1. Submodularity Reading Group More Examples of Matroids
M. Pawan Kumar

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

3. 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

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

5. Outline
Partition Matroid
Transversal Matroid
Matching Matroid
Gammoid

6. Partition
Set S
{1, 2, 3, 4, 5, 6, 7, 8, 9}
{Si}
{{1, 2, 3}, {4, 5, 6}, {7, 8}}?
Non-empty subsets {Si}
Partition
Mutually exclusive
Si ∩ Sj = ϕ, for all i ≠ j
Collectively exhaustive
∪i Si = S

7. Partition
Set S
{1, 2, 3, 4, 5, 6, 7, 8, 9}
{Si}
{{1, 2, 3}, {4, 5, 6, 7}, {7, 8, 9}}?
Non-empty subsets {Si}
Partition
Mutually exclusive
Si ∩ Sj = ϕ, for all i ≠ j
Collectively exhaustive
∪i Si = S

8. Partition
Set S
{1, 2, 3, 4, 5, 6, 7, 8, 9}
{Si}
{{1, 2, 3}, {4, 5, 6, 7}, {8, 9}}?
Non-empty subsets {Si}
Partition
Mutually exclusive
Si ∩ Sj = ϕ, for all i ≠ j
Collectively exhaustive
∪i Si = S

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

10. Limited Subset of Partition
Set S
{1, 2, 3, 4, 5, 6, 7, 8, 9}
Partition {Si}
{{1, 2, 3}, {4, 5, 6, 7}, {8, 9}}
Limits {li}
Limited Subset (LS) X ⊆ S
|X ∩ Si| ≤ li, for all i
{1, 2, 4, 5, 6, 8}?

11. Limited Subset of Partition
Set S
{1, 2, 3, 4, 5, 6, 7, 8, 9}
Partition {Si}
{{1, 2, 3}, {4, 5, 6, 7}, {8, 9}}
Limits {li}
Limited Subset (LS) X ⊆ S
|X ∩ Si| ≤ li, for all i
{1, 2, 4, 5, 8}?

12. Limited Subset of Partition
Set S
{1, 2, 3, 4, 5, 6, 7, 8, 9}
Partition {Si}
{{1, 2, 3}, {4, 5, 6, 7}, {8, 9}}
Limits {li}
Limited Subset (LS) X ⊆ S
|X ∩ Si| ≤ li, for all i
{1, 2, 4, 5}?

13. Limited Subset of Partition
Set S
{1, 2, 3, 4, 5, 6, 7, 8, 9}
Partition {Si}
{{1, 2, 3}, {4, 5, 6, 7}, {8, 9}}
Limits {li}
Limited Subset (LS) X ⊆ S
|X ∩ Si| ≤ li, for all i
Subset of an LS is an LS
Subset system

14. 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

15. Subset System Set S {Si, i = 1, 2, …, n} is a partition
{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

16. Outline
Partition Matroid
Transversal Matroid
Matching Matroid
Gammoid

17. Partial Transversal Set S {1, 2, 3, 4, 5, 6, 7, 8, 9} {Si}
{{1, 2, 3}, {4, 5, 6, 7}, {7, 8, 9}}
S1, S2, …, Sn ⊆ S (not necessarily disjoint)
X ⊆ S is a partial transversal (PT) of {Si}
X = {x1,…,xk}, each xj chosen from a distinct Si
{1, 4, 7, 8}?

18. Partial Transversal Set S {1, 2, 3, 4, 5, 6, 7, 8, 9} {Si}
{{1, 2, 3}, {4, 5, 6, 7}, {7, 8, 9}}
S1, S2, …, Sn ⊆ S (not necessarily disjoint)
X ⊆ S is a partial transversal (PT) of {Si}
X = {x1,…,xk}, each xj chosen from a distinct Si
{1, 7, 8}?

19. Partial Transversal Set S {1, 2, 3, 4, 5, 6, 7, 8, 9} {Si}
{{1, 2, 3}, {4, 5, 6, 7}, {7, 8, 9}}
S1, S2, …, Sn ⊆ S (not necessarily disjoint)
X ⊆ S is a partial transversal (PT) of {Si}
X = {x1,…,xk}, each xj chosen from a distinct Si
{1, 7}?

20. Partial Transversal Set S {1, 2, 3, 4, 5, 6, 7, 8, 9} {Si}
{{1, 2, 3}, {4, 5, 6, 7}, {7, 8, 9}}
S1, S2, …, Sn ⊆ S (not necessarily disjoint)
X ⊆ S is a partial transversal (PT) of {Si}
X = {x1,…,xk}, each xj chosen from a distinct Si
{7}?

21. Partial Transversal Set S {1, 2, 3, 4, 5, 6, 7, 8, 9} {Si}
{{1, 2, 3}, {4, 5, 6, 7}, {7, 8, 9}}
S1, S2, …, Sn ⊆ S (not necessarily disjoint)
X ⊆ S is a partial transversal (PT) of {Si}
X = {x1,…,xk}, each xj chosen from a distinct Si
Subset of a PT is a PT
Subset system

22. 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

23. Outline
Partition Matroid
Transversal Matroid
Matching Matroid
Gammoid

24. Matching G = (V, E) Matching is a set of disjoint edges.
No two edges in a matching share an endpoint.

25. ✓ Matching G = (V, E) Matching is a set of disjoint edges.
No two edges in a matching share an endpoint.

26. ✗ Matching G = (V, E) Matching is a set of disjoint edges.
No two edges in a matching share an endpoint.

27. Matching Matroid G = (V, E) S = V X ⊆S ∈I if a matching covers X
(S, I) is a matroid?
Matching Matroid

28. Outline
Partition Matroid
Transversal Matroid
Matching Matroid
Gammoid

29. Directed Graph
D = (V, A) v0 v8 v1 v2 v3 v4 v5 v6 v7 v9

30. t-s Path
D = (V, A) t v0 v8 v1 v2 v3 v4 v5 v6 s v7 v9
Walk from t to s consisting of distinct vertices

31. t-s Path
D = (V, A) t v0 v8 v1 v2 v3 v4 v5 v6 s v7 v9
Walk from t to s consisting of distinct vertices

32. T-S Path T S A t-s path where t  T and s  S, T, S ⊆V v0 v8 v1 v2 v3

33. T-S Path T S A t-s path where t  T and s  S, T, S ⊆V v0 v8 v1 v2 v3

34. T-S Path T S A t-s path where t  T and s  S, T, S ⊆V v0 v8 v1 v2 v3

35. Vertex Disjoint T-S Paths
Set of T-S Paths with no common vertex

36. Vertex Disjoint T-S Paths✓ v4 v5 v6 S v7 v9
Set of T-S Paths with no common vertex

37. Vertex Disjoint T-S Paths
Common
Vertex v7 v1 v2 v3 ✗ v4 v5 v6 S v7 v9
Set of T-S Paths with no common vertex

38. Vertex Disjoint T-S Paths
Common
Vertex v0 v1 v2 v3 ✗ v4 v5 v6 S v7 v9
Set of T-S Paths with no common vertex

39. Gammoid v0 v8
S ⊆V
X ⊆ S v1 v2 v3 X v4 v5 v6
X∈I? v7 v9 S
X∈I if some vertex disjoint T-X paths cover X

40. Gammoid v0 v8
S ⊆V
X ⊆ S v1 v2 v3 X v4 v5 v6
X∈I? v7 v9 S
X∈I if some vertex disjoint T-X paths cover X

41. Gammoid v0 v8
S ⊆V
X ⊆ S v1 v2 v3 X v4 v5 v6
X∈I? v7 v9 S
X∈I if some vertex disjoint T-X paths cover X

42. Strict Gammoid v0 v8
S = V
X ⊆ S v1 v2 v3 v4 v5 v6 v7 v9
X∈I if some vertex disjoint T-X paths cover X

43. Gammoid S S ⊆ V (S, I) matroid? X ⊆ S
X∈I if some vertex disjoint T-X paths cover X