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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 NP-completeness 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 coNP-complete, then L is NP-complete. 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 polynomial-time 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 NP-complete. 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 NP-complete it immediately follows that Minimum Leaf Spanning Tree is NP-complete (this is the restriction method of proof)....
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