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hw6sols

# hw6sols - CO250/CM340 INTRODUCTION TO OPTIMIZATION HW 6...

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Unformatted text preview: CO250/CM340 INTRODUCTION TO OPTIMIZATION - HW 6 SOLUTIONS Exercise 1 (10 marks) . (a) If the answer is NO then the max flow has value at most M . The Max Flow - Min Cut theorem then implies that the min cut has capacity at most M . So there is a cut C of capacity at most M . This cut can be essentially the certificate. “Essentially” because we have not finished yet: we need to be able to check that C is a cut in polynomial time. This is not so obvious if C is given as a set of edges: how does one check quickly that this set intersects all st-paths? To make this easy we can take the certificate not to contain just the cut C but also the set of vertices U which are reachable from paths starting at s without using edges of C . We can check in polynomial time (say O ( n 2 ) that no edge of G goes from in U to outside U except for edges of C . We also check that t / ∈ C . This verifies C is a cut. Then we can compute the capacity of C also in time O ( n 2 ) . Then check the capacity is at most M . (b) Let’s denote H by the graph obtained from G by deleting just the edge e . If all of G ’s maximum weight matchings contain e , then the weight of the max matching in H will be less than the one in G . This also holds vice versa. So the empty certificate is enough: to check a YES answer, we just have to use Xenia’s algorithm to find the max weight matching in G and that in H , and check that they have different weights. Exercise 2 (15 marks) . (a) (P) is min n X i =1 x i subject to x i + x j ≥ 1 for each i,j such that v i v j is one of the edges e k , 1 ≤ k ≤ m x i ≥ ( i ∈ { 1 , 2 ,...,n } ) . (b) Let δ ( i ) denote the set of k such that edge e k is incident with vertex i . The i-th column of the constraint matrix contains 1 in row k whenever edge e k is incident with vertex i , that is, k ∈ δ ( i ) . So when we take the transpose, there are 1’s in row i at each column k in δ ( i ) . Thus, (D) is 1 2 max m X k =1 y i subject to X k ∈ δ ( i ) y k ≤ 1 for each 1 ≤ i ≤...
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hw6sols - CO250/CM340 INTRODUCTION TO OPTIMIZATION HW 6...

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