1 1 Find a maximum Matching ? 2 2 3 3 4 4 5 5 Women Men.

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Presentation transcript:

1 1 Find a maximum Matching ? 2 2 3 3 4 4 5 5 Women Men

1 1 Find a maximum matching by eyes. 2 2 3 3 4 4 5 5 Women Men

Find a maximum matching by computer. 1 1 Find a maximum matching by computer. 2 2 Restrictions: Find Max 3 3 4 4 5 5 Women Men

Not a good way!! 1 1 Find a maximum matching by computer. 2 2 Restriction: x12+x13=1, x23=1, x33+x34+x35=1, x41+x42+x44=1, x55=1, x41=1, x12+x42=1, x13+x23+x31=1, x34+x44=1, x35+x55=1, xij=0 or 1. Find Max 3 3 4 4 5 5 Not a good way!! Women Men

1 1 Find a maximum Matching by computer again? 2 2 3 3 This is a maximal matching, but not a maximum matching. 4 4 5 5 Women Men

1 1 Find a maximum Matching by computer again? 2 2 3 1. Find an augmentative path. 3 4 4 5 5 Women Men

1 1 Find a maximum Matching by computer again? 2 2 3 1. Find an augmentative path. 2. Add the edges in the path, but not in matching. 3 4 4 5 5 Women Men

1 1 Find a maximum Matching by computer again? 2 2 3 1. Find an augmentative path. 2. Add the edges in the path, but not in matching. 3. Delete the edges in the path and in the matching. 3 4 4 5 5 Women Men

1 1 Find a maximum Matching by computer again? 2 2 3 Now we have 4 pairs 3 4 4 5 5 Women Men

1 1 Find a maximum Matching by computer again? 2 2 1. Keep doing this procedure. 3 3 4 4 5 5 Women Men

1 1 Find a maximum Matching by computer again? 2 2 1. Keep doing this procedure. 3 3 4 4 Wrong!! This is not an augmentative path. 5 5 Women Men

1 1 Find a maximum Matching by computer again? 2 2 1. Keep doing this procedure. 3 3 4 4 5 5 Women Men

1 1 Find a maximum Matching by computer again? 2 2 1. Keep doing this procedure. 3 3 4 4 5 5 Women Men

1 1 Find a maximum Matching by computer again? 2 2 Keep doing this procedure. When there exists no augmentative path, we find a maximum matching. 3 3 4 4 5 5 Women Men

3 1 1 1 5 3 2 3 4 2 1 2 3 5 Find maximum water flow with given capacities in the one-way pipes.

3/3 1/1 1/1 1/1 5/3 3 3 2/1 3/2 4 2 1 2 3 5 Find maximum water flow with given capacities by eyes.

3/3 1/1 1/1 1/1 5/3 3 3+3 2/1+1 3/2+1 4/3 2 1 2/2 3/2 5 Find maximum water flow with given capacities by eyes.

3/3 5 1/1 1/1 1/1 5/3 3 3+3+1 2/1+1 3/2+1 4/3 2 1/1 2/2 3/2+1 5/1 Find maximum water flow 7 with given capacities by eyes.

3/3 1/1 1/1 1/1 5 3/3 4 2 3 4/4 2/1 1 2/2 3/3 5/1 Flow 4 is maximal, but not a maximum flow.

3/3 1/1 1/1 1/1 5 3/3 4 2 3 4/4 2/1 1 2/2 3/3 5/1 Blue path is an augmentative path with increasing value 2.

3/3 1/1 1/1 1/1 5/2 3/3-2 4+2 2/2 3/2 4/4 2/1 1 2/2 3/3 5/1 Increasing or decreasing flow 2 in pipes according to augmentative path

3/3 1/1 1/1 1/1 5/2 3/3-2 4+2+1 2/2 3/2+1 4/4 2/1-1 1/1 2/2 3/3 5/1 The same way to increase the pink flow 1, and to obtain the maximum flow 7.

Relation to optimization problems c 3 b 1 1 1 5 3 2 3 a 4 d 2 1 2 3 5

Final comments There always exists a trivial solution, 0 flow. We don’t need to assume that xab is an integer, since each time we only increase integral flow value. 3. x<3 iff –x > -3 4. x<3 iff x+y=3 and y>0 5. Max x iff Min –x 6. Max xy is related to Max x+y, or Max x under the assumption x=cy.