Building Roads. Page 2. I = {;, a, b, c, d, e, ab, ac, ad, ae, bc, bd, be, cd, ce, de, abd, abe, acd, ace, bcd, bce, bde}

Transcription

1 Page

2 Building Roads Page I = {;, a, b, c, d, e, ab, ac, ad, ae, bc, bd, be, cd, ce, de, abd, abe, acd, ace, bcd, bce, bde}

3 Building Roads Page 3 2 a d 3 c b e I = {;, a, b, c, d, e, ab, ac, ad, ae, bc, bd, be, cd, ce, de, abd, abe, acd, ace, bcd, bce, bde} 4

4 Building Roads Page 4 2 a d 3 c b e I = {;, a, b, c, d, e, ab, ac, ad, ae, bc, bd, be, cd, ce, de, abd, abe, acd, ace, bcd, bce, bde} 4

5 Assigning Jobs Page 5 a W b c d e Loader Rock Truck Excavator I = {;, a, b, c, d, e, ab, ac, ad, ae, bc, bd, be, cd, ce, de, abd, abe, acd, ace, bcd, bce, bde}

6 Assigning Jobs Page 6 a W b c d e Loader Rock Truck Excavator I = {;, a, b, c, d, e, ab, ac, ad, ae, bc, bd, be, cd, ce, de, abd, abe, acd, ace, bcd, bce, bde}

7 Assigning Jobs Page 7 a W b c d e Loader Rock Truck Excavator I = {;, a, b, c, d, e, ab, ac, ad, ae, bc, bd, be, cd, ce, de, abd, abe, acd, ace, bcd, bce, bde}

8 Vectors ~a A ~ b A ~c A ~ d A ~e A Page 8

9 Vectors Page 9 ~a A ~ b A ~e ~ d ~c A ~ d A ~e A ~a ~ b ~c

10 ~a A ~ b A ~c A ~ d A ~e A I = {;, a, b, c, d, e, ab, ac, ad, ae, bc, bd, be, cd, ce, de, abd, abe, acd, ace, bcd, bce, bde} ~a ~e Vectors ~ b ~ d ~c Page

11 ~a A ~ b A ~c A ~ d A ~e A I = {;, a, b, c, d, e, ab, ac, ad, ae, bc, bd, be, cd, ce, de, abd, abe, acd, ace, bcd, bce, bde} ~a ~e Vectors ~ b ~ d ~c Page

12 ~a A ~ b A ~c A ~ d A ~e A I = {;, a, b, c, d, e, ab, ac, ad, ae, bc, bd, be, cd, ce, de, abd, abe, acd, ace, bcd, bce, bde} ~a ~e Vectors ~ b ~ d ~c Page 2

13 ~a A ~ b A ~c A ~ d A ~e A I = {;, a, b, c, d, e, ab, ac, ad, ae, bc, bd, be, cd, ce, de, abd, abe, acd, ace, bcd, bce, bde} ~a ~e Vectors ~ b ~ d ~c Page 3

14 ~a A ~ b A ~c A ~ d A ~e A I = {;, a, b, c, d, e, ab, ac, ad, ae, bc, bd, be, cd, ce, de, abd, abe, acd, ace, bcd, bce, bde} a e Vectors b d c Page 4

15 Common Structure Page 5 What was a common structure among the three scenarios?

16 Common Structure Page 6 What was a common structure among the three scenarios? Building Roads, Assigning Jobs, and Vectors all had the same feasible sets":

17 Common Structure Page 7 What was a common structure among the three scenarios? Building Roads, Assigning Jobs, and Vectors all had the same feasible sets": I = {;, a, b, c, d, e, ab, ac, ad, ae, bc, bd, be, cd, ce, de, abd, abe, acd, ace, bcd, bce, bde}

18 Common Structure Page 8 What was a common structure among the three scenarios? Building Roads, Assigning Jobs, and Vectors all had the same feasible sets": I = {;, a, b, c, d, e, ab, ac, ad, ae, bc, bd, be, cd, ce, de, abd, abe, acd, ace, bcd, bce, bde} Notice:

19 Common Structure Page 9 What was a common structure among the three scenarios? Building Roads, Assigning Jobs, and Vectors all had the same feasible sets": I = {;, a, b, c, d, e, ab, ac, ad, ae, bc, bd, be, cd, ce, de, abd, abe, acd, ace, bcd, bce, bde} Notice: (I) ;2I;

20 Common Structure Page 2 What was a common structure among the three scenarios? Building Roads, Assigning Jobs, and Vectors all had the same feasible sets": I = {;, a, b, c, d, e, ab, ac, ad, ae, bc, bd, be, cd, ce, de, abd, abe, acd, ace, bcd, bce, bde} Notice: (I) ;2I; (I2) If I 2Iand J I, then J 2I;

21 Common Structure Page 2 What was a common structure among the three scenarios? Building Roads, Assigning Jobs, and Vectors all had the same feasible sets": I = {;, a, b, c, d, e, ab, ac, ad, ae, bc, bd, be, cd, ce, de, abd, abe, acd, ace, bcd, bce, bde} Notice: (I) ;2I; (I2) If I 2Iand J I, then J 2I; (I3) If I, J 2Iand I > J, then there exists x 2 I such that J [{x} 2I.

22 Matroid Page 22 Definition Let E be a finite set and let I be a collection of subsets of E such that I satisfies properties (I), (I2), and (I3). Then the pair (E, I) is a matroid M. A subset of E that is in I is called an independent set in M. A set that is not independent is called dependent.

23 Matroid Definition Let E be a finite set and let I be a collection of subsets of E such that I satisfies properties (I), (I2), and (I3). Then the pair (E, I) is a matroid M. A subset of E that is in I is called an independent set in M. A set that is not independent is called dependent. Examples: Page 23

24 Matroid Definition Let E be a finite set and let I be a collection of subsets of E such that I satisfies properties (I), (I2), and (I3). Then the pair (E, I) is a matroid M. A subset of E that is in I is called an independent set in M. A set that is not independent is called dependent. Examples: Page 24 A graph, where E is the set of edges and I consists of acyclic subsets of edges. A matroid that can be depicted in this way is called a graphic matroid.

25 Matroid Definition Let E be a finite set and let I be a collection of subsets of E such that I satisfies properties (I), (I2), and (I3). Then the pair (E, I) is a matroid M. A subset of E that is in I is called an independent set in M. A set that is not independent is called dependent. Examples: Page 25 A graph, where E is the set of edges and I consists of acyclic subsets of edges. A matroid that can be depicted in this way is called a graphic matroid. A bipartite graph on vertices (X, Y ), where E = X and I consists of subsets of X that can appear together in a matching. A matroid that can be depicted in this way is called a transversal matroid.

26 Matroid Definition Let E be a finite set and let I be a collection of subsets of E such that I satisfies properties (I), (I2), and (I3). Then the pair (E, I) is a matroid M. A subset of E that is in I is called an independent set in M. A set that is not independent is called dependent. Examples: Page 26 A graph, where E is the set of edges and I consists of acyclic subsets of edges. A matroid that can be depicted in this way is called a graphic matroid. A bipartite graph on vertices (X, Y ), where E = X and I consists of subsets of X that can appear together in a matching. A matroid that can be depicted in this way is called a transversal matroid. A finite set E of vectors from a vector space, where I consists of subsets of E that are linearly independent. A matroid that can be depicted in this way is called a representable matroid.

27 Page 27

28 Page 28