practice midterm 1 soln

practice midterm 1 soln - CME305 Sample Midterm I 1....

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CME305 Sample Midterm I 1. Matchings and Independent Sets Assume that you are given graph G ( V,E ), a matching M in G and inde- pendent S in G . Show that | M | + | S | ≤ | V | . Solution: Let M be a maximum matching and S be a maximum in- dependent set. For each of the | M | matched pairs, S can include at most one of the vertex, and therefore | S | ≤ | V | − | M | , or equivalently | M | + | S | ≤ | V | . Since M and S are maximum, we know | M | + | S | ≤ | V | holds for all M and S . 2. Unique Minimum s-t Cut Given a network G ( V,E,s,t ), give a polynomial time algorithm to deter- mine whether G has a unique minimum s-t cut. Solution: First compute a minimum s - t cut C , and define its volume by | C | . Let e 1 ,e 2 ,...,e k be the edges in C . For each e i , try increasing the capacity of e i by 1 and compute a minimum cut in the new graph. Let the new minimum cut be C i , and denote its volume (in the new graph) as | C i | . If | C
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practice midterm 1 soln - CME305 Sample Midterm I 1....

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