EECS 495: Combinatorial Optimization Lecture 8 Matroid Intersection Reading: Schrijver, Chapter 41 • colorful spanning forests: for graph G with edges partitioned into color classes fE1;:::;Ekg, colorful spanning forest is Matroid Intersection a forest with edges of different colors. Define M1;M2 on ground set E with: Problem: { I1 = fF ⊂ E : F is acyclicg { I = fF ⊂ E : 8i; jF \ E j ≤ 1g • 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 = fJ : 8v 2 Vi; v incident to ≤ one e 2 Jg • 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 nF ) 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 2 I1 \I2 and A ⊆ S. Then Want max jIj = jV j − 1; I2I1\I2 J \ A 2 I1 and J \ A 2 I2 or equivalently so min(r1(F ) + r2(E n F )) = jV j − 1: F ⊆E jJj = jJ \ Aj + jJ \ Aj ≤ r1(A) + r2(A) Note Claim: (Edmonds, 1970) For matroids r1(F ) = jV j − c(F ) M ;M on S, 1 2 where c(F ) is num connected comp of G0 = (V; F ), so need max fjJjg = minfr1(A) + r2(A)g: J2I1\I2 A⊆S { num colors in EnF 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 2 Vi covered by F , Need: ν(G) = min(r1(F ) + r2(E n F )): F ⊆E 1. deletion as v 2 V1 covered by F plus v 2 V2 cov- 2. contraction ered by E n 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 jF j = min jB1 [ B2j F spans M1;M2 Bi basis of Mi or equivalently if for any S ⊂ T and i 62 T , = min (jB1j + jB2j − jB1 \ B2j) f(S [ fig) − f(S) ≥ f(T [ fig) − f(T ): Bi basis of Mi 2 [[Compare to concavity. ]] • since JA ⊆ JC know JA [feg 2 I (down- Claim: Above defns equivalent. ward closure) Proof: • therefore r(A [ feg) = r(A) + 1 : For A; B, if B ⊆ A, claim trivial. Else let !: exercise. B n A = fb1; : : : ; bkg 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 62 T , set A = S [ fig 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 [ feg) − r(A) 2 f0; 1g for all e; A • r(·) is submodular Proof: : First condition obvious. For second, fix A ⊆ C, e 62 C. Want: r(C [ feg) − r(C) ≤ r(A [ feg) − r(A): • since r(A [ feg) − r(A) ≥ 0 and r(C [ feg) − r(C) 2 f0; 1g, may assume r(C [ feg) − r(C) = 1 • let JA;JC be max indep sets in A; C with JA ⊆ JC • since r(C [ feg) = r(C) + 1 know JC [ feg 2 I (exchange property) 3.