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}

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

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

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

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}

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}

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}

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

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

~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

~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

~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

~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

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

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":

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}

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:

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;

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;

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.

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.

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.

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