Submodularity Reading Group More Examples of Matroids M. Pawan Kumar http://www.robots.ox.ac.uk/~oval/

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

Outline Partition Matroid Transversal Matroid Matching Matroid Gammoid

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

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

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

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

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} 3 2 1 Limited Subset (LS) X ⊆ S |X ∩ Si| ≤ li, for all i {1, 2, 4, 5, 6, 8}?

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} 3 2 1 Limited Subset (LS) X ⊆ S |X ∩ Si| ≤ li, for all i {1, 2, 4, 5, 8}?

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} 3 2 1 Limited Subset (LS) X ⊆ S |X ∩ Si| ≤ li, for all i {1, 2, 4, 5}?

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} 3 2 1 Limited Subset (LS) X ⊆ S |X ∩ Si| ≤ li, for all i Subset of an LS is an LS Subset system

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

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

Outline Partition Matroid Transversal Matroid Matching Matroid Gammoid

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}?

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}?

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}?

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}?

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

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

Outline Partition Matroid Transversal Matroid Matching Matroid Gammoid

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

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

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

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

Outline Partition Matroid Transversal Matroid Matching Matroid Gammoid

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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