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Lecture Note on MAS480A Matroid Theory

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Lecture Note on MAS480A Matroid Theory Lecture note on MAS480A Matroid Theory Donggyu Kim [email protected] 2019 Fall This lecture note is based on the lecture by Prof. Oum in KAIST in 2019 Fall. 1 Week01-1, 2019.09.02. Mon Motivations of the matroid theory are graphs and vector spaces. Example 1.1 (Matrices). Let A = R × E matrix over a field F. Let X ⊆ E. What are the properties of A[X], the submatrix of A by taking all columns in X? • Independency/dependency of column vectors • (column) Basis: maximal independent set of (column) vectors • Rank It is fact that all bases have the same size. Example 1.2 (Graphs). Let G = (V; E) be a graph. It is fact that all maximal sets of edges inducing no cycles have the same size. Note that such sets are edge- sets of spanning forests of G. Moreover, the size is (#vertices − #components). Example 1.3 (Matchings in a bipartite graph). Let G be a bipartite graph with a bipartition (A; B). We say that X ⊂ A is matchable to B if there is a matching (a set of edges pairwise vertex-disjoint) M covering X. It is fact that all maximal matchable subsets of A have the same size. Proof. Let X; Y ⊆ A be maximally matchable to B. Suppose jXj < jY j. Let M; N be matchings covering X; Y , respectively. Note that by the maximality jXj = jMj and jY j = jNj. Let consider H := G[M [ N]. Each component of H is a path or a cycle. Moreover, each path of length (= #edges) ≥ 2 or cycle is alternatively labeled by M and N. A component isomorphic to K2, i.e., just an edge, is possibly labeled by both M and N. If there is an edge e 2 N − M, then we can directly add it into M. It contradicts to the maximality of M. If there is no such an edge, we can find a path P which is a component of H and its two end-edges are contained in N since jNj > jMj. Let M 0 = M −(M \P )+(N \P ). It is also a matching in G, and gives X0 ⊆ A which is matchable to B. Note that X ( X0, so it is a contradiction. 1 If we solve problems in matroid, we can think it (matroid) as one of them (vector spaces/graphs/matchings). Then we may get idea to solve. Definition 1.1. A matroid is a pair M = (E; I) of a finite set E and a set I of subsets of E satisfying the following: (I1) ; 2 I, (I2) X 2 I, Y ⊆ X ) Y 2 I, and (I3) X; Y 2 I, jXj < jY j ) 9e 2 Y − X s.t. X [ feg 2 I. Let us say that • X is independent if X 2 I, • X is dependent if X 62 I, • X is a base if X is a maximal independent set, and • X is a circuit if X is a minimal dependent set. The definition of matroid is possibly extended to infinite E. However, there are no settled one. The extensions are messy and hard to handle. See Example 1.2, an independent set is a forest of G. A base is a spanning forest. A dependent set is an edge-set containing a cycle, and a circuit is a cycle. Theorem 1.1. For X ⊆ E, all maximal independent subset of X have the same size. Proof. Let A; B 2 I with A; B ⊆ X. Suppose jAj < jBj. By the axiom (I3), there is e 2 B − A such that A [ feg 2 I, which is a subset of X. It contradicts to a maximality of A. Definition 1.2. The rank function of a matroid M is rM (X) := (a size of a maximal independent subset of X): It is well-deinfed by the previous theorem. The rank of a matroid is r(M) := rM (E): E(M) = E is the ground set of the matroid M. Example 1.4 (Basic classes of matroids). Now we will introduce some basic calsses of matroids. 1. Uniform matroid, Ur;n (0 ≤ r ≤ n). jEj = n. X is independent iff jXj ≤ r. It satisfies all axioms of the matroid. 2. Vector matroid (or linear matroid). A := R × E matrix over F. X is independent iff the column vectors of A[X] are linearly independent. Here, we denote M = M(A). 2 Theorem 1.2. A vector matroid is a matroid. Proof. (I1) and (I2) are trivial. Also, (I3) is obvious by an argument of dimen- sions, but we will explain it more precisely. By a permutation of E, WMA first jX \ Y j columns of A correspond to X \ Y , next jX − Y j columns correspond to X − Y , next jY − Xj columns correspond to Y − X, and remainings corre- sponds to E − (X [ Y ). Note that elementary row operation does not affect on dependency of column vectors. So we can change first jXj columns to IjXj. By the independency of Y , next jY − Xj > 0 columns cannot be a zero matrix. This implies that we can find such e from these column vectors. In this case, we say that M is representable over F or F-representable. A is a representation of M. Similarly, we can define matroids from a graph G (Example 1.2 and 1.3). Setting E = E(G) and X 2 I iff X is a forest, then M = (E; I) is a matroid. We call it as a graphic matroid or a cycle matroid. Assume G is bipartite with a bipartition (A; B). Setting E = A and X 2 I iff X is matchable (to B), then M is also a matroid. We call it as a transversal matroid or a mathcing matroid. 1.1 Whitney, 1935 M Definition 1.3. Let M be a finite set. Let r be a function from 2 to Z≥0. Then we call a system (M; r) a matroid if satisfies below three axioms: (R1) r(;) = 0, (R2) for any N ⊆ M and e 2 M − N, r(N + e) = r(N) or = r(N) + 1, and (R3) for any N ⊆ M and e1; e2 2 M − N, if r(N + e1) = r(N + e2) = r(N) then r(N + e1 + e2) = r(N). We call r a rank function, and r(N) a rank of N. Without doubt, it is fact that for any N ⊆ a system (N; rj2N ) is a matroid. We call it a submatroid of (M; r). When we write a matroid, we can omit a rank function if it is apparent in a context. M Definition 1.4. Let define a two functions ρ, n : 2 ! Z≥0 as ρ(N) = jNj; n(N) = ρ(N) − r(N): ρ(N) is just a cardinality of N. We call n a nullity function, and n(N) a nullity of N. • N is independent if n(N) = 0, • N is dependent if n(N) > 0, • N is a base if it is a maximal independent set (M − N is called a base complement), and 3 • N is a circuit if it is a minimal dependent set. By the axiom (R1) and (R2), 0 ≤ r(N) ≤ ρ(N). So 0 ≤ n(N) ≤ ρ(N). Lemma 1.3. For N ⊆ N 0 ⊆ M, r(N) ≤ r(N 0) and n(N) ≤ r(N 0). Proof. By (R2), r(N) ≤ r(N 0). Also the axiom implies that r(N 0) ≤ r(N) + ρ(N 0) − ρ(N). It is equivalent with n(N) ≤ n(N 0). Lemma 1.4. Any subset of an independent set is independent. Proof. Let N ⊆ M be an independent set, i.e., n(N) = 0. Let N 0 ⊆ N. By the previous theorem, 0 ≤ n(N 0) ≤ n(N) = 0. Therefore, n(N 0) = 0. Theorem 1.5. N is independent iff N is contained in a base iff N does not contain any circuit. Proof. (1)2) Let N be an independent set. If N is a base, done. If not, there is e 2 M −N such that N +e is independent. Repeat this work until get a base. It actually finishes since M is finite. (2)1) By the previous lemma, done. (1)3) Suppose N contains a circuit C. Then 0 < n(C) ≤ n(N), which is a contradiction. (3)1) We will show a contraposition. Let N be a dependent set. If N is a circuit, done. If not, there is e 2 N such that N − e is a dependent. Repeat this work until get a circuit. Theorem 1.6. A circuit is a minimal submatroid contained in no bases, i.e., containing at least one element from each base complement. A base is a maximal submatroid containing no circuit. A base complement is a minimal submatroid containing at least one element from each circuit. Proof. By the previous theorem, N is `contained in no bases' iff it is dependent, and N is `containing no circuit' iff it is independent. Then the first two state- ments are obvious. The last one is obtained by taking the complement operator for each set in the second statement. We can observe the reciprocal relation ship between circuits and base com- plements. Do we derive a dual concept of a matroid from this? Definition 1.5. ∆(N; N 0) = r(N + N 0) − r(N): Proposition 1.7. ∆(A; B + C) = ∆(A + C; B) − ∆(A + C; C): 4 Proof. δ(A; B + C) = r(A + B + C) − r(A) = r(A + B + C) − r(A + C) − r(A) + r(A + C) = ∆(A + C; B) − ∆(A + C; C): Lemma 1.8.
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