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An Introduction to Matroids An introduction to Matroids An introduction to Matroids E.M.VRETTA Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki 3rd of May 2017 rd E.M.VRETTA | AUTH | 3 of May 2017 1 / 29 An introduction to Matroids Outline of the talk 1 Definition of a Matroid Independence axioms Circuit axioms Rank axioms 2 Structures that give rise to matroids Matrices Graphs Signed graphs Transversals Optimization problems rd E.M.VRETTA | AUTH | 3 of May 2017 2 / 29 An introduction to Matroids | Definition of a Matroid | Independence axioms Definition of Matroids (Independence axioms) Set system (E, I) is a tuple of a finite set E and a family of subsets of E, I =(Si : i ∈ I ) where the index mapping is φ : I → 2A. An independence system (E, I) is a finite set E together with a collection I of subsets of E closed under inclusion. Definition (Independence axioms) Given some finite set E, the set system (E, I) is a matroid if the following are satisfied: (I1) ∅∈I, (I2) If I ∈I and I ′ ⊆ I , then I ′ ∈I, (I3) If I1 and I2 are in I and |I1| < |I2|, then there is an element e of I2 − I1 such that I1 ∪ e ∈I. We write M =(E, I) or simply M if E and I are self-evident. The members of I are the independent sets of M. Basis is a maximal independent set. A subset of E that is not in I is called dependent. rd E.M.VRETTA | AUTH | 3 of May 2017 3 / 29 An introduction to Matroids | Definition of a Matroid | Circuit axioms Definition of Matroids (Circuit axioms) Matroids were introduced by Whitney in 1935 to try to capture abstractly the essence of dependence. Definition (Circuit axioms) Given some finite set E, the set system (E, C) is a matroid if the following are satisfied: (C1) ∅ ∈/ C, (C2) If C1, C2 ∈ C and C1 ⊆ C2, then C1 = C2, (C3) If C1, C2 ∈ C and e ∈ C1 ∩ C2, then there is a member C3 of C such that C3 ⊆ (C1 ∪ C2) − e. We write M =(E, C) or simply M if E and C are self-evident. E is called the ground set of M. Circuit is a minimal dependent set of M. C is the family of circuits of M. rd E.M.VRETTA | AUTH | 3 of May 2017 4 / 29 An introduction to Matroids | Definition of a Matroid | Rank axioms Definition of Matroids (Rank axioms) The rank of a matrix is the dimension of the vector space generated (or spanned) by its columns. The rank of a matrix is defined as the maximum number of linearly independent column vectors in the matrix. matroidal generalization: rank function of a matroid. Definition (Rank axioms) Let E be a set and r be a function that maps 2E into the set of non-negative integers and satisfies (R1) If X ⊆ E, then 0 ≤ r(X ) ≤ |X |. (R2) If X ⊆ Y ⊆ E, then r(X ) ≤ r(Y ). (R3) If X and Y are subsets of E, then r(X ∪ Y )+ r(X ∩ Y ) ≤ r(X )+ r(Y ). Let I be the collection of subsets X of E for which r(X )= |X |. Then (E, I) is a matroid having rank function r. rd E.M.VRETTA | AUTH | 3 of May 2017 5 / 29 An introduction to Matroids | Structures that give rise to matroids | Matrices Matroids can arise from matrices Let E be the set of column labels of an m × n matrix A over a field F, and let I be the set of subsets X of E for which the set of columns is a linearly independent set in the vector space V (m, F). Then M[A]=(E, I) is the vector matroid of A. Consider the matrix over the field R of real numbers: 12345 10011 A = 01001 M[A]=(E, I) where E = {1, 2, 3, 4, 5} and I = {∅, {1}, {2}, {4}, {5}, {1, 2}, {1, 5}, {2, 4}, {2, 5}, {4, 5}}. C is the set of subsets X of E for which the set of columns is a minimal linearly dependent set in the vector space V (m, F). M[A]=(E, C) where E = {1, 2, 3, 4, 5} and C = {{3}, {1, 4}, {1, 2, 5}, {2, 4, 5}}. r(M[A]) = rank(A)= 2. rd E.M.VRETTA | AUTH | 3 of May 2017 6 / 29 An introduction to Matroids | Structures that give rise to matroids | Graphs Graphs Definition A graph G =(V , E) is an ordered pair of a finite set of vertices V and a set of edges E ⊆ V ∪ V 2. G loop Given two vertices u, v of a graph G we define four types of edges: e = {u, v} is a link, half-edge e = {v, v} is a loop, e = {v} is a half-edge, link e = ∅ is a loose-edge. V (G) the set of vertices and E(G) the set of edges of G A graph H is a subgraph of a graph G if V (H) and E(H) are subsets of V (G) and E(G), respectively. rd E.M.VRETTA | AUTH | 3 of May 2017 7 / 29 An introduction to Matroids | Structures that give rise to matroids | Graphs Walk, Path, Cycle and tree in a graph v0 − vn walk is an alternating sequence W = {v0, e1, v1, e2,..., vn−1, en, vn} of vertices and edges such that ei = {vi−1, vi }. G v2 e5 e3 v0 e1 e4 e2 v1 rd E.M.VRETTA | AUTH | 3 of May 2017 8 / 29 An introduction to Matroids | Structures that give rise to matroids | Graphs Walk, Path, Cycle and tree in a graph v0 − vn walk is an alternating sequence W = {v0, e1, v1, e2,..., vn−1, en, vn} of vertices and edges such that ei = {vi−1, vi }. G A walk with all distinct vertices is called v2 path. e5 e3 v0 e1 e4 e2 v1 rd E.M.VRETTA | AUTH | 3 of May 2017 8 / 29 An introduction to Matroids | Structures that give rise to matroids | Graphs Walk, Path, Cycle and tree in a graph v0 − vn walk is an alternating sequence W = {v0, e1, v1, e2,..., vn−1, en, vn} of vertices and edges such that ei = {vi−1, vi }. G A walk with all distinct vertices is called v2 path. e5 e3 A path with the same endvertices is v0 e1 called cycle. e4 e2 v1 rd E.M.VRETTA | AUTH | 3 of May 2017 8 / 29 An introduction to Matroids | Structures that give rise to matroids | Graphs Walk, Path, Cycle and tree in a graph v0 − vn walk is an alternating sequence W = {v0, e1, v1, e2,..., vn−1, en, vn} of vertices and edges such that ei = {vi−1, vi }. A walk with all distinct vertices is called G path. e A path with the same endvertices is 5 e3 called cycle. e1 e4 A graph G is connected if for any two vertices v, w ∈ V (G) there is a v − w e2 path. rd E.M.VRETTA | AUTH | 3 of May 2017 8 / 29 An introduction to Matroids | Structures that give rise to matroids | Graphs Walk, Path, Cycle and tree in a graph v0 − vn walk is an alternating sequence W = {v0, e1, v1, e2,..., vn−1, en, vn} of vertices and edges such that ei = {vi−1, vi }. G A walk with all distinct vertices is called v2 path. e5 e3 A path with the same endvertices is v0 e1 called cycle. e4 A graph G is connected if for any two e vertices v, w ∈ V (G) there is a v − w 2 v path. 1 A tree is a connected acyclic graph. rd E.M.VRETTA | AUTH | 3 of May 2017 8 / 29 An introduction to Matroids | Structures that give rise to matroids | Graphs Walk, Path, Cycle and tree in a graph v0 − vn walk is an alternating sequence W = {v0, e1, v1, e2,..., vn−1, en, vn} of vertices and edges such that ei = {vi−1, vi }. G A walk with all distinct vertices is called v2 path. e5 e3 A path with the same endvertices is v0 e1 called cycle. e4 A graph G is connected if for any two e vertices v, w ∈ V (G) there is a v − w 2 v path. 1 A tree is a connected acyclic graph. A subgraph X of G is spanning if V (X )= V (G). rd E.M.VRETTA | AUTH | 3 of May 2017 8 / 29 An introduction to Matroids | Structures that give rise to matroids | Graphs Matroids can arise from graphs Let E be the set of edges of a graph G and C be the set of edge sets of cycles of G. Then C is the set of circuits of a matroid on E which is called the cycle matroid of G. Consider the graph G: G E = {e1, e2, e3, e4, e5} C = {{e3}, {e1, e4}, e5 e3 {e1, e2, e5}, {e2, e4, e5}} e1 e4 M(G)=(E, C) e2 I = {X ⊆ E : X does not contain the edges of a cycle of G} M(G)=(E, I) where I = {∅, {e1}, {e2}, {e4}, {e5}, {e1, e2}, {e1, e5}, {e2, e4}, {e2, e5}, {e4, e5}}. rd E.M.VRETTA | AUTH | 3 of May 2017 9 / 29 An introduction to Matroids | Structures that give rise to matroids | Graphs Matroids can arise from graphs Let E be the set of edges of a graph G and C be the set of edge sets of cycles of G.
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