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Unformatted text preview: Solutions 2 Some solutions are only sketches: you should be able to fill in the details. Note again that if you are asked to prove NPcompleteness the first thing you must do is prove membership of NP . As this is normally very easy, it is omitted here. Question 1 Show that if a language L is coNPcomplete, then L is NPcomplete. We need to show that there is a reduction from every language in NP to L . Choose any language L ∈ NP . Then L is in coNP . So there is a polynomialtime reduction R from L to L . Then R is also a reduction from L to L : x ∈ L ⇔ x 6∈ L ⇔ R ( x ) 6∈ L ⇔ R ( x ) ∈ L . Question 2 Prove that Graph Expanders is in coNP . Recall that a problem is in coNP if, when the answer is “No”, we can find a certificate that can be verified in polynomial time. If a graph is not an expander, then there is a set of vertices S ,  S  ≤  V  / 2, such that there are fewer than  S  vertices with neighbours in S . The set S is the certificate and it is easy to verify that it has this property. Question 3 Prove that Minimum Leaf Spanning Tree is NPcomplete. Let K = 2. A spanning tree with exactly two leaves (obviously you cannot have less than two leaves) is a Hamiltonian path. So Hamiltonian Path is a special case of Minimum Leaf Spanning Tree . As we know Hamiltonian Path is NPcomplete it immediately follows that Minimum Leaf Spanning Tree is NPcomplete (this is the restriction method of proof)....
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This note was uploaded on 01/29/2012 for the course ECE 345 taught by Professor Veneris during the Fall '10 term at University of Toronto.
 Fall '10
 Veneris
 Algorithms, Data Structures

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