 2 items Greedy Algorithm €1000

€400 €700 Steal at most 1 item Greedy Algorithm €1000 €1700

€400 Steal at most 0 items Greedy Algorithm €1700 Success

€1000 €400 €700 2 kg 1 kg 1.5 kg Steal at most 2.5 kg Greedy Algorithm (Most Expensive) € kg

€400 €700 1 kg 1.5 kg Steal at most 0.5 kg Greedy Algorithm (Most Expensive) € kg Failure

€1000 €400 €700 2 kg 1 kg 1.5 kg Steal at most 2.5 kg Greedy Algorithm (Best Ratio) € kg

€400 €700 1 kg 1.5 kg Steal at most 0.5 kg Greedy Algorithm (Best Ratio) € kg Failure Why?

Matroids Examples of Matroids Dual Matroid Outline

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

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

Example Set S = {1,2,…,m} I = Set of all X ⊆ S such that |X| ≤ k Is (S, I ) a subset system? Yes

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

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

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

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

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

Matroids (S, I ) is a matroid (S, I ) admits an optimal greedy algorithm

Matroids (S, I ) is a matroid (S, I ) admits an optimal greedy algorithm Why? We will find out by the end of the lecture

Matroids –Connection to Linear Algebra –Connection to Graph Theory Examples of Matroids Dual Matroid Outline

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 ✗

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 ✓

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 ✓

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 ✓

Matrix ASubset of columns {a 1,a 2,…,a k } Subset of LI columns are LI Define a subset system

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?

Answer Yes Matroids connected to Linear Algebra Inspires some naming conventions Linear Matroid

Independent Set Matroid M = (S, I ) X ⊆ S is independent if X  I X ⊆ S is dependent if X ∉ I

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

Independent Sets of Uniform Matroid X ⊆ S is independent if |X| ≤ k S = {1,2,…,m} X ⊆ S

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

Base of a Subset (Linear Matroid) U Is X a base of U? ✗

Base of a Subset (Linear Matroid) U ✗ Is X a base of U?

Base of a Subset (Linear Matroid) U ✓ Is X a base of U?

Base of a Subset (Linear Matroid) U ✗ Is X a base of U?

Base of a Subset (Linear Matroid) U ✓ Is X a base of U?

Base of a Subset (Linear Matroid) U Is X a base of U? ✓

Base of a Subset (Linear Matroid) U Base of U?

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

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

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?

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?

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

Rank of a Subset Matroid M = (S, I ) U ⊆ S r M (U) = Size of a base of U

Rank of a Subset (Linear Matroid) U r M (U)? 2

Rank of a Subset (Linear Matroid) U r M (U)? 1

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

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

Base of a Matroid Matroid M = (S, I ) X is a base S

Base of a Linear Matroid Is X a base? ✗

Base of a Linear Matroid Is X a base? ✓

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

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

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|

Rank of a Matroid Matroid M = (S, I ) r M = Rank of S

Rank of a Linear Matroid rM?rM? 3

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

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

Spanning Subset Matroid M = (S, I ) U ⊆ S U is spanning if it contains a base of the matroid

True or False A base is an inclusionwise minimal spanning subset TRUE

Spanning Subsets of Linear Matroid Is X a spanning subset? ✗

Spanning Subsets of Linear Matroid Is X a spanning subset? ✓

Spanning Subsets of Linear Matroid Is X a spanning subset? ✓

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

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

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?

Matroids –Connection to Linear Algebra –Connection to Graph Theory Examples of Matroids Dual Matroid Outline

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

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 }

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 }

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?

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?

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?

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 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?

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?

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?

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

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?

Answer Yes Matroids connected to Graph Theory Inspires some naming conventions Cycle Matroid Graphic matroids (isomorphic to cycle matroid)

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

Circuit of a Graphic Matroid v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 Is this a circuit?

Circuit of a Graphic Matroid v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 Is this a circuit?

Circuit of a Graphic Matroid v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 Is this a circuit?

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

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

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

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

Loop Matroid M = (S, I ) Element s ∈ S {s} is a circuit

Loop of a Graphic Matroid v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 Any loops in the matroid?

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

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

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

Parallel Elements Matroid M = (S, I ) Elements s,t ∈ S {s,t} is a circuit

v1v1 v1v1 v0v0 v0v0 v2v2 v2v2 v6v6 v6v6 v4v4 v4v4 v5v5 v5v5 v3v3 v3v3 Any parallel elements? 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

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

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

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?

Recap Circuit? Parallel elements? Loop?

Matroids Examples of Matroids Dual Matroid Outline

Uniform Matroid S = {1,2,…,m} X ⊆ S I = Set of all X ⊆ S such that |X| ≤ k

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

Graphic Matroid G = (V, E), S = E X ⊆ S X ∈ I if X is a forest

Matroids Examples of Matroids –Partition Matroid –Transversal Matroid –Matching Matroid Dual Matroid Outline

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 }

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 }

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 }

Partition Set S{1, 2, 3, 4, 5, 6, 7, 8, 9} {{1, 2, 3}, {4, 5, 6, 7}, {8, 9}} Partition {S i }

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 } Limited Subset (LS) X ⊆ S |X ∩ S i | ≤ l i, for all i {1, 2, 4, 5, 6, 8}?

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 } Limited Subset (LS) X ⊆ S |X ∩ S i | ≤ l i, for all i {1, 2, 4, 5, 8}?

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 } Limited Subset (LS) X ⊆ S {1, 2, 4, 5}? |X ∩ S i | ≤ l i, for all i

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 } Limited Subset (LS) X ⊆ S Subset of an LS is an LSSubset system |X ∩ S i | ≤ l i, for all i

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

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

Matroids Examples of Matroids –Partition Matroid –Transversal Matroid –Matching Matroid Dual Matroid Outline

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}?

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

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

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

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

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

Matroids Examples of Matroids –Partition Matroid –Transversal Matroid –Matching Matroid Dual Matroid Outline

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.

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. ✓

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. ✗

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

Matroids Examples of Matroids Dual Matroid Outline

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

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

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

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

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

Dual Matroid is a Subset System Proof?

Dual Matroid is a Matroid Proof?

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

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

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

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’

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’

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’

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’

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

Dual of Dual Matroid is the Matroid Proof?

Ranking Functions of M and M* M = (S, I )M* = (S, I *) r M* (U) = |U| + r M (S\U) - r M (S) Proof?

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*}

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