Matroid Theory. with a focus on operations. Congduan Li. November 21, Drexel University

Transcription

1 Matroid Theory with a focus on operations Congduan Li Drexel University November 21, 2011

2 Outline Matroid Matroid Review Dual Contraction & Deletion Excluded Minors

3 Matroid Review Definition Definition AmatroidM is an ordered pair (E, I) consisting of a finite set E and a collection I of subsets of E having the following three properties: 1. φ I; 2. Hereditary: If I I and I I, then I I; 3. Augmentation: If I 1 and I 2 are in I and I 1 < I 2, then there is an element e of I 2 I 1 such that I 1 e I. Demo Augmentation

4 Matroid Review Base/Basis Bases B(M) of a matroid are the maximal independent sets. Properties Same cardinality for all bases; The matroid is in the span of any base; Consider A = Bases: {1,2,3}, {1,2,4}, {1,2,5}, {1,3,4}, {1,4,5}, {2,3,4}, {2,3,5}, {3,4,5}

5 Matroid Review Base/Basis AcollectionofsubsetsB 2 E of a ground set E are the bases of a matroid if and only if: B is non-empty; base exchange: If B 1, B 2 B, for any x (B 1 B 2 ),thereis an element y (B 2 B 1 ) such that (B 1 x) y B. Consider A = Bases: {1,2,3}, {1,2,4}, {1,2,5}, {1,3,4}, {1,4,5}, {2,3,4}, {2,3,5}, {3,4,5}

6 Matroid Review Circuit The circuits C(M) of a matroid are the minimal dependent sets. Property If any one element is deleted from a circuit, it will become an independent set. Independent sets I(M) does not contain any element in C(M). Consider the graph 4 Circuits: {1,2,3,4}, {1,3,5}, {2,4,5}, {6}

7 Matroid Review Circuit A collection of sets C is the collection of circuits of a matroid if and only if: φ/ C if C 1, C 2 Cwith C 1 C 2 then C 1 = C 2 if C 1, C 2 C, C 1 = C 2 and e C 1 C 2 then there is some C 3 C with C 3 (C 1 C 2 ) e. Consider the graph 4 Circuits: {1,2,3,4}, {1,3,5}, {2,4,5}, {6}

8 Matroid Review Rank Function Rank: maximun number of linearly independent vectors. For any base of a matroid, r(b) =r(m). Formally, a rank function is r : 2 E N {0}, forwhich For X E, 0 r(x ) X If X Y E, r(x ) r(y ) X, Y E, r(x Y )+r(x Y ) r(x )+r(y ) Consider A = Bases: {1,2,3}, {1,2,4}, {1,2,5}, {1,3,4}, {1,4,5}, {2,3,4}, {2,3,5}, {3,4,5}; Rank: r(m) =r(b) =3

9 Matroid Review Closure Given a rank function of a matroid, the closure operation cl : 2 E 2 E is defined as cl X = {x E r(x x) =r(x )}. Closure is a span of a subspace. cl B = E(M); Independent Set: I = {X E x / cl(x x) x X } AsubsetX of E(M) is said to be a flat if cl X = X. Consider A = cl {1,2}={1,2,} flat cl {1,3}={1,3,5}

10 Matroid Review Closure Afunctioncl : 2 E 2 E is closure for amatroidm =(E, I) iff If X E then X clx If X Y E, thenclx cly If X E then cl(cl(x )) = clx If X E and x E and y cl(x x) cl(x ) then x cl(x y) Consider A = cl {1,2}={1,2,} flat cl {1,3}={1,3,5}

11 Matroid Review Greedy Algorithm Ground set E, I is a collection of subsets of E. Amappingfunction ω : E R. ForX I W (X )= x X ω(x) An optimization solution in I is a greedy process: 1. X 0 = 0, j = 0 2. If e j+1 (E X j ) such that X j e j+1 I and e j+1 has the max (min) weight among the elements left, then X j+1 = X j e j+1, j = j + 1, repeat. Otherwise, stop. Such a collection I is the collection of independent sets iff 1. φ I 2. If I 1 I and I 2 I 1,then I 2 I 3. For all weight mapping functions, the above greedy algorithm is able to find an optimization solution in I.

12 Representable Matroids Binary Amatroidthatisisomorphictoa vector matroid, where all elements take values from field F, iscalled F-representable. Fano matroid is only representable over field of characteristic 2 (Binary). Binary matroids have excluded minor: U 2,4

13 Representable Matroids Ternary Ternary matroids are those who can be represented over field of characteristic 3. Regular Matroids Representable over every field Both Binary and Ternary U 2,k is F-representable if and only if F k 1. U 2,4 is 3-representable (Ternary) B = Ternary matroids have excluded minor: U 2,5, U 3,5, F 7 and F 7.

14 Representable Matroids Graphic Matroid K 5 matroid and K 3,3 matroid: In a graph G =(V, E), letc 2 E be the sets of the graph cycles. Then C forms a set of circuits for a matroid with ground set E. Thematroid M(G) derived in this manner is called acyclematroid(graphicmatroid). Figure: K 5 matroid Excluded minors for graphic matroids: U 2,4, F 7, F 7, M (K 5 ), M (K 3,3 ) Figure: K 3,3 matroid

15 Representable Matroids Graphic Graphic Matroids: Independent sets: no cycles are contained Rank function: r(x )= V (G[X ]) w(g[x ]) Flat: contains no cycle that is contained in but exactly one edge Hyperplane: E(G) H is a minimal set of edges whose removal from G increases the number of connected components

16 Representable Matroids Cographic and Planar Amatroidiscalledcographicifits dual is graphic. Ex: M (K 5 ) and M (K 3,3 ) Abothgraphicandcographic matroid is called planar, isomorphic to a cycle matroid derived from a planar. In a planar, all edges can be drawn on a plane without intersections. Excluded minors for cographic: U 2,4, F 7, F 7, M(K 5), M(K 3,3 ) Excluded minors for planar: those excluded minors of graphic and cographic

17 Dual Definition The dual matroid M to a matroid M is the matroid with bases that are complements of the bases of M. B(M )={E B B B(M)}. (M ) = M and X is a base if and only if E X is a cobase G G*

18 Dual Independent Sets X is independent if and only if E X is cospanning X is spanning if and only if E X is coindependent

19 Dual Circuits X is a circuit if and only if E X is a cohyperplane X is a hyperplane if and only if E X is a cocircuit

20 Dual Rank Function Rank function of the dual matroid M : r (X )= X + r(e X ) r(m).

21 Dual Representable Case C = I r(m) A r(m) ( E r(m)) C = A T I E r(m)

22 Dual Graphic Case Circuits X of M (G) are cocircuits of M(G); E X is a hyperplane in M(G) Complement of a Hyperplane is a minimal collection of edges (Bonds) whose removal from G increases the number of connected components Circuits of M (G) are the bonds of G G G*

23 Contraction & Deletion Definition Restriction is a shrink of domain; Deletion a subset T is a restriction on (E-T). M\T = M (E T ) I(M\T )={I T : I I(M)} C(M\T )={C T : C C(M)} r M\T (X )=r M (X ) Example.

24 Contraction & Deletion Definition & Graphic case Contraction is dual operation of deletion. M/T =(M \T ) In a graphic matroid, M(G)/T = M(G/T ), whichmeansthe matroid contracting T is the matroid on the subgraph obtained from G by contracting edges T. Example.

25 Contraction & Deletion Representable case Intuitively, contraction means delete elements (edges) and merge the corresponding end points into single points if viewed in a graph. While deletion of an element does not do the combination. In Vectors, contracting an element means remove the corresponding component from the other vectors. Contracting T from a representable matroid: Find a base B T and represent other vectors with B, thendeletet. V3 V2 V1

26 Contraction & Deletion Rank Function Rank function: r M\T (X )=r M (X ) r M/T (X )=r M (X T ) r M (T ) Also compare these with dual operation, since the above equation can be obtained from r M\T and the following one r (X )= X + r(e X ) r(e)

27 Contraction & Deletion Independent sets Independent sets I(M\T )={I E T : I I(M)} I(M/T )={I E T : I B T I(M)} = {I I(M) :I cl M (X )=φ} where B T is the base of M T. M T has a base B such that B I I.

28 Contraction & Deletion Bases Bases: B(M\T ) is the set of maximal members of {B T : B B(M)}; B(M/T )={B E T : B B T B(M)}

29 Contraction & Deletion Circuits Circuits: C(M\T )={C E T : C C(M)} C(M/T )=min{c T E T : C C(M)}

30 Contraction & Deletion Closure Closure: cl M\T (X )=cl M (X ) T cl M/T (X )=cl M (X T ) T The closures also show the difference between deletion and contraction. That is, deletion does not affect the relationships between remaining elements.

31 Contraction & Deletion Other theorems M\T = M/T iff r(t )+r(e T )=r(m), T is a separator of M. M\e = M/e iff e is a loop or a coloop of M In a matroid M, lett 1 and T 2 be disjoint subsets of E(M). Then TFAE: (M\T1 )\T 2 = M\(T 1 T 2 )=(M\T 2 )\T 1 (M/T 1 )/T 2 = M/(T 1 T 2 )=(M/T 2 )/T 1 (M\T 1 )/T 2 =(M/T 2 )\T 1

32 Excluded Minors Definition Definition Minor: If a matroid M can be obtained by operating combination of restrictions and contractions on a matroid M, thenm is called a minor of M. Examples: U 2,4 is excluded by binary; F 7 &F7 are excluded by ternary.

33 Excluded Minors Summary Relationships between various classes of matroids: "#$%&'()!! ;/3+.%#'7! 0'-#%1!!!!!!!!! *+,%+)+-$#./+! 2%#-):+%)#/!!!!!!!!!!!! 5%#,6'7!!!!!!!!!! 2+%-#%1! 2%#-):+%)#/! *+34/#%! 2%#-):+%)#/ 9/#-#%!!!!!!!!!!!!!!!!!! 8&3%#,6'7! Figure: Relationships between various classes matroids: U 2,4, F 7, etc. in the graph are examples of the matroid class in which they lie. If an example is not contained in one class, it is called an excluded minor of this class.