EECS 495: Combinatorial Optimization Lecture 8 Matroid Intersection

Reading: Schrijver, Chapter 41 • colorful spanning forests: for graph G with edges partitioned into color classes {E1,...,Ek}, colorful spanning forest is Matroid Intersection a forest with edges of different colors. Define M1,M2 on ground set E with: Problem: – I1 = {F ⊂ E : F is acyclic} – I = {F ⊂ E : ∀i, |F ∩ E | ≤ 1} • Given: matroids M1 = (S, I1) and M2 = 2 i (S, I ) on same ground set S 2 Then • Find: max weight (cardinality) common – these are matroids (graphic, parti- independent set J ⊆ I ∩ I 1 2 tion) – common independent sets = color- Applications ful spanning forests

Generalizes: Note: In second two examples, (S, I1 ∩ I2) not a matroid, so more general than matroid • max weight independent set of M (take optimization. M = M1 = M2) Claim: Matroid intersection of three ma- troids NP-hard. • matching in bipartite graphs Proof: Reduction from directed Hamilto- For bipartite graph G = ((V1,V2),E), let nian path: given digraph D = (V,E) and Mi = (E, Ii) where vertices s, t, is there a path from s to t that goes through each vertex exactly once. Ii = {J : ∀v ∈ Vi, v incident to ≤ one e ∈ J}

• M1 graphic matroid of underlying undi- Then rected graph

– these are matroids (partition ma- • M2 partition matroid in which F ⊆ E troids) indep if each v has at most one incoming edge in F , except s which has none – common independent sets = match-

ings of G • M3 partition ..., except t which has none

1 Intersection is set of vertex-disjoint directed = r1(E) +r2(E)− min(r1(F ) +r2(E \F ) paths with one starting at s and one ending F ⊆E at t, so Hamiltonian path iff max cardinality which is number of vertices minus min intersection has size n − 1. vertex cover = max indep set. hhGeneral statement about spanning sets inii matroids. Min-Max Theorem • gives nec and suff conditions for exis- tence of colorful spanning tree: Natural upper bound: Let J ∈ I1 ∩ I2 and A ⊆ S. Then Want max |I| = |V | − 1, I∈I1∩I2 J ∩ A ∈ I1 and J ∩ A ∈ I2 or equivalently so min(r1(F ) + r2(E \ F )) = |V | − 1. F ⊆E |J| = |J ∩ A| + |J ∩ A| ≤ r1(A) + r2(A) Note Claim: (Edmonds, 1970) For matroids r1(F ) = |V | − c(F ) M ,M on S, 1 2 where c(F ) is num connected comp of G0 = (V,F ), so need max {|J|} = min{r1(A) + r2(A)}. J∈I1∩I2 A⊆S – num colors in E\F at least c(F )−1 Applications: – iff removing any t colors leaves at most t + 1 conn comp • generalizes Konig (in bipartite graphs, max matching = min vertex cover): Proof Since ri(F ) is number of v ∈ Vi covered by F , Need:

ν(G) = min(r1(F ) + r2(E \ F )). F ⊆E 1. deletion

as v ∈ V1 covered by F plus v ∈ V2 cov- 2. contraction ered by E \ F form a vertex cover. 3. submodularity of rank function • generalizes Konig-Rado (in bipartite graphs, max indep set equals min edge Def: A function f is submodular if for any cover): A, B, Edge cover is f(A ∪ B) + f(A ∩ B) ≤ f(A) + r(B)

min |F | = min |B1 ∪ B2| F spans M1,M2 Bi basis of Mi or equivalently if for any S ⊂ T and i 6∈ T ,

= min (|B1| + |B2| − |B1 ∩ B2|) f(S ∪ {i}) − f(S) ≥ f(T ∪ {i}) − f(T ). Bi basis of Mi

2 [[Compare to concavity. ]] • since JA ⊆ JC know JA ∪{e} ∈ I (down- Claim: Above defns equivalent. ward closure) Proof: • therefore r(A ∪ {e}) = r(A) + 1 ←: For A, B, if B ⊆ A, claim trivial. Else let →: exercise. B \ A = {b1, . . . , bk}

k X f(A ∪ B) − f(A) = (f(A + b1 + ... + bi) i=1 −f(A + b1 + ... + bi−1)) k X ≤ (f(A ∩ B + b1 + ... + bi) i=1 −f(A ∩ B + b1 + ... + bi−1)) = f(B) − f(A ∩ B)

→: For S ⊆ T , i 6∈ T , set A = S ∪ {i} and B = T : f(T + i) + f(S) = f(A ∪ B) + f(A ∩ B) ≤ f(A) + f(B) = f(S + i) + f(T )

Claim: r(·) is rank func of a matroid iff

• r(∅) = 0 and r(A ∪ {e}) − r(A) ∈ {0, 1} for all e, A • r(·) is submodular

Proof: ←: First condition obvious. For second, fix A ⊆ C, e 6∈ C. Want: r(C ∪ {e}) − r(C) ≤ r(A ∪ {e}) − r(A).

• since r(A ∪ {e}) − r(A) ≥ 0 and r(C ∪ {e}) − r(C) ∈ {0, 1}, may assume r(C ∪ {e}) − r(C) = 1

• let JA,JC be max indep sets in A, C with JA ⊆ JC

• since r(C ∪ {e}) = r(C) + 1 know JC ∪ {e} ∈ I (exchange property)

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