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sol10 - MATH239 Assignment 10 Solutions 1(10 points...

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Assignment 10 Solutions 1. (10 points) Consider the following bipartite graph G with bipartition ( A,B ) where A = { 1 ,..., 9 } and B = { a,b,. ..,i } . i 2 3 4 5 6 7 8 9 a b c d e f g h 1 i. Let M = { 3 a, 4 b, 6 c, 7 d, 9 h, 8 i } . Using the bipartite matching algorithm, de- termine the sets X 0 , X , and Y from the XY construction. Find either an M -augmenting path or cover C with | C | = | M | . Solution. X 0 = { 1 , 2 , 5 } . So 1 , 2 , 5 X . The following table gives the vertices in X and Y in the order added. new vertex a 3 c 6 b 4 d 7 e h 9 g f i 8 added to set Y X Y X Y X Y X Y Y X Y Y Y X predecessor 1 a 5 c 3 b 5 d 7 7 h 7 6 9 i Thus X = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } and Y = { a,b,c,d,e,f,g,h,i } . Note that g Y is M -exposed, and P = (5 ,d, 7 ,g ) is an M -augmenting path. ii. Let M = { 1 a, 2 c, 5 d, 4 b, 6 f, 8 i, 9 h, 7 e } . Using the bipartite matching algorithm, determine the sets X 0 , X , and Y from the XY construction. Find either an M -augmenting path or cover C with | C | = | M | . Solution. X 0 = { 3 } . So 3 X . The following table gives the vertices in X and Y in the order added. new vertex a b c d 1 2 4 5 added to set Y Y Y Y X X X X predecessor 3 3 3 3 a c b d Thus X = { 1 , 2 , 3 , 4 , 5 } and Y = { a,b,c,d } . Note that Y contains no M - exposed vertex. Let C
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sol10 - MATH239 Assignment 10 Solutions 1(10 points...

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