EECS 495: Combinatorial Optimization Lecture 7 Representation, Matroid Optimization

Reading: Schrijver, Chapters 39 and 40 graph and S = E. A set F ⊆ E is indepen- dent if it is acyclic. Food for thought: can two non-isomorphic graphs give isomorphic matroid structure?

Recap Representation Def: A matroid M = (S, I) is a finite ground Def: For a field F , a matroid M is repre- set S together with a collection of indepen- sentable over F if it is isomorphic to a linear dent sets I ⊆ 2S satisfying: matroid with A and linear indepen- dence taken over F . • downward closed: if I ∈ I and J ⊆ I, 2 then J ∈ I, and Example: Is U4 binary? Need: matrix A with entries in {0, 1} s.t. no • exchange property: if I,J ∈ I and |J| > column is the zero vector, no two rows sum |I|, then there exists an element z ∈ J \I to zero over GF(2), any three rows sum to s.t. I ∪ {z} ∈ I. GF(2).

Def: A basis is a maximal independent set. • if so, can assume A is 2×4 with columns The cardinality of a basis is the of the 1/2 being (0, 1) and (1, 0) and remaining matroid. two vectors with entries in 0, 1 neither all k Def: Uniform matroids Un are given by |S| = zero. n, I = {I ⊆ S : |I| ≤ k}. • only three such non-zero vectors, so can’t Def: Linear matroids: Let F be a field, A ∈ have all pairs indep. F m×n an m×n matrix over F , S = {1, . . . , n} be index set of columns of A. Then I ⊆ S is 2 independent if the corresponding columns are Question: representation of U4 ? linearly independent. (1, 0), (0, 1), (1, −1), (1, 1) in <. Note: WLOG any linear matroids can be Def: A is a matroid repre- sentable over GF (2). written as A = [Im|B] where m is rank of matroid and B is an (n−m)×m matrix over Def: A is representable over F . any field. Def: Graphic matroids: Let G = (V,E) be a Example: Graphic matroids are regular.

1 Proof: Take A to be vertex/edge incidence Conjecture (Rota, 1971): Matroids repre- matrix with +1/ − 1 in each column in any sentable over a finite field can be character- order. ized by a finite list of excluded minors. Much like planar graphs are those with no • Minimally dependent sets sum to zero K or K as a minor. perhaps with multiplying by −1. 3,3 5 • Works over any field with +1 as multi- Matroid Optimization plicative identity and −1 additive inverse of +1. Given: Matroid M = (S, I) and weights c : S → R Note: Have graphic ⊂ binary ⊂ regular ⊂ linear. Find: max-weight (or min-weight) basis ""Recall Kruskal’s Alg for min spanning## Note: There are matroids that are not linear tree: select edges in increasing order of (MacLane, 1936; Lazarson, 1958). weight Algorithm: Greedy Matroid Operations • Set J = ∅. Def: (from last lecture): The dual M ∗ of ma- • Order S s.t. c ≥ ... ≥ c . troid M = (S, I) is the matroid with ground 1 n set S whose independent sets I are such that • For i = 1 to n, if J ∪ {i} is independent, S \ I contains a basis of M. J := J ∪ {i} Def: The deletion M \ Z of matroid M = If weights are non-neg, this is max-weight (S, I) and subset Z ⊂ S is the matroid with indep set; otherwise stop selecting elts ground set S \ Z and independent sets {I ⊆   when ci becomes negative for max-weight S \ Z : I ∈ I}. indep set. Example: Take graph, delete edges, take Claim: Greedy finds maximal-weight basis. acyclic subsets of remaining edges. [[First rephrase second axiom. ]] Def: The contraction M/Z of ... is ... (M ∗ \ Proof: Clearly a basis. Suppose not max- Z)∗. weight, i.e., for greedy set J and opt J 0, ""So for X ⊆ Z maximal independent set## c(J) < c(J 0). of M, I independent in M/Z if I ∪ X

independent in M. • Let J = {e1, . . . , el} be greedy set la- 0 Def: If a matroid M arises from M by a beled according to chosen order so ce1 ≥ 0 series of deletions and contractions, then M ... ≥ cel .

is a minor of M. 0 • Let J = {q1, . . . , qk} be max-weight ba- Claim: (Tutte, 1958) A matroid is binary if sis labeled s.t. cq ≥ ... ≥ cq . 2 1 k and only if it has no U4 minor. • Let i be smallest index s.t. c > c (if ""Similar characterization of ternary ma-## qi ei no such index, must have k > l so let troids as those that exclude the so-called i = l + 1). Fano matroid and its dual as a minor. 2 • Consider independent sets I = Let OP ,OD be primal/dual value. To prove 0 n {e1, . . . , ei−1} and I = {q1, . . . , qi}. TDI need for any w ∈ Z exists opt dual soln that’s integral. • since |I0| > |I| exchange property says Recall TDI means for integral cost vector ∃z ∈ I0 s.t. I + z independent c s.t. primal soln finite, there exists in-   • but each elt in I0 has greater weight than tegral opt dual. Furthermore if polytope I and z was available to greedy at step i is TDI and b is integral, then polytope is integral. by above, so greedy can’t have chosen ei over z. • WLOG w non-negative (else discard neg In fact, matroids are precisely set systems elts and note dual constraint satisfied on which greedy works, see book. since y ≥ 0. What about running time? Depends on • Let J be independent set found by  to test if I + z in-   greedy. dependent. Want poly in |S| given indep   set oracle, or sometimes given sucinct • Note w(J) ≤ maxI∈I w(I) ≤ OP = OD.   representation of M like in graphs (note   • Find integral y s.t. dual value equals listing all indep sets is exponential in   w(J) hence proving both claims. Label |S|). Question, is there a matroid with a sucinct rep in which checking indepen- elts in decreasing order of weight and let dence is hard? Ui = {s1, . . . , si}. y = w(s ) − w(s ) Matroid Polytopes Ui i i+1

yUn = w(sn) Variables: xs for each s ∈ S Constraints: yU = 0, otherwise

xS ≥ 0, ∀s ∈ S – feasible: for any si ∈ S, P Pn X yU = yU x ≤ r(U), ∀U ⊆ S U:si∈U j=i j s Pn−1 s∈U = j=i (w(si)+w(si+1))+w(sn) = w(si). Claim: Greedy is optimal. – optimal: Claim: Matroid polytope integral. n−1 X X Proof: Consider primal objective r(U)y = r(U )(w(s ) − w(s )) max P w(s)x . Dual is: U i i i+1 s∈S S U⊆S i=1 X +r(Un)w(sn) min r(U)yU = w(s1)r(U1) n U⊆S X + w(si)(r(Ui) − r(Ui−1)) X s.t. y ≥ w(s), ∀s ∈ S i=2 U = w(J) U:s∈U

yU ≥ 0, ∀U ⊆ S

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